Subgroups — C₁₁₃×#22476 #43290496

Order 28928.

1696 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#546×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	D₄×C₂²⋊C₄×C₁₁₃	C₂	1	2	yes	maximal, complemented
3	C₁₁₃×#1996	C₂	1	2	yes	maximal, complemented
4	C₁₁₃×#1898	C₂	1	4	yes	maximal, complemented
5	C₁₁₃×#1819	C₂	1	2	yes	maximal, complemented
6	C₁₁₃×#1799	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#99×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#342×C₁₁₃	C₂	1	2	yes	maximal, complemented
9	C₂²×#41×C₁₁₃	C₄	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
11	#546×C₁₁₃	—	2	2	no	—
12	#546×C₁₁₃	—	2	4	no	—
13	#546×C₁₁₃	—	2	2	no	—
14	C₂×#124×C₁₁₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#123×C₁₁₃	C₄	1	2	yes	complemented
16	C₂×C₂²⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂×#121×C₁₁₃	C₄	1	1	yes	characteristic, complemented
18	C₂×#121×C₁₁₃	C₄	1	2	yes	complemented
19	C₂×#120×C₁₁₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₁₁₃	—	2	2	no	—
21	C₂×C₄×D₄×C₁₁₃	C₄	1	1	yes	characteristic, complemented
22	C₂×C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
25	C₂×#99×C₁₁₃	C₂²	1	2	yes	—
26	C₂×#99×C₁₁₃	C₂²	1	2	yes	—
27	#390×C₁₁₃	—	2	2	no	—
28	#386×C₁₁₃	—	2	4	no	—
29	#380×C₁₁₃	C₂²	1	2	yes	—
30	#379×C₁₁₃	C₂²	1	4	yes	—
31	C₄×C₂²⋊C₄×C₁₁₃	C₂²	1	2	yes	—
32	#342×C₁₁₃	C₂²	1	2	yes	—
33	#342×C₁₁₃	C₂²	1	1	yes	characteristic
34	#342×C₁₁₃	C₂²	1	4	yes	—
35	#342×C₁₁₃	C₂²	1	4	yes	—
36	#342×C₁₁₃	C₂²	1	1	yes	characteristic
37	C₂⁵×C₁₁₃	C₂³	1	2	yes	—
38	C₂×#41×C₁₁₃	—	2	1	no	—
39	C₂×#41×C₁₁₃	—	2	2	no	—
40	C₂×#41×C₁₁₃	—	2	1	no	—
41	C₂×#41×C₁₁₃	—	4	4	no	—
42	C₂²×Q₈×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
43	C₂²×D₄×C₁₁₃	—	4	4	no	—
44	C₂²×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	—
46	C₂²×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
48	C₂²×D₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
49	C₂²×D₄×C₁₁₃	D₄	1	2	yes	—
50	C₂²×D₄×C₁₁₃	D₄	1	4	yes	—
51	C₂²×D₄×C₁₁₃	D₄	1	2	yes	—
52	C₂²×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₁₁₃	—	2	4	no	—
54	C₂³×C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
55	C₂³×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂³×C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
57	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
58	#124×C₁₁₃	—	2	2	no	—
59	#124×C₁₁₃	—	2	2	no	—
60	#123×C₁₁₃	—	2	4	no	—
61	#123×C₁₁₃	—	2	4	no	—
62	C₂²⋊Q₈×C₁₁₃	—	2	2	no	—
63	C₂²⋊Q₈×C₁₁₃	—	2	2	no	—
64	#121×C₁₁₃	—	2	2	no	—
65	#121×C₁₁₃	—	2	2	no	—
66	#121×C₁₁₃	—	2	4	no	—
67	#121×C₁₁₃	—	2	4	no	—
68	#120×C₁₁₃	—	2	4	no	—
69	#120×C₁₁₃	—	2	4	no	—
70	C₄×D₄×C₁₁₃	—	2	2	no	—
71	C₄×D₄×C₁₁₃	—	2	1	no	—
72	C₄×D₄×C₁₁₃	—	2	1	no	—
73	C₄×D₄×C₁₁₃	—	2	2	no	—
74	C₄×D₄×C₁₁₃	—	2	2	no	—
75	C₄×D₄×C₁₁₃	—	4	8	no	—
