Subgroups — #13507325

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11735893	C₂	1	1	yes	maximal, characteristic
2	C₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727035	C₂	1	1	yes	maximal, characteristic, complemented
5	#11727034	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727033	C₂	1	1	yes	maximal, characteristic, complemented
7	#11727031	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726856	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₇)⋊C₁₆	C₂	1	2	yes	maximal, complemented
10	#11726828	C₂	1	2	yes	maximal, complemented
11	#11726826	C₂	1	2	yes	maximal, complemented
12	#90170	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	#89883	C₄	1	2	yes	—
15	C₈×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	#89698	C₄	1	1	yes	characteristic
19	C₂×#6372	C₂²	1	2	yes	—
20	C₂×D₇×C₁₆	C₂²	1	2	yes	—
21	#89462	C₄	1	2	yes	—
22	#89461	C₂²	1	2	yes	—
23	#89460	C₄	1	1	yes	characteristic
24	C₈×D₂₈	C₂²	1	1	yes	characteristic
25	#89451	C₄	1	1	yes	characteristic
26	C₈×Dic₁₄	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
30	C₇×#346	C₂²	1	2	yes	—
31	(C₂×D₇)⋊C₁₆	C₂²	1	2	yes	—
32	(C₂×D₇)⋊C₁₆	C₂²	1	2	yes	—
33	#89297	C₂²	1	2	yes	—
34	Dic₇⋊C₁₆	C₂²	1	2	yes	—
35	Dic₇⋊C₁₆	C₂²	1	2	yes	—
36	C₁₆×Dic₇	C₂²	1	2	yes	—
37	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
41	#33480	—	7	1	no	maximal, Sylow
42	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
45	#6493	C₈	1	1	yes	characteristic
46	#6492	C₈	1	2	yes	—
47	C₄×C₂²⋊D₇	C₂×C₄	1	2	yes	—
48	#6487	C₈	1	1	yes	characteristic
49	C₂×#1131	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₇	C₂³	1	2	yes	—
51	#6439	C₈	1	2	yes	—
52	#6438	C₈	1	1	yes	characteristic
53	C₄×D₂₈	C₂×C₄	1	1	yes	characteristic
54	#6435	C₂×C₄	1	2	yes	—
55	#6433	C₈	1	1	yes	characteristic
56	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₇×C₁₆	C₂³	1	2	yes	—
58	C₂×C₇×C₁₆	C₂³	1	2	yes	—
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	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₇)⋊C₈	C₂×C₄	1	2	yes	—
66	#6389	C₂×C₄	1	2	yes	—
67	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
68	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
69	C₈×Dic₇	C₂³	1	2	yes	—
70	C₂×C₇⋊C₁₆	—	2	2	no	—
71	C₂×C₇⋊C₁₆	—	2	2	no	—
72	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
76	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
77	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
78	#6377	C₂×C₄	1	2	yes	—
79	#6372	—	2	4	no	—
80	D₇×C₁₆	—	2	4	no	—
81	#2966	—	7	1	no	—
82	#2183	—	7	1	no	—
83	Q₈×C₁₆	—	7	1	no	—
84	#2169	—	7	1	no	—
85	#2168	—	7	2	no	—
86	D₄×C₁₆	—	7	1	no	—
87	D₄×C₁₆	—	7	2	no	—
88	#2164	—	7	2	no	—
89	#2163	D₇	1	1	yes	characteristic, complemented
90	#2108	—	7	2	no	—
91	C₂×C₄×C₁₆	—	7	1	no	—
92	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₇	C₂×C₈	1	2	yes	—
96	C₂×D₂₈	C₂×C₈	1	1	yes	characteristic
97	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₄	C₂×C₈	1	1	yes	characteristic
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	1	no	—
101	C₂×C₇×C₈	—	2	1	no	—
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
106	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
108	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²⋊Dic₇	C₂×C₈	1	2	yes	—
111	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
112	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
113	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
114	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
115	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
116	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
117	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
118	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
119	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
120	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
