Subgroups — C₁₁₃×#22996 #43291016

Order 28928.

1030 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#461×C₁₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₁₁₃	C₂	1	2	yes	maximal
3	C₁₁₃×#2365	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic
5	C₁₁₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₁₃×#1933	C₂	1	2	yes	maximal, complemented
8	C₁₁₃×#1877	C₂	1	2	yes	maximal
9	C₁₁₃×#1865	C₂	1	2	yes	maximal, complemented
10	C₈×C₄⋊C₄×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#102×C₁₁₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₁₁₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₁₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₁₁₃	—	2	1	no	—
18	#461×C₁₁₃	C₄	1	8	yes	complemented
19	C₄×SD₁₆×C₁₁₃	—	2	8	no	—
20	C₂×#106×C₁₁₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
22	C₂×#103×C₁₁₃	C₂²	1	2	yes	—
23	C₂×#103×C₁₁₃	C₂²	1	2	yes	—
24	C₂×#103×C₁₁₃	C₂²	1	2	yes	—
25	C₂×#102×C₁₁₃	C₂²	1	1	yes	characteristic
26	C₂×#102×C₁₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₁₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
29	#391×C₁₁₃	C₂²	1	2	yes	—
30	#389×C₁₁₃	—	2	1	no	—
31	#389×C₁₁₃	C₂²	1	2	yes	—
32	#387×C₁₁₃	—	2	2	no	—
33	#385×C₁₁₃	—	2	2	no	—
34	#384×C₁₁₃	C₂²	1	1	yes	characteristic
35	#380×C₁₁₃	—	2	1	no	—
36	C₄×C₄⋊C₄×C₁₁₃	—	2	1	no	—
37	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
39	#340×C₁₁₃	C₂²	1	1	yes	characteristic
40	#340×C₁₁₃	C₂²	1	2	yes	—
41	#340×C₁₁₃	C₂²	1	1	yes	characteristic
42	#336×C₁₁₃	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₁₃	C₂³	1	2	yes	—
44	C₂²×D₄×C₁₁₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₁₁₃	—	2	1	no	—
46	C₂³×C₄×C₁₁₃	—	2	2	no	—
47	C₂×SD₁₆×C₁₁₃	—	2	4	no	—
48	C₂×SD₁₆×C₁₁₃	—	2	4	no	—
49	C₂×SD₁₆×C₁₁₃	C₂×C₄	1	8	yes	—
50	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₁₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₁₁₃	—	2	2	no	—
54	C₄⋊Q₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
55	C₄×Q₈×C₁₁₃	—	2	4	no	—
56	C₄×Q₈×C₁₁₃	—	4	2	no	—
57	C₄×D₄×C₁₁₃	—	2	4	no	—
58	C₄×D₄×C₁₁₃	—	2	2	no	—
59	C₄×D₄×C₁₁₃	—	4	2	no	—
60	C₄×D₄×C₁₁₃	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₁₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₁₁₃	—	4	4	no	—
68	C₂×C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₁₁₃	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₁₁₃	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₁₁₃	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
