Subgroups — #13507321

Order 1792.

558 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	C₈×#1161	C₂	1	1	yes	maximal, characteristic
2	C₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
3	#11728703	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727033	C₂	1	1	yes	maximal, characteristic, complemented
5	#11727032	C₂	1	2	yes	maximal, complemented
6	#11727028	C₂	1	2	yes	maximal, complemented
7	#11727028	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726829	C₂	1	2	yes	maximal, complemented
9	#11726825	C₂	1	2	yes	maximal, complemented
10	#11726809	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#1161	C₄	1	1	yes	characteristic
13	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
14	Q₈×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₂×#6372	C₂²	1	2	yes	—
16	C₂×#6372	C₂²	1	2	yes	—
17	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
19	#89606	C₄	1	2	yes	—
20	#89606	C₄	1	1	yes	characteristic
21	#89605	C₄	1	2	yes	—
22	#89605	C₄	1	1	yes	characteristic
23	C₈×D₂₈	C₂²	1	2	yes	—
24	C₈×D₂₈	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₇	C₂²	1	2	yes	—
27	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₇×#346	C₂²	1	2	yes	—
30	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
31	(C₂×D₇)⋊C₁₆	C₂²	1	2	yes	—
32	(C₂×D₇)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₇)⋊C₁₆	C₂²	1	2	yes	—
34	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇	C₂²	1	2	yes	—
36	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
37	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
39	#89256	C₂²	1	2	yes	—
40	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
41	#33479	—	7	1	no	maximal, Sylow
42	C₂×#1161	C₈	1	1	yes	characteristic
43	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
44	Q₈×Dic₇	C₈	1	1	yes	characteristic
45	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₇	C₂³	1	2	yes	—
49	#6456	C₈	1	2	yes	—
50	#6456	C₈	1	1	yes	characteristic
51	#6455	C₈	1	2	yes	—
52	#6455	C₈	1	1	yes	characteristic
53	C₄×D₂₈	C₂×C₄	1	2	yes	—
54	C₄×D₂₈	C₂×C₄	1	1	yes	characteristic
55	C₄²×D₇	C₂×C₄	1	2	yes	—
56	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₁₆	C₂³	1	2	yes	—
59	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
60	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₇×C₈	C₂³	1	2	yes	—
64	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
66	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
67	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₇	C₂×C₄	1	2	yes	—
69	C₈×Dic₇	C₂³	1	1	yes	characteristic
70	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
72	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
74	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
75	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
77	#6372	—	2	4	no	—
78	#6372	—	2	4	no	—
79	D₇×C₁₆	—	2	2	no	—
80	D₇×C₁₆	—	2	2	no	—
81	C₈×#41	—	7	1	no	—
82	#2183	—	7	1	no	—
83	#2183	D₇	1	1	yes	characteristic, complemented
84	#2169	—	7	1	no	—
85	#2169	—	7	1	no	—
86	#2168	—	7	2	no	—
87	#2168	—	7	2	no	—
88	#2162	—	7	2	no	—
89	#2162	—	7	1	no	—
90	C₄×M₃₂	—	7	2	no	—
91	C₂×C₄×C₁₆	—	7	1	no	—
92	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
93	#1161	—	2	4	no	—
94	C₂×D₂₈	C₂×C₈	1	2	yes	—
95	C₂×D₂₈	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₇	C₂×C₈	1	2	yes	—
97	C₂×C₄×D₇	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
99	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
105	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
106	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
109	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