76	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
78	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
79	C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
80	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
82	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
83	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
84	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
85	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₁₁₃	—	4	4	no	—
87	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
88	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
89	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
90	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	1	2	yes	complemented
92	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
94	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₁₁₃	C₂³	1	2	yes	—
97	C₂×C₄²×C₁₁₃	—	2	2	no	—
98	C₂×C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic, complemented
99	#99×C₁₁₃	—	2	4	no	—
100	#99×C₁₁₃	—	4	2	no	—
101	#95×C₁₁₃	—	2	2	no	—
102	#95×C₁₁₃	—	2	2	no	—
103	C₂⁴×C₁₁₃	—	2	4	no	—
104	C₂⁴×C₁₁₃	—	2	2	no	—
105	C₂⁴×C₁₁₃	—	2	2	no	—
106	C₂⁴×C₁₁₃	—	4	4	no	—
107	C₂⁴×C₁₁₃	—	8	2	no	—
108	C₂⁴×C₁₁₃	C₂²×C₄	1	2	yes	—
109	C₂⁴×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂⁴×C₁₁₃	C₂²×C₄	1	2	yes	—
111	C₂⁴×C₁₁₃	C₂×D₄	1	4	yes	—
112	C₂⁴×C₁₁₃	C₂×D₄	1	4	yes	—
113	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
114	#41×C₁₁₃	—	4	8	no	—
115	#41×C₁₁₃	—	4	4	no	—
116	#41×C₁₁₃	—	4	4	no	—
117	C₂×Q₈×C₁₁₃	—	2	1	no	—
118	C₂×Q₈×C₁₁₃	—	2	1	no	—
119	C₂×Q₈×C₁₁₃	—	4	2	no	—
120	C₂×D₄×C₁₁₃	—	2	8	no	—
121	C₂×D₄×C₁₁₃	—	2	4	no	—
122	C₂×D₄×C₁₁₃	—	2	2	no	—
123	C₂×D₄×C₁₁₃	—	2	2	no	—
124	C₂×D₄×C₁₁₃	—	2	2	no	—
125	C₂×D₄×C₁₁₃	—	2	2	no	—
126	C₂×D₄×C₁₁₃	—	2	1	no	—
127	C₂×D₄×C₁₁₃	—	2	1	no	—
128	C₂×D₄×C₁₁₃	—	2	4	no	—
129	C₂×D₄×C₁₁₃	—	4	2	no	—
130	C₂×D₄×C₁₁₃	—	4	4	no	—
131	C₂×D₄×C₁₁₃	—	4	4	no	—
132	C₂×D₄×C₁₁₃	—	4	8	no	—
133	C₂×D₄×C₁₁₃	—	4	4	no	—
134	C₂×D₄×C₁₁₃	—	8	2	no	—
135	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
136	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
138	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
139	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	4	yes	—
140	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
141	C₂×D₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₁₁₃	—	2	2	no	—
143	C₂²×C₄×C₁₁₃	—	2	2	no	—
144	C₂²×C₄×C₁₁₃	—	2	2	no	—
145	C₂²×C₄×C₁₁₃	—	2	2	no	—
146	C₂²×C₄×C₁₁₃	—	2	2	no	—
147	C₂²×C₄×C₁₁₃	—	2	1	no	—
148	C₂²×C₄×C₁₁₃	—	2	1	no	—
149	C₂²×C₄×C₁₁₃	—	2	2	no	—
150	C₂²×C₄×C₁₁₃	—	2	4	no	—
151	C₂²×C₄×C₁₁₃	—	4	2	no	—
152	C₂²×C₄×C₁₁₃	—	4	2	no	—
153	C₂²×C₄×C₁₁₃	—	4	8	no	—
154	C₂²×C₄×C₁₁₃	—	4	4	no	—
155	C₂²×C₄×C₁₁₃	—	4	8	no	—
156	C₂²×C₄×C₁₁₃	—	8	4	no	—
157	C₂²×C₄×C₁₁₃	—	8	2	no	—