76	C₂×C₄²×C₁₁₃	—	2	2	no	—
77	C₂×C₄²×C₁₁₃	—	2	1	no	—
78	C₂×C₄²×C₁₁₃	C₂³	1	2	yes	—
79	C₂×C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
80	#106×C₁₁₃	—	2	4	no	—
81	C₄⋊C₈×C₁₁₃	C₂×C₄	1	4	yes	complemented
82	#103×C₁₁₃	—	2	4	no	—
83	#103×C₁₁₃	—	4	2	no	—
84	#103×C₁₁₃	C₂×C₄	1	8	yes	complemented
85	#102×C₁₁₃	—	2	2	no	—
86	#102×C₁₁₃	D₄	1	4	yes	—
87	C₄×C₈×C₁₁₃	—	2	4	no	—
88	C₄×C₈×C₁₁₃	—	2	2	no	—
89	#95×C₁₁₃	—	2	2	no	—
90	#95×C₁₁₃	—	2	1	no	—
91	#95×C₁₁₃	—	2	2	no	—
92	#95×C₁₁₃	—	2	2	no	—
93	#95×C₁₁₃	D₄	1	2	yes	—
94	C₂⁴×C₁₁₃	—	2	1	no	—
95	C₂×Q₈×C₁₁₃	—	2	2	no	—
96	C₂×Q₈×C₁₁₃	—	2	2	no	—
97	C₂×Q₈×C₁₁₃	—	2	2	no	—
98	C₂×Q₈×C₁₁₃	—	2	2	no	—
99	C₂×Q₈×C₁₁₃	—	2	2	no	—
100	C₂×Q₈×C₁₁₃	C₂²×C₄	1	4	yes	—
101	C₂×D₄×C₁₁₃	—	2	1	no	—
102	C₂×D₄×C₁₁₃	—	2	1	no	—
103	C₂×D₄×C₁₁₃	—	2	1	no	—
104	C₂×D₄×C₁₁₃	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₁₁₃	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₁₁₃	#41	1	2	yes	—
107	C₂²×C₄×C₁₁₃	—	2	2	no	—
108	C₂²×C₄×C₁₁₃	—	2	1	no	—
109	C₂²×C₄×C₁₁₃	—	2	1	no	—
110	C₂²×C₄×C₁₁₃	—	2	4	no	—
111	C₂²×C₄×C₁₁₃	—	2	1	no	—
112	C₂²×C₄×C₁₁₃	—	2	2	no	—
113	C₂²×C₄×C₁₁₃	—	4	4	no	—
114	C₂²×C₄×C₁₁₃	—	4	2	no	—
115	C₂²×C₄×C₁₁₃	—	4	2	no	—
116	C₂²×C₄×C₁₁₃	—	4	4	no	—
117	C₂²×C₄×C₁₁₃	—	8	2	no	—
118	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
123	SD₁₆×C₁₁₃	—	2	16	no	—
124	C₂×C₈×C₁₁₃	—	2	2	no	—
125	C₂×C₈×C₁₁₃	—	2	1	no	—
126	C₂×C₈×C₁₁₃	—	2	2	no	—
127	C₂×C₈×C₁₁₃	—	2	1	no	—
128	C₂×C₈×C₁₁₃	—	2	1	no	—
129	C₂×C₈×C₁₁₃	—	2	1	no	—
130	C₂×C₈×C₁₁₃	—	2	1	no	—
131	C₂×C₈×C₁₁₃	C₂²×C₄	1	4	yes	—
132	C₂×C₈×C₁₁₃	C₂×D₄	1	2	yes	—
133	C₄⋊C₄×C₁₁₃	—	2	4	no	—
134	C₄⋊C₄×C₁₁₃	—	2	4	no	—
135	C₄⋊C₄×C₁₁₃	—	2	2	no	—
136	C₄⋊C₄×C₁₁₃	—	2	1	no	—
137	C₄⋊C₄×C₁₁₃	—	2	1	no	—
138	C₄⋊C₄×C₁₁₃	—	2	2	no	—
139	C₄⋊C₄×C₁₁₃	—	2	1	no	—
140	C₄⋊C₄×C₁₁₃	—	2	2	no	—
141	C₄⋊C₄×C₁₁₃	—	2	1	no	—
142	C₄⋊C₄×C₁₁₃	—	4	4	no	—
143	C₄⋊C₄×C₁₁₃	—	4	4	no	—
144	C₄⋊C₄×C₁₁₃	—	4	2	no	—
145	C₄⋊C₄×C₁₁₃	—	8	1	no	—
146	C₄⋊C₄×C₁₁₃	SD₁₆	1	4	yes	complemented
147	C₄⋊C₄×C₁₁₃	C₂²×C₄	1	2	yes	—
148	C₄⋊C₄×C₁₁₃	C₂×D₄	1	2	yes	—
149	C₂²⋊C₄×C₁₁₃	—	2	4	no	—
150	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
151	C₄²×C₁₁₃	—	2	2	no	—
152	C₄²×C₁₁₃	—	2	2	no	—
153	C₄²×C₁₁₃	—	4	4	no	—
154	C₄²×C₁₁₃	—	4	2	no	—
155	C₄²×C₁₁₃	C₂²×C₄	1	2	yes	—
156	C₄²×C₁₁₃	#41	1	2	yes	—
157	C₂³×C₁₁₃	—	2	2	no	—
158	C₂³×C₁₁₃	—	2	2	no	—
159	C₂³×C₁₁₃	—	2	2	no	—
160	C₂³×C₁₁₃	—	4	1	no	—