110	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
111	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
112	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
113	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₇	C₂×C₈	1	2	yes	—
115	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
116	C₄×Dic₇	C₂×C₈	1	2	yes	—
117	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
118	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
121	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₇	—	2	1	no	—
123	C₈×D₇	—	2	1	no	—
124	C₈×D₇	—	2	2	no	—
125	C₈×D₇	—	2	2	no	—
126	C₈×D₇	—	2	2	no	—
127	C₈×D₇	—	2	4	no	—
128	C₇×C₁₆	—	2	1	no	—
129	C₇×C₁₆	—	2	2	no	—
130	C₇×C₁₆	#41	1	2	yes	complemented
131	C₇⋊C₁₆	—	2	1	no	—
132	C₇⋊C₁₆	—	2	2	no	—
133	C₇⋊C₁₆	#41	1	2	yes	complemented
134	C₄×#41	—	7	1	no	—
135	C₂×M₃₂	—	7	2	no	—
136	C₂×M₃₂	—	7	2	no	—
137	C₂²×C₁₆	—	7	1	no	—
138	C₂²×C₁₆	—	7	1	no	—
139	C₈×Q₈	—	7	1	no	—
140	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
141	C₈×D₄	—	7	1	no	—
142	C₈×D₄	—	7	2	no	—
143	C₈×D₄	—	7	2	no	—
144	C₈×D₄	—	7	1	no	—
145	#431	—	7	2	no	—
146	#431	—	7	1	no	—
147	C₂×C₄×C₈	—	7	1	no	—
148	C₂×C₄×C₈	—	7	2	no	—
149	C₄⋊C₁₆	—	7	2	no	—
150	C₄⋊C₁₆	—	7	1	no	—
151	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
152	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
153	C₂²⋊C₁₆	—	7	2	no	—
154	C₂²⋊C₁₆	—	7	2	no	—
155	C₂²⋊C₁₆	—	7	2	no	—
156	#346	—	7	2	no	—
157	#346	C₂×D₇	1	2	yes	—
158	C₄×C₁₆	—	7	1	no	—
159	C₄×C₁₆	—	7	1	no	—
160	C₄×C₁₆	—	7	2	no	—
161	C₄×C₁₆	—	7	1	no	—
162	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
163	C₂²×D₇	C₂²×C₈	1	2	yes	—
164	C₂²×D₇	C₂²×C₈	1	1	yes	characteristic
165	C₇×Q₈	—	2	2	no	—
166	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
167	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
169	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×Dic₇	C₂²×C₈	1	2	yes	—
172	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
173	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
174	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
175	D₂₈	—	2	4	no	—
176	D₂₈	—	2	2	no	—
177	C₄×D₇	—	2	1	no	—
178	C₄×D₇	—	2	2	no	—
179	C₄×D₇	—	2	1	no	—
180	C₄×D₇	—	2	2	no	—
181	C₄×D₇	—	2	4	no	—
182	C₄×D₇	—	2	2	no	—
183	C₄×D₇	—	2	4	no	—
184	C₄×D₇	M₃₂	1	4	yes	complemented
185	C₇×C₈	—	2	1	no	—
186	C₇×C₈	—	2	2	no	—
187	C₇×C₈	C₂×#41	1	1	yes	characteristic
188	C₇×C₈	C₂×#41	1	1	yes	characteristic
189	C₇⋊C₈	—	2	1	no	—
190	C₇⋊C₈	—	2	2	no	—
191	C₇⋊C₈	#131	1	2	yes	—
192	C₂×#41	—	7	1	no	—
193	C₂²×C₈	—	7	1	no	—
194	C₂²×C₈	—	7	2	no	—
195	C₂²×C₈	—	7	1	no	—
196	C₂²×C₈	—	7	2	no	—
197	C₄×Q₈	—	7	1	no	—
198	C₄×Q₈	C₄×D₇	1	1	yes	characteristic
199	C₄×D₄	—	7	2	no	—
200	C₄×D₄	—	7	1	no	—
201	C₄×D₄	—	7	2	no	—
202	C₄×D₄	—	7	1	no	—
203	#117	—	7	1	no	—
204	#117	—	7	2	no	—
205	C₂×C₄²	—	7	1	no	—
206	C₂×C₄²	—	7	2	no	—
207	M₃₂	—	14	4	no	—
208	M₃₂	—	14	4	no	—
209	C₂×C₁₆	—	7	1	no	—
210	C₂×C₁₆	—	7	1	no	—
211	C₂×C₁₆	—	7	2	no	—
212	C₂×C₁₆	—	14	2	no	—
213	C₂×C₁₆	—	14	2	no	—
214	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×D₇	1	2	yes	—
216	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
217	C₄⋊C₈	—	7	2	no	—
218	C₄⋊C₈	—	7	1	no	—
219	C₄⋊C₈	C₄×D₇	1	2	yes	—
220	C₄⋊C₈	C₄×D₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	7	2	no	—
222	C₂²⋊C₈	—	7	2	no	—
223	C₂²⋊C₈	—	7	2	no	—
224	C₄×C₈	—	7	1	no	—
225	C₄×C₈	—	7	2	no	—
226	C₄×C₈	—	7	1	no	—
227	C₄×C₈	—	7	1	no	—
228	C₄×C₈	—	7	2	no	—
229	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₇	1	2	yes	—
231	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
232	C₂×D₇	—	2	1	no	—