158	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
162	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
167	C₄⋊C₄×C₁₁₃	—	2	1	no	—
168	C₄⋊C₄×C₁₁₃	—	2	1	no	—
169	C₄⋊C₄×C₁₁₃	—	2	2	no	—
170	C₄⋊C₄×C₁₁₃	—	2	4	no	—
171	C₄⋊C₄×C₁₁₃	—	4	4	no	—
172	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
173	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
174	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
175	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
176	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
177	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
178	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
179	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
180	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
181	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
182	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
183	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
184	C₂²⋊C₄×C₁₁₃	—	2	2	no	—
185	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
186	C₂²⋊C₄×C₁₁₃	—	4	8	no	—
187	C₂²⋊C₄×C₁₁₃	—	8	4	no	—
188	C₂²⋊C₄×C₁₁₃	#41	1	4	yes	complemented
189	C₄²×C₁₁₃	—	2	1	no	—
190	C₄²×C₁₁₃	—	2	1	no	—
191	C₄²×C₁₁₃	—	4	4	no	—
192	C₂³×C₁₁₃	—	2	2	no	—
193	C₂³×C₁₁₃	—	2	2	no	—
194	C₂³×C₁₁₃	—	2	4	no	—
195	C₂³×C₁₁₃	—	2	4	no	—
196	C₂³×C₁₁₃	—	2	8	no	—
197	C₂³×C₁₁₃	—	2	4	no	—
198	C₂³×C₁₁₃	—	2	2	no	—
199	C₂³×C₁₁₃	—	2	2	no	—
200	C₂³×C₁₁₃	—	2	4	no	—
201	C₂³×C₁₁₃	—	2	4	no	—
202	C₂³×C₁₁₃	—	2	1	no	—
203	C₂³×C₁₁₃	—	2	1	no	—
204	C₂³×C₁₁₃	—	2	1	no	—
205	C₂³×C₁₁₃	—	4	1	no	—
206	C₂³×C₁₁₃	—	4	2	no	—
207	C₂³×C₁₁₃	—	4	4	no	—
208	C₂³×C₁₁₃	—	4	1	no	—
209	C₂³×C₁₁₃	—	4	4	no	—
210	C₂³×C₁₁₃	—	4	2	no	—
211	C₂³×C₁₁₃	—	4	4	no	—
212	C₂³×C₁₁₃	—	4	2	no	—
213	C₂³×C₁₁₃	—	4	1	no	—
214	C₂³×C₁₁₃	—	4	1	no	—
215	C₂³×C₁₁₃	—	8	8	no	—
216	C₂³×C₁₁₃	—	8	4	no	—
217	C₂³×C₁₁₃	—	8	4	no	—
218	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	4	yes	—
219	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂³×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂³×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
222	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
223	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
224	C₂³×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂³×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
226	Q₈×C₁₁₃	—	4	2	no	—
227	Q₈×C₁₁₃	—	4	2	no	—
228	D₄×C₁₁₃	—	2	8	no	—
229	D₄×C₁₁₃	—	2	4	no	—
230	D₄×C₁₁₃	—	2	4	no	—
231	D₄×C₁₁₃	—	2	2	no	—
232	D₄×C₁₁₃	—	4	2	no	—
233	D₄×C₁₁₃	—	4	4	no	—
234	D₄×C₁₁₃	—	4	4	no	—
235	D₄×C₁₁₃	—	4	2	no	—
236	D₄×C₁₁₃	—	4	2	no	—
237	D₄×C₁₁₃	—	4	4	no	—
238	D₄×C₁₁₃	—	8	4	no	—
239	D₄×C₁₁₃	#99	1	4	yes	complemented
240	C₂×C₄×C₁₁₃	—	2	4	no	—
241	C₂×C₄×C₁₁₃	—	2	2	no	—
242	C₂×C₄×C₁₁₃	—	2	1	no	—