161	C₂³×C₁₁₃	—	4	1	no	—
162	C₂³×C₁₁₃	—	4	1	no	—
163	C₂³×C₁₁₃	—	4	1	no	—
164	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₁₁₃	—	2	4	no	—
166	Q₈×C₁₁₃	—	2	4	no	—
167	Q₈×C₁₁₃	—	4	2	no	—
168	Q₈×C₁₁₃	—	4	2	no	—
169	D₄×C₁₁₃	—	2	2	no	—
170	D₄×C₁₁₃	—	2	2	no	—
171	D₄×C₁₁₃	—	2	2	no	—
172	D₄×C₁₁₃	C₄×D₄	1	4	yes	—
173	C₂×C₄×C₁₁₃	—	2	2	no	—
174	C₂×C₄×C₁₁₃	—	2	1	no	—
175	C₂×C₄×C₁₁₃	—	2	2	no	—
176	C₂×C₄×C₁₁₃	—	2	1	no	—
177	C₂×C₄×C₁₁₃	—	2	2	no	—
178	C₂×C₄×C₁₁₃	—	2	4	no	—
179	C₂×C₄×C₁₁₃	—	2	2	no	—
180	C₂×C₄×C₁₁₃	—	2	2	no	—
181	C₂×C₄×C₁₁₃	—	2	2	no	—
182	C₂×C₄×C₁₁₃	—	2	2	no	—
183	C₂×C₄×C₁₁₃	—	4	2	no	—
184	C₂×C₄×C₁₁₃	—	4	2	no	—
185	C₂×C₄×C₁₁₃	—	4	1	no	—
186	C₂×C₄×C₁₁₃	—	4	1	no	—
187	C₂×C₄×C₁₁₃	—	4	1	no	—
188	C₂×C₄×C₁₁₃	—	4	1	no	—
189	C₂×C₄×C₁₁₃	—	4	8	no	—
190	C₂×C₄×C₁₁₃	—	4	2	no	—
191	C₂×C₄×C₁₁₃	—	4	1	no	—
192	C₂×C₄×C₁₁₃	—	4	2	no	—
193	C₂×C₄×C₁₁₃	—	4	1	no	—
194	C₂×C₄×C₁₁₃	—	4	2	no	—
195	C₂×C₄×C₁₁₃	—	4	1	no	—
196	C₂×C₄×C₁₁₃	—	8	8	no	—
197	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
198	C₂×C₄×C₁₁₃	C₄×D₄	1	2	yes	—
199	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	4	yes	—
200	C₂×C₄×C₁₁₃	C₂×SD₁₆	1	2	yes	—
201	C₂×C₄×C₁₁₃	#135	1	2	yes	—
202	C₂×C₄×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
208	C₈×C₁₁₃	—	2	4	no	—
209	C₈×C₁₁₃	—	2	2	no	—
210	C₈×C₁₁₃	—	4	1	no	—
211	C₂²×C₁₁₃	—	2	4	no	—
212	C₂²×C₁₁₃	—	4	2	no	—
213	C₂²×C₁₁₃	—	4	2	no	—
214	C₂²×C₁₁₃	—	4	2	no	—
215	C₂²×C₁₁₃	—	4	2	no	—
216	C₂²×C₁₁₃	—	4	2	no	—
217	C₂²×C₁₁₃	—	4	2	no	—
218	C₂²×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₁₁₃	D₄²	1	1	yes	characteristic
220	C₂²×C₁₁₃	#548	1	1	yes	characteristic
221	C₂²×C₁₁₃	C₂²×SD₁₆	1	1	yes	characteristic
222	C₂²×C₁₁₃	C₂×#135	1	1	yes	characteristic
223	C₂²×C₁₁₃	#577	1	1	yes	characteristic
224	C₂²×C₁₁₃	#578	1	1	yes	characteristic
225	C₄×C₁₁₃	—	2	2	no	—
226	C₄×C₁₁₃	—	2	4	no	—
227	C₄×C₁₁₃	—	4	4	no	—
228	C₄×C₁₁₃	—	4	2	no	—
229	C₄×C₁₁₃	—	4	2	no	—
230	C₄×C₁₁₃	—	4	2	no	—
231	C₄×C₁₁₃	—	8	1	no	—
232	C₄×C₁₁₃	—	8	2	no	—
233	C₄×C₁₁₃	C₄×SD₁₆	1	4	yes	complemented
234	C₄×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
235	C₄×C₁₁₃	#520	1	1	yes	characteristic
236	C₄×C₁₁₃	#547	1	1	yes	characteristic
237	C₄×C₁₁₃	#553	1	1	yes	characteristic
238	#22996	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂×C₁₁₃	—	4	4	no	—
240	C₂×C₁₁₃	#2361	1	1	yes	characteristic
241	C₂×C₁₁₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
