Subgroups — #13325494

Order 1792.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736739	C₂	1	2	yes	maximal, complemented
2	C₄×#6467	C₂	1	1	yes	maximal, characteristic
3	C₇×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727608	C₂	1	2	yes	maximal, complemented
5	#11727485	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725842	C₂	1	4	yes	maximal, complemented
7	C₈×Dic₂₈	C₂	1	2	yes	maximal, complemented
8	#11725806	C₂	1	2	yes	maximal, complemented
9	C₂×#6467	C₄	1	1	yes	characteristic
10	C₄×#1157	C₂²	1	2	yes	—
11	C₇×#431	C₂²	1	1	yes	characteristic
12	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	#89885	C₄	1	2	yes	—
15	#89878	C₄	1	1	yes	characteristic
16	#89696	C₂²	1	2	yes	—
17	#89609	—	2	2	no	—
18	#89605	—	2	2	no	—
19	#89501	—	2	4	no	—
20	C₄×Dic₂₈	C₄	1	1	yes	characteristic
21	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
22	C₄×D₅₆	C₄	1	1	yes	characteristic
23	C₄×D₅₆	C₂²	1	1	yes	characteristic
24	C₄×#1132	C₄	1	2	yes	—
25	C₄×#1132	C₂²	1	2	yes	—
26	#89460	C₂²	1	4	yes	—
27	C₈×Dic₁₄	C₂²	1	4	yes	—
28	C₇×#322	C₂²	1	2	yes	—
29	C₇×C₈²	C₂²	1	2	yes	—
30	#89254	C₂²	1	4	yes	—
31	#89253	C₂²	1	2	yes	—
32	#89253	C₂²	1	2	yes	—
33	#89251	C₂²	1	4	yes	—
34	#18059	—	7	1	no	maximal, Sylow
35	C₂×#1157	C₂×C₄	1	2	yes	—
36	C₂×C₇×M₁₆	—	2	1	no	—
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₄×C₂²⋊D₇	—	2	2	no	—
41	C₄×C₂²⋊D₇	—	2	2	no	—
42	#6489	C₂×C₄	1	2	yes	—
43	C₂×#1136	—	2	2	no	—
44	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
45	#6467	—	2	4	no	—
46	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#1132	C₂×C₄	1	2	yes	—
48	C₂×#1131	—	2	4	no	—
49	C₄×D₂₈	C₂×C₄	1	2	yes	—
50	C₄×D₂₈	C₂³	1	2	yes	—
51	#6435	—	2	2	no	—
52	C₄²×D₇	—	2	2	no	—
53	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₄	C₂³	1	2	yes	—
55	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
56	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
57	C₇×C₂²⋊C₈	D₄	1	2	yes	—
58	C₇×#97	—	2	1	no	—
59	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂³	1	2	yes	—
65	#6395	C₂×C₄	1	2	yes	—
66	#6395	C₂×C₄	1	2	yes	—
67	(C₂×D₇)⋊C₈	—	2	4	no	—
68	#6392	C₂×C₄	1	1	yes	characteristic
69	#6392	C₂×C₄	1	1	yes	characteristic
70	#6391	C₂×C₄	1	2	yes	—
71	#6390	C₂×C₄	1	2	yes	—
72	#6390	C₂×C₄	1	2	yes	—
73	Dic₇⋊C₈	—	2	4	no	—
74	C₈×Dic₇	—	2	4	no	—
75	C₄⋊(C₇⋊C₈)	—	2	2	no	—
76	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
77	#2991	—	7	2	no	—
78	C₄×#135	—	7	1	no	—
79	#1589	—	7	2	no	—
80	#1588	—	7	4	no	—
81	C₈×Q₁₆	—	7	2	no	—
82	#1567	—	7	1	no	—
83	#1489	—	7	2	no	—
84	C₈×M₁₆	D₇	1	1	yes	characteristic, complemented
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	#41	1	1	yes	characteristic
88	C₂×C₂²⋊D₇	—	2	2	no	—
89	#1157	—	2	4	no	—
90	#1157	—	4	2	no	—
91	C₂×D₂₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₇	—	2	2	no	—
93	C₂×C₄×D₇	—	2	2	no	—
94	C₂×C₄×D₇	—	2	2	no	—
95	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
96	C₇×M₁₆	—	2	4	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	4	1	no	—
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
106	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	#41	1	1	yes	characteristic
109	C₄²×C₇	C₂⁴	1	1	yes	characteristic
110	C₂²⋊Dic₇	—	2	2	no	—