243	C₂×C₄×C₁₁₃	—	2	1	no	—
244	C₂×C₄×C₁₁₃	—	2	2	no	—
245	C₂×C₄×C₁₁₃	—	2	1	no	—
246	C₂×C₄×C₁₁₃	—	2	2	no	—
247	C₂×C₄×C₁₁₃	—	2	1	no	—
248	C₂×C₄×C₁₁₃	—	2	1	no	—
249	C₂×C₄×C₁₁₃	—	2	4	no	—
250	C₂×C₄×C₁₁₃	—	4	1	no	—
251	C₂×C₄×C₁₁₃	—	4	2	no	—
252	C₂×C₄×C₁₁₃	—	4	2	no	—
253	C₂×C₄×C₁₁₃	—	4	2	no	—
254	C₂×C₄×C₁₁₃	—	4	4	no	—
255	C₂×C₄×C₁₁₃	—	4	2	no	—
256	C₂×C₄×C₁₁₃	—	4	4	no	—
257	C₂×C₄×C₁₁₃	—	4	2	no	—
258	C₂×C₄×C₁₁₃	—	4	1	no	—
259	C₂×C₄×C₁₁₃	—	4	2	no	—
260	C₂×C₄×C₁₁₃	—	4	1	no	—
261	C₂×C₄×C₁₁₃	—	4	2	no	—
262	C₂×C₄×C₁₁₃	—	4	2	no	—
263	C₂×C₄×C₁₁₃	—	4	4	no	—
264	C₂×C₄×C₁₁₃	—	4	2	no	—
265	C₂×C₄×C₁₁₃	—	4	4	no	—
266	C₂×C₄×C₁₁₃	—	4	4	no	—
267	C₂×C₄×C₁₁₃	—	4	2	no	—
268	C₂×C₄×C₁₁₃	—	8	2	no	—
269	C₂×C₄×C₁₁₃	—	8	2	no	—
270	C₂×C₄×C₁₁₃	—	8	16	no	—
271	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	2	yes	—
273	C₂×C₄×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂²×C₁₁₃	—	2	2	no	—
275	C₂²×C₁₁₃	—	2	2	no	—
276	C₂²×C₁₁₃	—	2	1	no	—
277	C₂²×C₁₁₃	—	2	2	no	—
278	C₂²×C₁₁₃	—	2	1	no	—
279	C₂²×C₁₁₃	—	2	2	no	—
280	C₂²×C₁₁₃	—	2	2	no	—
281	C₂²×C₁₁₃	—	2	4	no	—
282	C₂²×C₁₁₃	—	2	2	no	—
283	C₂²×C₁₁₃	—	2	4	no	—
284	C₂²×C₁₁₃	—	2	2	no	—
285	C₂²×C₁₁₃	—	2	4	no	—
286	C₂²×C₁₁₃	—	4	2	no	—
287	C₂²×C₁₁₃	—	4	1	no	—
288	C₂²×C₁₁₃	—	4	1	no	—
289	C₂²×C₁₁₃	—	4	8	no	—
290	C₂²×C₁₁₃	—	4	8	no	—
291	C₂²×C₁₁₃	—	4	4	no	—
292	C₂²×C₁₁₃	—	4	4	no	—
293	C₂²×C₁₁₃	—	4	1	no	—
294	C₂²×C₁₁₃	—	4	1	no	—
295	C₂²×C₁₁₃	—	4	1	no	—
296	C₂²×C₁₁₃	—	8	4	no	—
297	C₂²×C₁₁₃	—	8	4	no	—
298	C₂²×C₁₁₃	—	8	2	no	—
299	C₂²×C₁₁₃	—	8	2	no	—
300	C₂²×C₁₁₃	—	8	1	no	—
301	C₂²×C₁₁₃	—	8	1	no	—
302	C₂²×C₁₁₃	C₂×#99	1	4	yes	—
303	C₂²×C₁₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
304	C₂²×C₁₁₃	C₄×#41	1	1	yes	characteristic
305	C₂²×C₁₁₃	#520	1	1	yes	characteristic
306	C₂²×C₁₁₃	#546	1	2	yes	—
307	C₂²×C₁₁₃	#547	1	2	yes	—
308	C₄×C₁₁₃	—	2	1	no	—
309	C₄×C₁₁₃	—	2	2	no	—
310	C₄×C₁₁₃	—	4	4	no	—
311	C₄×C₁₁₃	—	4	1	no	—
312	C₄×C₁₁₃	—	4	1	no	—
313	C₄×C₁₁₃	—	4	2	no	—
314	C₄×C₁₁₃	—	4	1	no	—
315	C₄×C₁₁₃	—	4	2	no	—
316	C₄×C₁₁₃	—	8	2	no	—
317	C₄×C₁₁₃	—	8	4	no	—
318	C₄×C₁₁₃	C₂×#99	1	2	yes	—
319	#22476	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
320	C₂×C₁₁₃	—	2	2	no	—
321	C₂×C₁₁₃	—	2	2	no	—
322	C₂×C₁₁₃	—	2	1	no	—
323	C₂×C₁₁₃	—	2	1	no	—
324	C₂×C₁₁₃	—	2	4	no	—
325	C₂×C₁₁₃	—	4	2	no	—
326	C₂×C₁₁₃	—	4	4	no	—
327	C₂×C₁₁₃	—	8	1	no	—
328	C₂×C₁₁₃	#2339	1	1	yes	characteristic
329	C₂×C₁₁₃	C₂²×#99	1	1	yes	characteristic
330	C₂×C₁₁₃	#2883	1	1	yes	characteristic
331	C₂×#546	C₂×C₁₁₃	1	1	yes	characteristic
332	D₄×C₂²⋊C₄	C₂×C₁₁₃	1	2	yes	complemented
333	#1996	C₂×C₁₁₃	1	2	yes	complemented
334	#1898	C₂×C₁₁₃	1	4	yes	complemented
335	#1819	C₂×C₁₁₃	1	2	yes	complemented
336	#1799	C₂×C₁₁₃	1	1	yes	characteristic, complemented
337	C₄×#99	C₂×C₁₁₃	1	1	yes	characteristic, complemented
