Subgroups — C₁₁₃×#16602 #43284622

Order 28928.

1706 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	C₂²×C₄⋊Q₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#393×C₁₁₃	C₂	1	2	yes	maximal
3	C₂×#388×C₁₁₃	C₂	1	1	yes	maximal, characteristic
4	C₁₁₃×#1446	C₂	1	2	yes	maximal, complemented
5	C₁₁₃×#1442	C₂	1	2	yes	maximal, complemented
6	C₁₁₃×#1441	C₂	1	2	yes	maximal, complemented
7	C₁₁₃×#1439	C₂	1	2	yes	maximal
8	C₁₁₃×#1439	C₂	1	2	yes	maximal
9	C₁₁₃×#1434	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₁₁₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₁₁₃	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₁₁₃	—	2	4	no	—
13	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
18	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊C₄×C₁₁₃	C₂²	1	4	yes	—
20	C₂²×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
21	#393×C₁₁₃	—	2	8	no	—
22	#388×C₁₁₃	—	2	4	no	—
23	C₂×#95×C₁₁₃	C₂²	1	1	yes	characteristic
24	C₂×#95×C₁₁₃	C₂²	1	1	yes	characteristic
25	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
26	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
27	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
28	C₂×#95×C₁₁₃	C₂²	1	2	yes	—
29	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
30	C₂×#95×C₁₁₃	C₂²	1	4	yes	—
31	C₂⁵×C₁₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₁₁₃	—	2	4	no	—
33	C₂²×Q₈×C₁₁₃	—	2	4	no	—
34	C₂²×Q₈×C₁₁₃	—	4	4	no	—
35	C₂²×Q₈×C₁₁₃	—	4	4	no	—
36	C₂²×Q₈×C₁₁₃	D₄	1	8	yes	—
37	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₁₁₃	C₂³	1	4	yes	—
41	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₁₁₃	—	2	32	no	—
46	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
47	C₂×C₄⋊C₄×C₁₁₃	—	2	8	no	—
48	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
49	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₁₁₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₁₃	—	2	4	no	—
52	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₁₁₃	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₁₁₃	—	2	8	no	—
57	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₁₁₃	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₁₁₃	—	4	4	no	—
60	C₂×C₂²⋊C₄×C₁₁₃	—	4	4	no	—
61	C₂×C₂²⋊C₄×C₁₁₃	Q₈	1	4	yes	complemented
62	C₂×C₄²×C₁₁₃	—	2	4	no	—
63	C₂×C₄²×C₁₁₃	D₄	1	4	yes	—
64	#95×C₁₁₃	—	2	2	no	—