111	(C₂×D₇)⋊C₄	—	2	2	no	—
112	(C₂×D₇)⋊C₄	—	2	2	no	—
113	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
114	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
115	Dic₇⋊C₄	—	2	2	no	—
116	Dic₇⋊C₄	—	2	2	no	—
117	C₄×Dic₇	—	2	2	no	—
118	C₄×Dic₇	—	2	2	no	—
119	C₄×Dic₇	—	2	2	no	—
120	#1136	—	8	2	no	—
121	C₂×C₇⋊C₈	—	2	4	no	—
122	Dic₂₈	M₁₆	1	4	yes	complemented
123	D₅₆	—	2	2	no	—
124	#1132	—	2	4	no	—
125	#1131	—	8	4	no	—
126	C₂×#135	—	7	1	no	—
127	C₄×#41	—	7	2	no	—
128	C₈×Q₈	—	7	4	no	—
129	C₄×Q₁₆	—	7	1	no	—
130	C₄×Q₁₆	—	7	1	no	—
131	C₄×SD₁₆	—	7	2	no	—
132	C₄×SD₁₆	—	7	2	no	—
133	C₄×D₈	—	7	1	no	—
134	C₄×D₈	—	7	1	no	—
135	#436	—	7	4	no	—
136	#436	—	14	4	no	—
137	#431	—	14	2	no	—
138	#431	C₂×D₇	1	1	yes	characteristic
139	#427	—	7	1	no	—
140	#423	—	7	2	no	—
141	#423	—	14	2	no	—
142	#416	—	7	2	no	—
143	C₄×M₁₆	C₂×D₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
145	#335	—	7	2	no	—
146	#335	—	7	2	no	—
147	Q₈⋊C₈	—	7	4	no	—
148	D₄⋊C₈	—	7	4	no	—
149	#322	C₂×D₇	1	2	yes	—
150	C₈²	C₂×D₇	1	2	yes	—
151	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
152	C₂²×D₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	1	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	1	no	—
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
164	C₂²⋊D₇	—	4	4	no	—
165	C₂×Dic₇	—	2	2	no	—
166	C₂×Dic₇	—	2	2	no	—
167	C₂×Dic₇	—	2	2	no	—
168	D₂₈	—	2	2	no	—
169	D₂₈	—	2	2	no	—
170	C₄×D₇	—	4	4	no	—
171	C₄×D₇	—	4	2	no	—
172	C₄×D₇	—	4	2	no	—
173	C₄×D₇	—	4	4	no	—
174	Dic₁₄	—	2	2	no	—
175	Dic₁₄	C₂×M₁₆	1	4	yes	—
176	C₇×C₈	—	2	4	no	—
177	C₇×C₈	—	2	1	no	—
178	C₇×C₈	—	2	1	no	—
179	C₇×C₈	—	2	1	no	—
180	C₇×C₈	—	2	2	no	—
181	C₇×C₈	—	2	2	no	—
182	C₇×C₈	C₂×M₁₆	1	2	yes	—
183	C₇⋊C₈	—	4	4	no	—
184	C₂×#41	—	7	2	no	—
185	#135	—	14	4	no	—
186	C₂×Q₁₆	—	7	1	no	—
187	C₂×SD₁₆	—	7	2	no	—
188	C₂×D₈	—	7	1	no	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	—	14	2	no	—
191	C₂×M₁₆	—	14	4	no	—
192	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
193	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
194	C₄×Q₈	—	7	2	no	—
195	C₄×Q₈	—	7	2	no	—
196	C₄×D₄	—	7	2	no	—
197	C₄×D₄	—	7	2	no	—
198	C₄×D₄	—	14	2	no	—
199	C₄×D₄	—	14	2	no	—
200	#117	—	7	2	no	—
201	#117	—	14	2	no	—
202	C₂×C₄²	—	14	2	no	—
203	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
204	#107	—	7	1	no	—
205	#107	—	7	1	no	—
206	#106	—	7	2	no	—
207	C₄⋊C₈	—	7	4	no	—
208	C₄⋊C₈	—	14	4	no	—
209	C₄⋊C₈	—	14	2	no	—
210	C₄⋊C₈	D₂₈	1	1	yes	characteristic
211	C₄⋊C₈	D₂₈	1	1	yes	characteristic
212	#103	—	7	2	no	—
213	#103	—	7	2	no	—
214	#102	—	7	2	no	—
215	#102	—	7	2	no	—
216	C₂²⋊C₈	—	14	4	no	—
217	C₂²⋊C₈	D₂₈	1	2	yes	—
218	#97	—	2	1	no	—
219	C₄×C₈	—	14	4	no	—
220	C₄×C₈	C₄×D₇	1	1	yes	characteristic
221	C₄×C₈	C₄×D₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₇	1	2	yes	—
223	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₇	1	2	yes	—
226	C₂²×C₇	—	2	1	no	—
227	C₂²×C₇	—	2	1	no	—
228	C₂²×C₇	—	2	1	no	—
229	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂×D₇	—	4	2	no	—
231	C₂×D₇	—	4	2	no	—
232	C₂×D₇	—	4	2	no	—
233	C₄×C₇	—	2	1	no	—
234	C₄×C₇	—	2	1	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	#443	1	2	yes	—
237	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