338	C₂×#342	C₂×C₁₁₃	1	2	yes	complemented
339	C₁₁₃	#22476	1	1	yes	characteristic, direct product factor, Sylow
340	C₂²×#41	C₄×C₁₁₃	1	1	yes	characteristic, complemented
341	C₂³×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
342	#546	—	2	2	no	—
343	#546	—	2	4	no	—
344	#546	—	2	2	no	—
345	C₂×#124	C₄×C₁₁₃	1	1	yes	characteristic, complemented
346	C₂×#123	C₄×C₁₁₃	1	2	yes	complemented
347	C₂×C₂²⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
348	C₂×#121	C₄×C₁₁₃	1	1	yes	characteristic, complemented
349	C₂×#121	C₄×C₁₁₃	1	2	yes	complemented
350	C₂×#120	C₂²×C₁₁₃	1	2	yes	—
351	C₂×C₄×D₄	—	2	2	no	—
352	C₂×C₄×D₄	C₄×C₁₁₃	1	1	yes	characteristic, complemented
353	C₂×C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
354	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
355	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	4	yes	—
356	C₂×#99	C₂²×C₁₁₃	1	2	yes	—
357	C₂×#99	C₂²×C₁₁₃	1	2	yes	—
358	#390	—	2	2	no	—
359	#386	—	2	4	no	—
360	#380	C₂²×C₁₁₃	1	2	yes	—
361	#379	C₂²×C₁₁₃	1	4	yes	—
362	C₄×C₂²⋊C₄	C₂²×C₁₁₃	1	2	yes	—
363	#342	C₂²×C₁₁₃	1	2	yes	—
364	#342	C₂²×C₁₁₃	1	1	yes	characteristic
365	#342	C₂²×C₁₁₃	1	4	yes	—
366	#342	C₂²×C₁₁₃	1	4	yes	—
367	#342	C₂²×C₁₁₃	1	1	yes	characteristic
368	C₂⁵	C₂³×C₁₁₃	1	2	yes	—
369	C₂×#41	—	2	1	no	—
370	C₂×#41	—	2	2	no	—
371	C₂×#41	—	2	1	no	—
372	C₂×#41	—	4	4	no	—
373	C₂²×Q₈	C₂×C₄×C₁₁₃	1	1	yes	characteristic, complemented
374	C₂²×D₄	—	4	4	no	—
375	C₂²×D₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
376	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	—
377	C₂²×D₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic
378	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
379	C₂²×D₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
380	C₂²×D₄	D₄×C₁₁₃	1	2	yes	—
381	C₂²×D₄	D₄×C₁₁₃	1	4	yes	—
382	C₂²×D₄	D₄×C₁₁₃	1	2	yes	—
383	C₂²×D₄	C₂³×C₁₁₃	1	1	yes	characteristic
384	C₂³×C₄	—	2	4	no	—
385	C₂³×C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
386	C₂³×C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic, complemented
387	C₂³×C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
388	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
389	#124	—	2	2	no	—
390	#124	—	2	2	no	—
391	#123	—	2	4	no	—
392	#123	—	2	4	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	—	2	2	no	—
397	#121	—	2	4	no	—
398	#121	—	2	4	no	—
399	#120	—	2	4	no	—
400	#120	—	2	4	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	4	8	no	—
407	C₂×C₄⋊C₄	—	2	2	no	—
408	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic, complemented
409	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic, complemented
410	C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
411	C₂×C₂²⋊C₄	—	2	4	no	—
412	C₂×C₂²⋊C₄	—	2	4	no	—
413	C₂×C₂²⋊C₄	—	2	8	no	—
414	C₂×C₂²⋊C₄	—	2	4	no	—