65	#95×C₁₁₃	—	2	2	no	—
66	#95×C₁₁₃	—	2	8	no	—
67	#95×C₁₁₃	—	2	4	no	—
68	#95×C₁₁₃	—	2	4	no	—
69	#95×C₁₁₃	—	2	4	no	—
70	#95×C₁₁₃	—	4	4	no	—
71	#95×C₁₁₃	—	4	4	no	—
72	C₂⁴×C₁₁₃	—	2	1	no	—
73	C₂⁴×C₁₁₃	—	2	2	no	—
74	C₂⁴×C₁₁₃	—	2	1	no	—
75	C₂⁴×C₁₁₃	—	2	2	no	—
76	C₂⁴×C₁₁₃	—	2	2	no	—
77	C₂⁴×C₁₁₃	—	2	2	no	—
78	C₂⁴×C₁₁₃	—	2	2	no	—
79	C₂⁴×C₁₁₃	—	2	2	no	—
80	C₂⁴×C₁₁₃	C₂×Q₈	1	2	yes	—
81	C₂⁴×C₁₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×Q₈×C₁₁₃	—	2	16	no	—
83	C₂×Q₈×C₁₁₃	—	4	16	no	—
84	C₂×Q₈×C₁₁₃	—	4	16	no	—
85	C₂×Q₈×C₁₁₃	—	4	8	no	—
86	C₂×Q₈×C₁₁₃	—	4	8	no	—
87	C₂²×C₄×C₁₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₁₃	—	2	2	no	—
92	C₂²×C₄×C₁₁₃	—	2	2	no	—
93	C₂²×C₄×C₁₁₃	—	2	1	no	—
94	C₂²×C₄×C₁₁₃	—	2	4	no	—
95	C₂²×C₄×C₁₁₃	—	2	4	no	—
96	C₂²×C₄×C₁₁₃	—	2	1	no	—
97	C₂²×C₄×C₁₁₃	—	2	2	no	—
98	C₂²×C₄×C₁₁₃	—	2	2	no	—
99	C₂²×C₄×C₁₁₃	—	2	4	no	—
100	C₂²×C₄×C₁₁₃	—	2	4	no	—
101	C₂²×C₄×C₁₁₃	—	2	4	no	—
102	C₂²×C₄×C₁₁₃	—	2	4	no	—
103	C₂²×C₄×C₁₁₃	—	2	2	no	—
104	C₂²×C₄×C₁₁₃	—	2	1	no	—
105	C₂²×C₄×C₁₁₃	—	2	4	no	—
106	C₂²×C₄×C₁₁₃	—	2	1	no	—
107	C₂²×C₄×C₁₁₃	—	2	4	no	—
108	C₂²×C₄×C₁₁₃	—	2	4	no	—
109	C₂²×C₄×C₁₁₃	—	2	2	no	—
110	C₂²×C₄×C₁₁₃	—	2	2	no	—
111	C₂²×C₄×C₁₁₃	—	2	2	no	—
112	C₂²×C₄×C₁₁₃	—	2	4	no	—
113	C₂²×C₄×C₁₁₃	—	2	4	no	—
114	C₂²×C₄×C₁₁₃	—	2	2	no	—
115	C₂²×C₄×C₁₁₃	—	2	2	no	—
116	C₂²×C₄×C₁₁₃	—	2	4	no	—
117	C₂²×C₄×C₁₁₃	—	2	4	no	—
118	C₂²×C₄×C₁₁₃	C₂×D₄	1	8	yes	—
119	C₂²×C₄×C₁₁₃	C₂×D₄	1	4	yes	—
120	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₁₁₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
124	C₂²×C₄×C₁₁₃	C₂×Q₈	1	2	yes	—
125	C₂²×C₄×C₁₁₃	#41	1	2	yes	—
126	C₂²×C₄×C₁₁₃	#41	1	4	yes	—
127	C₄⋊C₄×C₁₁₃	—	2	16	no	—
128	C₄⋊C₄×C₁₁₃	—	2	16	no	—
129	C₄⋊C₄×C₁₁₃	—	4	4	no	—
130	C₂²⋊C₄×C₁₁₃	—	2	8	no	—
131	C₂²⋊C₄×C₁₁₃	—	4	16	no	—
132	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
133	C₂²⋊C₄×C₁₁₃	—	4	4	no	—
134	C₄²×C₁₁₃	—	2	8	no	—
135	C₂³×C₁₁₃	—	2	2	no	—
136	C₂³×C₁₁₃	—	2	4	no	—
137	C₂³×C₁₁₃	—	2	4	no	—
138	C₂³×C₁₁₃	—	2	4	no	—
139	C₂³×C₁₁₃	—	4	1	no	—
140	C₂³×C₁₁₃	—	4	1	no	—
141	C₂³×C₁₁₃	—	4	4	no	—
142	C₂³×C₁₁₃	—	4	1	no	—
143	C₂³×C₁₁₃	—	4	1	no	—
144	C₂³×C₁₁₃	—	4	1	no	—
145	C₂³×C₁₁₃	—	4	1	no	—
146	C₂³×C₁₁₃	—	4	2	no	—
147	C₂³×C₁₁₃	—	4	4	no	—
148	C₂³×C₁₁₃	—	4	2	no	—
149	C₂³×C₁₁₃	—	4	4	no	—
150	C₂³×C₁₁₃	—	4	2	no	—
151	C₂³×C₁₁₃	—	4	4	no	—