238	C₄×C₇	#568	1	1	yes	characteristic
239	C₄×C₇	#576	1	1	yes	characteristic
240	C₄×C₇	#578	1	1	yes	characteristic
241	Dic₇	—	2	4	no	—
242	Dic₇	—	4	2	no	—
243	Dic₇	—	4	2	no	—
244	#41	—	14	4	no	—
245	#41	—	28	2	no	—
246	C₂×Q₈	—	7	2	no	—
247	C₂×D₄	—	7	2	no	—
248	C₂×D₄	—	14	2	no	—
249	C₂²×C₄	—	14	2	no	—
250	C₂²×C₄	—	14	2	no	—
251	C₂²×C₄	—	14	2	no	—
252	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
254	C₂²×C₄	#1157	1	1	yes	characteristic
255	Q₁₆	—	7	4	no	—
256	SD₁₆	—	14	4	no	—
257	D₈	—	14	2	no	—
258	M₁₆	—	2	4	no	—
259	M₁₆	—	56	4	no	—
260	M₁₆	—	56	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	14	4	no	—
265	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
269	C₂×C₈	C₂×D₂₈	1	2	yes	—
270	C₂×C₈	C₂×D₂₈	1	2	yes	—
271	C₄⋊C₄	—	7	2	no	—
272	C₄⋊C₄	—	7	2	no	—
273	C₄⋊C₄	—	14	2	no	—
274	C₄⋊C₄	—	14	2	no	—
275	C₂²⋊C₄	—	14	2	no	—
276	C₂²⋊C₄	—	14	2	no	—
277	C₂²⋊C₄	—	14	2	no	—
278	C₄²	—	14	2	no	—
279	C₄²	—	14	2	no	—
280	C₄²	—	14	2	no	—
281	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
282	C₄²	C₂×D₂₈	1	1	yes	characteristic
283	C₄²	#1157	1	1	yes	characteristic
284	C₄²	C₂³×D₇	1	1	yes	characteristic
285	C₂×C₇	—	4	1	no	—
286	C₂×C₇	D₄×M₁₆	1	1	yes	characteristic
287	C₂×C₇	#2951	1	1	yes	characteristic
288	C₂×C₇	C₂×#443	1	1	yes	characteristic
289	D₇	—	8	2	no	—
290	C₂³	—	14	2	no	—
291	C₂³	C₄×D₂₈	1	1	yes	characteristic
292	Q₈	—	7	4	no	—
293	Q₈	—	14	2	no	—
294	D₄	—	14	2	no	—
295	D₄	—	14	2	no	—
296	D₄	—	28	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	14	2	no	—
303	C₂×C₄	—	14	2	no	—
304	C₂×C₄	—	14	2	no	—
305	C₂×C₄	—	28	4	no	—
306	C₂×C₄	—	28	2	no	—
307	C₂×C₄	—	28	2	no	—
308	C₂×C₄	—	28	4	no	—
309	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
310	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
311	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
314	C₂×C₄	C₂×#1157	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	28	4	no	—
322	C₈	D₇×M₁₆	1	2	yes	—
323	C₇	#18059	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	28	2	no	—
328	C₂²	—	28	2	no	—
329	C₂²	—	28	2	no	—
330	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	14	4	no	—
335	C₄	—	28	2	no	—
336	C₄	—	28	2	no	—
337	C₄	#89472	1	2	yes	—
338	C₄	C₂×D₇×M₁₆	1	1	yes	characteristic
339	C₄	#90436	1	1	yes	characteristic
340	C₄	#90443	1	1	yes	characteristic
341	C₄	#90444	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	56	2	no	—
344	C₂	C₂×#89472	1	1	yes	characteristic
345	C₂	M₁₆×D₂₈	1	1	yes	characteristic
346	C₂	#11736007	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×C₈×M₁₆	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×#6467	C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₈×M₁₆	D₇
hypocenter	C₇	#18059
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #11736739	C₂	semidirect
3. C₇×C₈×M₁₆	C₂	semidirect
4. #11727608	C₂	semidirect
5. #11727485	C₂	semidirect
6. #11725842	C₂	semidirect
7. C₈×Dic₂₈	C₂	semidirect
8. #11725806	C₂	semidirect
84. C₈×M₁₆	D₇	semidirect
122. Dic₂₈	M₁₆	semidirect
323. C₇	#18059	semidirect

Maximal subgroups

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

subgroup	index
#11736739	2
C₄×#6467	2
C₇×C₈×M₁₆	2
#11727608	2
#11727485	2
#11725842	2
C₈×Dic₂₈	2
#11725806	2
#18059	7