415	C₂×C₂²⋊C₄	—	2	4	no	—
416	C₂×C₂²⋊C₄	—	2	4	no	—
417	C₂×C₂²⋊C₄	—	4	4	no	—
418	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
419	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
420	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
421	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	1	yes	characteristic, complemented
422	C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	1	2	yes	complemented
423	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
426	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	2	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₁₁₃	1	2	yes	—
428	C₂×C₄²	—	2	2	no	—
429	C₂×C₄²	C₂×C₄×C₁₁₃	1	1	yes	characteristic, complemented
430	#99	—	2	4	no	—
431	#99	—	4	2	no	—
432	#95	—	2	2	no	—
433	#95	—	2	2	no	—
434	C₂⁴	—	2	4	no	—
435	C₂⁴	—	2	2	no	—
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	4	4	no	—
438	C₂⁴	—	8	2	no	—
439	C₂⁴	C₂²×C₄×C₁₁₃	1	2	yes	—
440	C₂⁴	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
441	C₂⁴	C₂²×C₄×C₁₁₃	1	2	yes	—
442	C₂⁴	C₂×D₄×C₁₁₃	1	4	yes	—
443	C₂⁴	C₂×D₄×C₁₁₃	1	4	yes	—
444	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
445	#41	—	4	8	no	—
446	#41	—	4	4	no	—
447	#41	—	4	4	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	4	2	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	4	no	—
462	C₂×D₄	—	4	4	no	—
463	C₂×D₄	—	4	8	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	8	2	no	—
466	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
467	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
468	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	4	yes	—
469	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
470	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	4	yes	—
471	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	2	yes	—
472	C₂×D₄	C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	4	2	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	—	4	8	no	—
485	C₂²×C₄	—	4	4	no	—
486	C₂²×C₄	—	4	8	no	—
487	C₂²×C₄	—	8	4	no	—
488	C₂²×C₄	—	8	2	no	—
489	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
491	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
492	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
493	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
494	C₂²×C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
495	C₂²×C₄	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	4	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	4	8	no	—
517	C₂²⋊C₄	—	4	8	no	—
518	C₂²⋊C₄	—	8	4	no	—
519	C₂²⋊C₄	#41×C₁₁₃	1	4	yes	complemented
520	C₄²	—	2	1	no	—
521	C₄²	—	2	1	no	—