152	C₂³×C₁₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₂³×C₁₁₃	C₂²×D₄	1	2	yes	—
154	C₂³×C₁₁₃	C₂²×Q₈	1	1	yes	characteristic
155	C₂³×C₁₁₃	C₂×#41	1	1	yes	characteristic
156	C₂³×C₁₁₃	C₂×#41	1	2	yes	—
157	C₂³×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
158	C₂³×C₁₁₃	H(5,𝔽₂)	1	2	yes	—
159	C₂³×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
160	C₂³×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
161	Q₈×C₁₁₃	—	4	32	no	—
162	C₂×C₄×C₁₁₃	—	2	2	no	—
163	C₂×C₄×C₁₁₃	—	2	2	no	—
164	C₂×C₄×C₁₁₃	—	2	8	no	—
165	C₂×C₄×C₁₁₃	—	2	2	no	—
166	C₂×C₄×C₁₁₃	—	2	2	no	—
167	C₂×C₄×C₁₁₃	—	2	2	no	—
168	C₂×C₄×C₁₁₃	—	2	2	no	—
169	C₂×C₄×C₁₁₃	—	2	4	no	—
170	C₂×C₄×C₁₁₃	—	2	8	no	—
171	C₂×C₄×C₁₁₃	—	2	4	no	—
172	C₂×C₄×C₁₁₃	—	2	8	no	—
173	C₂×C₄×C₁₁₃	—	2	2	no	—
174	C₂×C₄×C₁₁₃	—	2	8	no	—
175	C₂×C₄×C₁₁₃	—	2	8	no	—
176	C₂×C₄×C₁₁₃	—	2	4	no	—
177	C₂×C₄×C₁₁₃	—	4	1	no	—
178	C₂×C₄×C₁₁₃	—	4	4	no	—
179	C₂×C₄×C₁₁₃	—	4	4	no	—
180	C₂×C₄×C₁₁₃	—	4	1	no	—
181	C₂×C₄×C₁₁₃	—	4	4	no	—
182	C₂×C₄×C₁₁₃	—	4	4	no	—
183	C₂×C₄×C₁₁₃	—	4	2	no	—
184	C₂×C₄×C₁₁₃	—	4	1	no	—
185	C₂×C₄×C₁₁₃	—	4	1	no	—
186	C₂×C₄×C₁₁₃	—	4	1	no	—
187	C₂×C₄×C₁₁₃	—	4	4	no	—
188	C₂×C₄×C₁₁₃	—	4	1	no	—
189	C₂×C₄×C₁₁₃	—	4	1	no	—
190	C₂×C₄×C₁₁₃	—	4	1	no	—
191	C₂×C₄×C₁₁₃	—	4	4	no	—
192	C₂×C₄×C₁₁₃	—	4	4	no	—
193	C₂×C₄×C₁₁₃	—	4	4	no	—
194	C₂×C₄×C₁₁₃	—	4	4	no	—
195	C₂×C₄×C₁₁₃	—	4	4	no	—
196	C₂×C₄×C₁₁₃	—	4	4	no	—
197	C₂×C₄×C₁₁₃	—	4	4	no	—
198	C₂×C₄×C₁₁₃	—	4	4	no	—
199	C₂×C₄×C₁₁₃	#120	1	4	yes	—
200	C₂×C₄×C₁₁₃	C₂²⋊Q₈	1	4	yes	—
201	C₂×C₄×C₁₁₃	#124	1	4	yes	—
202	C₂²×C₁₁₃	—	4	2	no	—
203	C₂²×C₁₁₃	—	4	2	no	—
204	C₂²×C₁₁₃	—	4	2	no	—
205	C₂²×C₁₁₃	—	4	8	no	—
206	C₂²×C₁₁₃	—	8	1	no	—
207	C₂²×C₁₁₃	—	8	1	no	—
208	C₂²×C₁₁₃	—	8	1	no	—
209	C₂²×C₁₁₃	—	8	1	no	—
210	C₂²×C₁₁₃	—	8	4	no	—
211	C₂²×C₁₁₃	C₂×#120	1	1	yes	characteristic
212	C₂²×C₁₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
213	C₂²×C₁₁₃	C₂×#124	1	1	yes	characteristic
214	C₂²×C₁₁₃	#534	1	4	yes	—
215	C₂²×C₁₁₃	#535	1	1	yes	characteristic
216	C₂²×C₁₁₃	#535	1	1	yes	characteristic
217	C₂²×C₁₁₃	#536	1	1	yes	characteristic
218	C₂²×C₁₁₃	#536	1	1	yes	characteristic
219	C₂²×C₁₁₃	#542	1	2	yes	—
220	C₂²×C₁₁₃	#542	1	4	yes	—
221	C₂²×C₁₁₃	#544	1	2	yes	—
222	C₂²×C₁₁₃	#548	1	2	yes	—
223	C₂²×C₁₁₃	#548	1	4	yes	—
224	C₂²×C₁₁₃	D₄×Q₈	1	2	yes	—
225	C₂²×C₁₁₃	#554	1	2	yes	—
226	C₂²×C₁₁₃	#555	1	2	yes	—