242	C₂×C₁₁₃	#2947	1	1	yes	characteristic
243	C₂×C₁₁₃	D₄×SD₁₆	1	1	yes	characteristic
244	C₂×C₁₁₃	#3283	1	1	yes	characteristic
245	C₂×C₁₁₃	#3359	1	1	yes	characteristic
246	C₂×C₁₁₃	#3360	1	1	yes	characteristic
247	C₂×#461	C₂×C₁₁₃	1	1	yes	characteristic
248	C₂×C₄×SD₁₆	C₂×C₁₁₃	1	2	yes	—
249	#2365	C₂×C₁₁₃	1	1	yes	characteristic, complemented
250	D₄×C₄⋊C₄	C₂×C₁₁₃	1	1	yes	characteristic
251	#1956	C₂×C₁₁₃	1	1	yes	characteristic, complemented
252	#1948	C₂×C₁₁₃	1	1	yes	characteristic, complemented
253	#1933	C₂×C₁₁₃	1	2	yes	complemented
254	#1877	C₂×C₁₁₃	1	2	yes	—
255	#1865	C₂×C₁₁₃	1	2	yes	complemented
256	C₈×C₄⋊C₄	C₂×C₁₁₃	1	1	yes	characteristic, complemented
257	C₄×#102	C₂×C₁₁₃	1	1	yes	characteristic
258	C₁₁₃	#22996	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²×SD₁₆	C₂²×C₁₁₃	1	2	yes	—
260	C₂×C₄⋊Q₈	C₂²×C₁₁₃	1	1	yes	characteristic
261	C₂×C₄×Q₈	C₂²×C₁₁₃	1	2	yes	—
262	C₂×C₄×D₄	C₂²×C₁₁₃	1	2	yes	—
263	C₂×C₄×D₄	C₂²×C₁₁₃	1	1	yes	characteristic
264	C₂²×C₄⋊C₄	—	2	1	no	—
265	#461	C₄×C₁₁₃	1	8	yes	complemented
266	C₄×SD₁₆	—	2	8	no	—
267	C₂×#106	C₂²×C₁₁₃	1	2	yes	—
268	C₂×C₄⋊C₈	C₂²×C₁₁₃	1	1	yes	characteristic
269	C₂×#103	C₂²×C₁₁₃	1	2	yes	—
270	C₂×#103	C₂²×C₁₁₃	1	2	yes	—
271	C₂×#103	C₂²×C₁₁₃	1	2	yes	—
272	C₂×#102	C₂²×C₁₁₃	1	1	yes	characteristic
273	C₂×#102	C₂²×C₁₁₃	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₁₁₃	1	2	yes	—
275	C₂×C₄×C₈	C₂²×C₁₁₃	1	1	yes	characteristic
276	#391	C₂²×C₁₁₃	1	2	yes	—
277	#389	—	2	1	no	—
278	#389	C₂²×C₁₁₃	1	2	yes	—
279	#387	—	2	2	no	—
280	#385	—	2	2	no	—
281	#384	C₂²×C₁₁₃	1	1	yes	characteristic
282	#380	—	2	1	no	—
283	C₄×C₄⋊C₄	—	2	1	no	—
284	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
285	C₄×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
286	#340	C₂²×C₁₁₃	1	1	yes	characteristic
287	#340	C₂²×C₁₁₃	1	2	yes	—
288	#340	C₂²×C₁₁₃	1	1	yes	characteristic
289	#336	C₂²×C₁₁₃	1	2	yes	—
290	C₂²×Q₈	C₂³×C₁₁₃	1	2	yes	—
291	C₂²×D₄	C₂³×C₁₁₃	1	1	yes	characteristic
292	C₂³×C₄	—	2	1	no	—
293	C₂³×C₄	—	2	2	no	—
294	C₂×SD₁₆	—	2	4	no	—
295	C₂×SD₁₆	—	2	4	no	—
296	C₂×SD₁₆	C₂×C₄×C₁₁₃	1	8	yes	—
297	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₁₁₃	1	2	yes	—
299	C₂²×C₈	C₂³×C₁₁₃	1	1	yes	characteristic
300	C₄⋊Q₈	—	2	2	no	—
301	C₄⋊Q₈	C₂×C₄×C₁₁₃	1	4	yes	complemented
302	C₄×Q₈	—	2	4	no	—
303	C₄×Q₈	—	4	2	no	—
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	4	2	no	—
307	C₄×D₄	C₂×C₄×C₁₁₃	1	4	yes	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	4	4	no	—