522	C₄²	—	4	4	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	8	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	1	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	1	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	4	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	4	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	4	1	no	—
545	C₂³	—	4	1	no	—
546	C₂³	—	8	8	no	—
547	C₂³	—	8	4	no	—
548	C₂³	—	8	4	no	—
549	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	4	yes	—
550	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
551	C₂³	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
552	C₂³	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
553	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
554	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
555	C₂³	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
556	C₂³	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
557	Q₈	—	4	2	no	—
558	Q₈	—	4	2	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	8	no	—
561	D₄	—	2	2	no	—
562	D₄	—	2	4	no	—
563	D₄	—	4	2	no	—
564	D₄	—	4	4	no	—
565	D₄	—	4	4	no	—
566	D₄	—	4	2	no	—
567	D₄	—	4	2	no	—
568	D₄	—	4	4	no	—
569	D₄	—	8	4	no	—
570	D₄	#99×C₁₁₃	1	4	yes	complemented
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	—	4	4	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	—	8	2	no	—
601	C₂×C₄	—	8	16	no	—
602	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
603	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	2	yes	—
604	C₂×C₄	C₂×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	2	2	no	—
612	C₂²	—	2	4	no	—
613	C₂²	—	2	2	no	—
614	C₂²	—	2	4	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	4	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	1	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	8	no	—
621	C₂²	—	4	8	no	—
622	C₂²	—	4	4	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	4	1	no	—
627	C₂²	—	8	4	no	—
628	C₂²	—	8	4	no	—
629	C₂²	—	8	2	no	—
630	C₂²	—	8	2	no	—
631	C₂²	—	8	1	no	—
632	C₂²	—	8	1	no	—
633	C₂²	C₂×#99×C₁₁₃	1	4	yes	—
634	C₂²	C₂²×C₂²⋊C₄×C₁₁₃	1	1	yes	characteristic
635	C₂²	C₄×#41×C₁₁₃	1	1	yes	characteristic
636	C₂²	#520×C₁₁₃	1	1	yes	characteristic
637	C₂²	#546×C₁₁₃	1	2	yes	—
638	C₂²	#547×C₁₁₃	1	2	yes	—
639	C₄	—	2	1	no	—
640	C₄	—	2	2	no	—
641	C₄	—	4	4	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	1	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	2	no	—
647	C₄	—	8	4	no	—
648	C₄	—	8	2	no	—
649	C₄	C₂×#99×C₁₁₃	1	2	yes	—
650	C₂	—	2	2	no	—
651	C₂	—	2	2	no	—
652	C₂	—	2	1	no	—
653	C₂	—	2	1	no	—
654	C₂	—	2	4	no	—
655	C₂	—	4	2	no	—
656	C₂	—	4	4	no	—
657	C₂	—	8	1	no	—
658	C₂	C₁₁₃×#2339	1	1	yes	characteristic
659	C₂	C₂²×#99×C₁₁₃	1	1	yes	characteristic
660	C₂	C₁₁₃×#2883	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
prime radical	C₂×#546×C₁₁₃	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁₃ ◁ C₂³×D₄×C₁₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. D₄×C₂²⋊C₄×C₁₁₃	C₂	semidirect