227	C₂²×C₁₁₃	#563	1	4	yes	—
228	C₄×C₁₁₃	—	2	4	no	—
229	C₄×C₁₁₃	—	2	4	no	—
230	C₄×C₁₁₃	—	2	4	no	—
231	C₄×C₁₁₃	—	4	2	no	—
232	C₄×C₁₁₃	—	4	8	no	—
233	C₄×C₁₁₃	—	8	1	no	—
234	C₄×C₁₁₃	—	8	1	no	—
235	C₄×C₁₁₃	—	8	4	no	—
236	#16602	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂×C₁₁₃	—	8	2	no	—
238	C₂×C₁₁₃	#2409	1	1	yes	characteristic
239	C₂×C₁₁₃	#2436	1	1	yes	characteristic
240	C₂×C₁₁₃	#2563	1	1	yes	characteristic
241	C₂×C₁₁₃	#2613	1	1	yes	characteristic
242	C₂×C₁₁₃	#2628	1	1	yes	characteristic
243	C₂×C₁₁₃	#2640	1	1	yes	characteristic
244	C₂×C₁₁₃	#2729	1	4	yes	—
245	C₂×C₁₁₃	#2731	1	4	yes	—
246	C₂×C₁₁₃	#2821	1	1	yes	characteristic
247	C₂²×C₄⋊Q₈	C₂×C₁₁₃	1	1	yes	characteristic, complemented
248	C₂×#393	C₂×C₁₁₃	1	2	yes	—
249	C₂×#388	C₂×C₁₁₃	1	1	yes	characteristic
250	#1446	C₂×C₁₁₃	1	2	yes	complemented
251	#1442	C₂×C₁₁₃	1	2	yes	complemented
252	#1441	C₂×C₁₁₃	1	2	yes	complemented
253	#1439	C₂×C₁₁₃	1	2	yes	—
254	#1439	C₂×C₁₁₃	1	2	yes	—
255	#1434	C₂×C₁₁₃	1	1	yes	characteristic, complemented
256	C₁₁₃	#16602	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³×Q₈	C₂²×C₁₁₃	1	2	yes	—
258	C₂×C₄⋊Q₈	—	2	8	no	—
259	C₂×C₄⋊Q₈	—	2	4	no	—
260	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
261	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	2	yes	—
262	C₂²×C₄⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
263	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
264	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
265	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	1	yes	characteristic
266	C₂²×C₂²⋊C₄	C₂²×C₁₁₃	1	4	yes	—
267	C₂²×C₄²	C₂²×C₁₁₃	1	1	yes	characteristic
268	#393	—	2	8	no	—
269	#388	—	2	4	no	—
270	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
271	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
272	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
273	C₂×#95	C₂²×C₁₁₃	1	2	yes	—
274	C₂×#95	C₂²×C₁₁₃	1	1	yes	characteristic
275	C₂×#95	C₂²×C₁₁₃	1	1	yes	characteristic
276	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
277	C₂×#95	C₂²×C₁₁₃	1	4	yes	—
278	C₂⁵	C₂³×C₁₁₃	1	1	yes	characteristic
279	C₂²×Q₈	—	2	4	no	—
280	C₂²×Q₈	—	2	4	no	—
281	C₂²×Q₈	—	4	4	no	—
282	C₂²×Q₈	—	4	4	no	—
283	C₂²×Q₈	D₄×C₁₁₃	1	8	yes	—
284	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