233	C₂×D₇	—	2	1	no	—
234	C₂×D₇	—	2	2	no	—
235	C₂×D₇	—	2	2	no	—
236	C₂×D₇	—	2	2	no	—
237	C₂×D₇	C₂×M₃₂	1	2	yes	—
238	C₄×C₇	—	2	2	no	—
239	C₄×C₇	—	2	1	no	—
240	C₄×C₇	—	2	2	no	—
241	C₄×C₇	C₂×M₃₂	1	2	yes	—
242	C₄×C₇	C₄×#41	1	1	yes	characteristic
243	C₄×C₇	C₂×#131	1	1	yes	characteristic
244	Dic₇	—	2	1	no	—
245	Dic₇	—	2	2	no	—
246	Dic₇	C₂×M₃₂	1	2	yes	—
247	Dic₇	#504	1	2	yes	—
248	#41	—	14	4	no	—
249	C₂×Q₈	C₈×D₇	1	1	yes	characteristic
250	C₂×D₄	—	7	2	no	—
251	C₂×D₄	—	7	1	no	—
252	C₂²×C₄	—	7	1	no	—
253	C₂²×C₄	—	7	2	no	—
254	C₂²×C₄	—	7	1	no	—
255	C₂²×C₄	—	7	2	no	—
256	C₂²×C₄	—	7	1	no	—
257	C₂²×C₄	—	7	2	no	—
258	C₂×C₈	—	7	1	no	—
259	C₂×C₈	—	7	1	no	—
260	C₂×C₈	—	7	2	no	—
261	C₂×C₈	—	14	2	no	—
262	C₂×C₈	—	14	4	no	—
263	C₂×C₈	—	14	1	no	—
264	C₂×C₈	—	14	2	no	—
265	C₂×C₈	—	14	1	no	—
266	C₂×C₈	—	14	2	no	—
267	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
269	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
270	C₄⋊C₄	—	7	2	no	—
271	C₄⋊C₄	—	7	1	no	—
272	C₄⋊C₄	C₈×D₇	1	2	yes	—
273	C₄⋊C₄	C₈×D₇	1	1	yes	characteristic
274	C₂²⋊C₄	—	7	2	no	—
275	C₂²⋊C₄	—	7	2	no	—
276	C₂²⋊C₄	—	7	2	no	—
277	C₄²	—	7	1	no	—
278	C₄²	—	7	2	no	—
279	C₄²	—	7	2	no	—
280	C₄²	—	7	1	no	—
281	C₄²	—	7	1	no	—
282	C₄²	C₂×C₄×D₇	1	2	yes	—
283	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	7	2	no	—
287	C₁₆	—	14	1	no	—
288	C₁₆	—	14	2	no	—
289	C₁₆	#1161	1	2	yes	complemented
290	C₂×C₇	C₈×#41	1	1	yes	characteristic
291	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
292	C₂×C₇	C₂×#504	1	1	yes	characteristic
293	D₇	—	2	2	no	—
294	D₇	—	4	2	no	—
295	C₂³	—	7	1	no	—
296	C₂³	—	7	2	no	—
297	Q₈	—	2	2	no	—
298	D₄	—	14	4	no	—
299	D₄	—	14	2	no	—
300	C₂×C₄	—	7	1	no	—
301	C₂×C₄	—	7	1	no	—
302	C₂×C₄	—	7	4	no	—
303	C₂×C₄	—	7	2	no	—
304	C₂×C₄	—	7	1	no	—
305	C₂×C₄	—	14	2	no	—
306	C₂×C₄	—	14	4	no	—
307	C₂×C₄	—	14	4	no	—
308	C₂×C₄	—	14	1	no	—
309	C₂×C₄	—	14	2	no	—
310	C₂×C₄	—	14	1	no	—
311	C₂×C₄	—	14	2	no	—
312	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
315	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	2	no	—
319	C₈	—	7	2	no	—
320	C₈	—	14	1	no	—
321	C₈	—	14	2	no	—
322	C₈	C₂×#1161	1	1	yes	characteristic
323	C₈	D₇×#41	1	1	yes	characteristic
324	C₇	#33479	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	7	2	no	—
326	C₂²	—	14	1	no	—
327	C₂²	—	14	1	no	—
328	C₂²	—	14	2	no	—
329	C₂²	—	14	2	no	—
330	C₂²	—	14	2	no	—
331	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	7	2	no	—
336	C₄	—	7	2	no	—
337	C₄	—	14	1	no	—
338	C₄	—	14	2	no	—
339	C₄	D₇×M₃₂	1	2	yes	—
340	C₄	C₄×#1161	1	1	yes	characteristic
341	C₄	D₇×#131	1	1	yes	characteristic
342	C₂	—	14	2	no	—
343	C₂	—	28	2	no	—
344	C₂	C₈×#1161	1	1	yes	characteristic
345	C₂	C₂×D₇×M₃₂	1	1	yes	characteristic
346	C₂	D₇×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×#2183	C₂
prime radical	C₂×#1161	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2183	D₇
hypocenter	C₇	#33479
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. C₇×#2183	C₂	semidirect
3. #11728703	C₂	semidirect
4. #11727033	C₂	semidirect
5. #11727032	C₂	semidirect
6. #11727028	C₂	semidirect
7. #11727028	C₂	semidirect
8. #11726829	C₂	semidirect
9. #11726825	C₂	semidirect
10. #11726809	C₂	semidirect
11. C₄×D₇×C₁₆	C₂	semidirect
83. #2183	D₇	semidirect
130. C₇×C₁₆	#41	semidirect
133. C₇⋊C₁₆	#41	semidirect
184. C₄×D₇	M₃₂	semidirect
289. C₁₆	#1161	semidirect
324. C₇	#33479	semidirect

Maximal subgroups

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

subgroup	index
C₈×#1161	2
C₇×#2183	2
#11728703	2
#11727033	2
#11727032	2
#11727028	2
#11726829	2
#11726825	2
#11726809	2
C₄×D₇×C₁₆	2
#33479	7