3. C₁₁₃×#1996	C₂	semidirect
4. C₁₁₃×#1898	C₂	semidirect
5. C₁₁₃×#1819	C₂	semidirect
6. C₁₁₃×#1799	C₂	semidirect
7. C₄×#99×C₁₁₃	C₂	semidirect
8. C₂×#342×C₁₁₃	C₂	semidirect
9. C₂²×#41×C₁₁₃	C₄	semidirect
14. C₂×#124×C₁₁₃	C₄	semidirect
15. C₂×#123×C₁₁₃	C₄	semidirect
17. C₂×#121×C₁₁₃	C₄	semidirect
18. C₂×#121×C₁₁₃	C₄	semidirect
21. C₂×C₄×D₄×C₁₁₃	C₄	semidirect
42. C₂²×Q₈×C₁₁₃	C₂×C₄	semidirect
47. C₂²×D₄×C₁₁₃	C₂×C₄	semidirect
48. C₂²×D₄×C₁₁₃	C₂×C₄	semidirect
54. C₂³×C₄×C₁₁₃	C₂×C₄	semidirect
55. C₂³×C₄×C₁₁₃	C₂×C₄	semidirect
56. C₂³×C₄×C₁₁₃	C₂×C₄	semidirect
77. C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	semidirect
78. C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	semidirect
79. C₂×C₄⋊C₄×C₁₁₃	C₂×C₄	semidirect
87. C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	semidirect
88. C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	semidirect
89. C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	semidirect
90. C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	semidirect
91. C₂×C₂²⋊C₄×C₁₁₃	C₂×C₄	semidirect
98. C₂×C₄²×C₁₁₃	C₂×C₄	semidirect
188. C₂²⋊C₄×C₁₁₃	#41	semidirect
239. D₄×C₁₁₃	#99	semidirect
319. #22476	C₁₁₃	direct (N × H)
332. D₄×C₂²⋊C₄	C₂×C₁₁₃	semidirect
333. #1996	C₂×C₁₁₃	semidirect
334. #1898	C₂×C₁₁₃	semidirect
335. #1819	C₂×C₁₁₃	semidirect
336. #1799	C₂×C₁₁₃	semidirect
337. C₄×#99	C₂×C₁₁₃	semidirect
338. C₂×#342	C₂×C₁₁₃	semidirect
339. C₁₁₃	#22476	direct (N × H)
340. C₂²×#41	C₄×C₁₁₃	semidirect
345. C₂×#124	C₄×C₁₁₃	semidirect
346. C₂×#123	C₄×C₁₁₃	semidirect
348. C₂×#121	C₄×C₁₁₃	semidirect
349. C₂×#121	C₄×C₁₁₃	semidirect
352. C₂×C₄×D₄	C₄×C₁₁₃	semidirect
373. C₂²×Q₈	C₂×C₄×C₁₁₃	semidirect
378. C₂²×D₄	C₂×C₄×C₁₁₃	semidirect
379. C₂²×D₄	C₂×C₄×C₁₁₃	semidirect
385. C₂³×C₄	C₂×C₄×C₁₁₃	semidirect
386. C₂³×C₄	C₂×C₄×C₁₁₃	semidirect
387. C₂³×C₄	C₂×C₄×C₁₁₃	semidirect
408. C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	semidirect
409. C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	semidirect
410. C₂×C₄⋊C₄	C₂×C₄×C₁₁₃	semidirect
418. C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	semidirect
419. C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	semidirect
420. C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	semidirect
421. C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	semidirect
422. C₂×C₂²⋊C₄	C₂×C₄×C₁₁₃	semidirect
429. C₂×C₄²	C₂×C₄×C₁₁₃	semidirect
519. C₂²⋊C₄	#41×C₁₁₃	semidirect
570. D₄	#99×C₁₁₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#546×C₁₁₃	2
D₄×C₂²⋊C₄×C₁₁₃	2
C₁₁₃×#1996	2
C₁₁₃×#1898	2
C₁₁₃×#1819	2
C₁₁₃×#1799	2
C₄×#99×C₁₁₃	2
C₂×#342×C₁₁₃	2
#22476	113

Subgroups — C113×#22476 #43290496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#22476 #43290496