286	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₁₁₃	1	4	yes	—
288	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
290	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
291	C₂³×C₄	C₂³×C₁₁₃	1	1	yes	characteristic
292	C₄⋊Q₈	—	2	32	no	—
293	C₂×C₄⋊C₄	—	2	8	no	—
294	C₂×C₄⋊C₄	—	2	8	no	—
295	C₂×C₄⋊C₄	—	2	4	no	—
296	C₂×C₄⋊C₄	—	2	4	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	4	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	8	no	—
304	C₂×C₂²⋊C₄	—	2	4	no	—
305	C₂×C₂²⋊C₄	—	2	4	no	—
306	C₂×C₂²⋊C₄	—	4	4	no	—
307	C₂×C₂²⋊C₄	—	4	4	no	—
308	C₂×C₂²⋊C₄	Q₈×C₁₁₃	1	4	yes	complemented
309	C₂×C₄²	—	2	4	no	—
310	C₂×C₄²	D₄×C₁₁₃	1	4	yes	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	—	2	8	no	—
314	#95	—	2	4	no	—
315	#95	—	2	4	no	—
316	#95	—	2	4	no	—
317	#95	—	4	4	no	—
318	#95	—	4	4	no	—
319	C₂⁴	—	2	1	no	—
320	C₂⁴	—	2	2	no	—
321	C₂⁴	—	2	1	no	—
322	C₂⁴	—	2	2	no	—
323	C₂⁴	—	2	2	no	—
324	C₂⁴	—	2	2	no	—
325	C₂⁴	—	2	2	no	—
326	C₂⁴	—	2	2	no	—
327	C₂⁴	C₂×Q₈×C₁₁₃	1	2	yes	—
328	C₂⁴	C₂⁴×C₁₁₃	1	1	yes	characteristic
329	C₂×Q₈	—	2	16	no	—
330	C₂×Q₈	—	4	16	no	—
331	C₂×Q₈	—	4	16	no	—
332	C₂×Q₈	—	4	8	no	—
333	C₂×Q₈	—	4	8	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	C₂×D₄×C₁₁₃	1	8	yes	—
366	C₂²×C₄	C₂×D₄×C₁₁₃	1	4	yes	—
367	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₁₁₃	1	2	yes	—
370	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
371	C₂²×C₄	C₂×Q₈×C₁₁₃	1	2	yes	—
372	C₂²×C₄	#41×C₁₁₃	1	2	yes	—
373	C₂²×C₄	#41×C₁₁₃	1	4	yes	—
374	C₄⋊C₄	—	2	16	no	—
375	C₄⋊C₄	—	2	16	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	2	8	no	—
378	C₂²⋊C₄	—	4	16	no	—
379	C₂²⋊C₄	—	4	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₄²	—	2	8	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	2	4	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	—	4	4	no	—
389	C₂³	—	4	1	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	4	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	4	no	—
399	C₂³	C₂²×D₄×C₁₁₃	1	1	yes	characteristic
400	C₂³	C₂²×D₄×C₁₁₃	1	2	yes	—
401	C₂³	C₂²×Q₈×C₁₁₃	1	1	yes	characteristic
402	C₂³	C₂×#41×C₁₁₃	1	1	yes	characteristic
403	C₂³	C₂×#41×C₁₁₃	1	2	yes	—
404	C₂³	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
405	C₂³	H(5,𝔽₂)×C₁₁₃	1	2	yes	—
406	C₂³	2₋¹⁺⁴×C₁₁₃	1	2	yes	—
407	C₂³	2₋¹⁺⁴×C₁₁₃	1	2	yes	—