315	C₂×C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
316	C₂×C₄⋊C₄	D₄×C₁₁₃	1	1	yes	characteristic
317	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
320	C₂×C₄⋊C₄	C₂³×C₁₁₃	1	1	yes	characteristic
321	C₂×C₂²⋊C₄	—	2	1	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	C₂³×C₁₁₃	1	2	yes	—
326	C₂×C₄²	C₂³×C₁₁₃	1	1	yes	characteristic
327	#106	—	2	4	no	—
328	C₄⋊C₈	C₂×C₄×C₁₁₃	1	4	yes	complemented
329	#103	—	2	4	no	—
330	#103	—	4	2	no	—
331	#103	C₂×C₄×C₁₁₃	1	8	yes	complemented
332	#102	—	2	2	no	—
333	#102	D₄×C₁₁₃	1	4	yes	—
334	C₄×C₈	—	2	4	no	—
335	C₄×C₈	—	2	2	no	—
336	#95	—	2	2	no	—
337	#95	—	2	1	no	—
338	#95	—	2	2	no	—
339	#95	—	2	2	no	—
340	#95	D₄×C₁₁₃	1	2	yes	—
341	C₂⁴	—	2	1	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	C₂²×C₄×C₁₁₃	1	4	yes	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	C₂²×C₄×C₁₁₃	1	2	yes	—
352	C₂×D₄	C₂×D₄×C₁₁₃	1	2	yes	—
353	C₂×D₄	#41×C₁₁₃	1	2	yes	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	2	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	8	2	no	—
365	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
366	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₁₁₃	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₁₁₃	1	1	yes	characteristic
370	SD₁₆	—	2	16	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	C₂²×C₄×C₁₁₃	1	4	yes	—
379	C₂×C₈	C₂×D₄×C₁₁₃	1	2	yes	—
380	C₄⋊C₄	—	2	4	no	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	4	4	no	—
390	C₄⋊C₄	—	4	4	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	—	8	1	no	—
393	C₄⋊C₄	SD₁₆×C₁₁₃	1	4	yes	complemented
394	C₄⋊C₄	C₂²×C₄×C₁₁₃	1	2	yes	—
395	C₄⋊C₄	C₂×D₄×C₁₁₃	1	2	yes	—
396	C₂²⋊C₄	—	2	4	no	—
397	C₂²⋊C₄	—	4	4	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	4	4	no	—
401	C₄²	—	4	2	no	—
402	C₄²	C₂²×C₄×C₁₁₃	1	2	yes	—
403	C₄²	#41×C₁₁₃	1	2	yes	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
412	Q₈	—	2	4	no	—
413	Q₈	—	2	4	no	—
414	Q₈	—	4	2	no	—
415	Q₈	—	4	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	C₄×D₄×C₁₁₃	1	4	yes	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	8	8	no	—
444	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
445	C₂×C₄	C₄×D₄×C₁₁₃	1	2	yes	—
446	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	4	yes	—
447	C₂×C₄	C₂×SD₁₆×C₁₁₃	1	2	yes	—
448	C₂×C₄	#135×C₁₁₃	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₁₁₃	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