408	Q₈	—	4	32	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	8	no	—
422	C₂×C₄	—	2	8	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	4	no	—
439	C₂×C₄	—	4	4	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	#120×C₁₁₃	1	4	yes	—
447	C₂×C₄	C₂²⋊Q₈×C₁₁₃	1	4	yes	—
448	C₂×C₄	#124×C₁₁₃	1	4	yes	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	8	4	no	—
458	C₂²	C₂×#120×C₁₁₃	1	1	yes	characteristic
459	C₂²	C₂×C₂²⋊Q₈×C₁₁₃	1	1	yes	characteristic
460	C₂²	C₂×#124×C₁₁₃	1	1	yes	characteristic
461	C₂²	#534×C₁₁₃	1	4	yes	—
462	C₂²	#535×C₁₁₃	1	1	yes	characteristic
463	C₂²	#535×C₁₁₃	1	1	yes	characteristic
464	C₂²	#536×C₁₁₃	1	1	yes	characteristic
465	C₂²	#536×C₁₁₃	1	1	yes	characteristic
466	C₂²	#542×C₁₁₃	1	2	yes	—
467	C₂²	#542×C₁₁₃	1	4	yes	—
468	C₂²	#544×C₁₁₃	1	2	yes	—
469	C₂²	#548×C₁₁₃	1	2	yes	—
470	C₂²	#548×C₁₁₃	1	4	yes	—
471	C₂²	D₄×Q₈×C₁₁₃	1	2	yes	—
472	C₂²	#554×C₁₁₃	1	2	yes	—
473	C₂²	#555×C₁₁₃	1	2	yes	—
474	C₂²	#563×C₁₁₃	1	4	yes	—
475	C₄	—	2	4	no	—
476	C₄	—	2	4	no	—
477	C₄	—	2	4	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	8	no	—
480	C₄	—	8	1	no	—
481	C₄	—	8	1	no	—
482	C₄	—	8	4	no	—
483	C₂	—	8	2	no	—
484	C₂	C₁₁₃×#2409	1	1	yes	characteristic
485	C₂	C₁₁₃×#2436	1	1	yes	characteristic
486	C₂	C₁₁₃×#2563	1	1	yes	characteristic
487	C₂	C₁₁₃×#2613	1	1	yes	characteristic
488	C₂	C₁₁₃×#2628	1	1	yes	characteristic
489	C₂	C₁₁₃×#2640	1	1	yes	characteristic
490	C₂	C₁₁₃×#2729	1	4	yes	—
491	C₂	C₁₁₃×#2731	1	4	yes	—
492	C₂	C₁₁₃×#2821	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₁₃	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₁₃
prime radical	C₂⁵×C₁₁₃	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₁₁₃	C₂	semidirect
4. C₁₁₃×#1446	C₂	semidirect
5. C₁₁₃×#1442	C₂	semidirect
6. C₁₁₃×#1441	C₂	semidirect
9. C₁₁₃×#1434	C₂	semidirect
61. C₂×C₂²⋊C₄×C₁₁₃	Q₈	semidirect
236. #16602	C₁₁₃	direct (N × H)
247. C₂²×C₄⋊Q₈	C₂×C₁₁₃	semidirect
250. #1446	C₂×C₁₁₃	semidirect
251. #1442	C₂×C₁₁₃	semidirect
252. #1441	C₂×C₁₁₃	semidirect
255. #1434	C₂×C₁₁₃	semidirect
256. C₁₁₃	#16602	direct (N × H)
308. C₂×C₂²⋊C₄	Q₈×C₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₁₁₃	2
C₂×#393×C₁₁₃	2
C₂×#388×C₁₁₃	2
C₁₁₃×#1446	2
C₁₁₃×#1442	2
C₁₁₃×#1441	2
C₁₁₃×#1439	2
C₁₁₃×#1434	2
#16602	113

Subgroups — C113×#16602 #43284622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#16602 #43284622