451	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₁₁₃	1	1	yes	characteristic
453	C₂×C₄	H(5,𝔽₂)×C₁₁₃	1	1	yes	characteristic
454	C₂×C₄	2₋¹⁺⁴×C₁₁₃	1	1	yes	characteristic
455	C₈	—	2	4	no	—
456	C₈	—	2	2	no	—
457	C₈	—	4	1	no	—
458	C₂²	—	2	4	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
466	C₂²	D₄²×C₁₁₃	1	1	yes	characteristic
467	C₂²	#548×C₁₁₃	1	1	yes	characteristic
468	C₂²	C₂²×SD₁₆×C₁₁₃	1	1	yes	characteristic
469	C₂²	C₂×#135×C₁₁₃	1	1	yes	characteristic
470	C₂²	#577×C₁₁₃	1	1	yes	characteristic
471	C₂²	#578×C₁₁₃	1	1	yes	characteristic
472	C₄	—	2	2	no	—
473	C₄	—	2	4	no	—
474	C₄	—	4	4	no	—
475	C₄	—	4	2	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	8	1	no	—
479	C₄	—	8	2	no	—
480	C₄	C₄×SD₁₆×C₁₁₃	1	4	yes	complemented
481	C₄	C₂×C₄×D₄×C₁₁₃	1	1	yes	characteristic
482	C₄	#520×C₁₁₃	1	1	yes	characteristic
483	C₄	#547×C₁₁₃	1	1	yes	characteristic
484	C₄	#553×C₁₁₃	1	1	yes	characteristic
485	C₂	—	4	4	no	—
486	C₂	C₁₁₃×#2361	1	1	yes	characteristic
487	C₂	C₂×C₄×SD₁₆×C₁₁₃	1	1	yes	characteristic
488	C₂	C₁₁₃×#2947	1	1	yes	characteristic
489	C₂	D₄×SD₁₆×C₁₁₃	1	1	yes	characteristic
490	C₂	C₁₁₃×#3283	1	1	yes	characteristic
491	C₂	C₁₁₃×#3359	1	1	yes	characteristic
492	C₂	C₁₁₃×#3360	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₁₃×#2365	C₂	semidirect
5. C₁₁₃×#1956	C₂	semidirect
6. C₁₁₃×#1948	C₂	semidirect
7. C₁₁₃×#1933	C₂	semidirect
9. C₁₁₃×#1865	C₂	semidirect
10. C₈×C₄⋊C₄×C₁₁₃	C₂	semidirect
18. #461×C₁₁₃	C₄	semidirect
54. C₄⋊Q₈×C₁₁₃	C₂×C₄	semidirect
81. C₄⋊C₈×C₁₁₃	C₂×C₄	semidirect
84. #103×C₁₁₃	C₂×C₄	semidirect
146. C₄⋊C₄×C₁₁₃	SD₁₆	semidirect
233. C₄×C₁₁₃	C₄×SD₁₆	semidirect
238. #22996	C₁₁₃	direct (N × H)
249. #2365	C₂×C₁₁₃	semidirect
251. #1956	C₂×C₁₁₃	semidirect
252. #1948	C₂×C₁₁₃	semidirect
253. #1933	C₂×C₁₁₃	semidirect
255. #1865	C₂×C₁₁₃	semidirect
256. C₈×C₄⋊C₄	C₂×C₁₁₃	semidirect
258. C₁₁₃	#22996	direct (N × H)
265. #461	C₄×C₁₁₃	semidirect
301. C₄⋊Q₈	C₂×C₄×C₁₁₃	semidirect
328. C₄⋊C₈	C₂×C₄×C₁₁₃	semidirect
331. #103	C₂×C₄×C₁₁₃	semidirect
393. C₄⋊C₄	SD₁₆×C₁₁₃	semidirect
480. C₄	C₄×SD₁₆×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#461×C₁₁₃	2
C₂×C₄×SD₁₆×C₁₁₃	2
C₁₁₃×#2365	2
D₄×C₄⋊C₄×C₁₁₃	2
C₁₁₃×#1956	2
C₁₁₃×#1948	2
C₁₁₃×#1933	2
C₁₁₃×#1877	2
C₁₁₃×#1865	2
C₈×C₄⋊C₄×C₁₁₃	2
C₄×#102×C₁₁₃	2
#22996	113

Subgroups — C113×#22996 #43291016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#22996 #43291016