121	C₄×Dic₇	C₂²×C₄	1	2	yes	—
122	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
123	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
124	#1131	—	2	4	no	—
125	C₈×D₇	—	2	2	no	—
126	C₈×D₇	—	2	2	no	—
127	C₇×C₁₆	—	2	2	no	—
128	C₇×C₁₆	—	2	2	no	—
129	C₇⋊C₁₆	—	2	1	no	—
130	C₇⋊C₁₆	—	2	1	no	—
131	C₇⋊C₁₆	#41	1	2	yes	—
132	C₇⋊C₁₆	#41	1	2	yes	—
133	#529	—	7	1	no	—
134	C₂×M₃₂	—	7	2	no	—
135	C₂²×C₁₆	—	7	1	no	—
136	C₂²×C₁₆	—	7	1	no	—
137	C₂²×C₁₆	—	7	2	no	—
138	#446	—	7	1	no	—
139	C₈×Q₈	—	7	1	no	—
140	#436	—	7	1	no	—
141	#435	—	7	2	no	—
142	C₈×D₄	—	7	2	no	—
143	C₈×D₄	—	7	1	no	—
144	#433	—	7	2	no	—
145	#432	—	7	1	no	—
146	#405	—	7	2	no	—
147	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
148	C₄⋊C₁₆	—	7	2	no	—
149	C₄⋊C₁₆	—	7	2	no	—
150	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
151	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
152	C₂²⋊C₁₆	—	7	2	no	—
153	C₂²⋊C₁₆	—	7	2	no	—
154	C₂²⋊C₁₆	C₂×D₇	1	2	yes	—
155	#346	—	7	2	no	—
156	#346	C₂×D₇	1	2	yes	—
157	C₄×C₁₆	—	7	2	no	—
158	C₄×C₁₆	—	7	1	no	—
159	C₄×C₁₆	—	7	1	no	—
160	C₄×C₁₆	—	7	1	no	—
161	C₄×C₁₆	—	7	1	no	—
162	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
163	C₂²×D₇	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₇	—	2	2	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	—	2	1	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
174	C₂²⋊D₇	—	2	4	no	—
175	C₂×Dic₇	C₂²×C₈	1	2	yes	—
176	C₂×Dic₇	C₂²×C₈	1	2	yes	—
177	C₂×Dic₇	C₂²×C₈	1	2	yes	—
178	D₂₈	—	2	2	no	—
179	C₄×D₇	—	2	2	no	—
180	C₄×D₇	—	2	2	no	—
181	Dic₁₄	—	2	2	no	—
182	C₇×C₈	—	2	1	no	—
183	C₇×C₈	—	2	1	no	—
184	C₇×C₈	#131	1	2	yes	—
185	C₇×C₈	C₂×#41	1	1	yes	characteristic
186	C₇×C₈	C₂×#41	1	1	yes	characteristic
187	C₇⋊C₈	—	2	2	no	—
188	C₇⋊C₈	—	2	2	no	—
189	C₂×M₁₆	—	7	2	no	—
190	C₂²×C₈	—	7	2	no	—
191	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
193	#126	—	7	2	no	—
194	#125	—	7	1	no	—
195	#124	—	7	1	no	—
196	#123	—	7	2	no	—
197	C₂²⋊Q₈	—	7	1	no	—
198	#121	—	7	1	no	—
199	C₄×Q₈	—	7	1	no	—
200	C₄×D₄	—	7	2	no	—
201	C₄×D₄	—	7	1	no	—
202	#117	—	7	2	no	—
203	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
204	M₃₂	—	14	4	no	—
205	C₂×C₁₆	—	7	1	no	—
206	C₂×C₁₆	—	7	1	no	—
207	C₂×C₁₆	—	7	1	no	—
208	C₂×C₁₆	—	7	1	no	—
209	C₂×C₁₆	—	14	4	no	—
210	C₂×C₁₆	—	14	2	no	—
211	C₂×C₁₆	—	14	2	no	—
212	C₂×C₁₆	C₂²×D₇	1	2	yes	—
213	C₂×C₁₆	C₂²×D₇	1	2	yes	—
214	C₄⋊C₈	—	7	2	no	—
215	C₄⋊C₈	—	7	2	no	—
216	C₄⋊C₈	—	7	1	no	—
217	C₄⋊C₈	—	7	1	no	—
218	C₂²⋊C₈	—	7	2	no	—
219	C₂²⋊C₈	—	7	2	no	—
220	C₂²⋊C₈	—	7	2	no	—
221	#97	—	7	2	no	—
222	#97	—	7	2	no	—
223	C₄×C₈	—	7	2	no	—
224	C₄×C₈	C₄×D₇	1	1	yes	characteristic
225	C₄×C₈	C₄×D₇	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
228	C₂²×C₇	—	2	1	no	—
229	C₂²×C₇	—	2	1	no	—
230	C₂²×C₇	#504	1	2	yes	—
231	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
232	C₂×D₇	—	2	2	no	—
233	C₂×D₇	—	2	2	no	—
234	C₂×D₇	—	2	2	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	—	2	1	no	—
239	C₄×C₇	#504	1	2	yes	—
240	C₄×C₇	C₄×#41	1	1	yes	characteristic
241	C₄×C₇	C₂×#131	1	1	yes	characteristic
242	Dic₇	—	2	2	no	—
243	Dic₇	—	2	2	no	—
244	Dic₇	—	2	2	no	—
245	C₂×Q₈	—	7	1	no	—
246	C₂×D₄	—	7	2	no	—
247	C₂×D₄	—	7	1	no	—
248	C₂²×C₄	—	7	2	no	—
249	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
250	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
252	M₁₆	—	14	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	7	2	no	—
257	C₂×C₈	—	7	2	no	—
258	C₂×C₈	—	14	2	no	—
259	C₂×C₈	—	14	2	no	—
260	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
264	C₄⋊C₄	—	7	2	no	—
265	C₄⋊C₄	—	7	2	no	—
266	C₄⋊C₄	—	7	2	no	—
267	C₄⋊C₄	—	7	2	no	—
268	C₄⋊C₄	—	7	1	no	—
269	C₄⋊C₄	—	7	1	no	—
270	C₂²⋊C₄	—	7	2	no	—
271	C₂²⋊C₄	—	7	2	no	—
272	C₂²⋊C₄	—	7	2	no	—
273	C₂²⋊C₄	—	7	2	no	—
274	C₂²⋊C₄	—	7	2	no	—
275	C₄²	—	7	2	no	—
276	C₄²	C₈×D₇	1	1	yes	characteristic
277	C₄²	C₈×D₇	1	1	yes	characteristic
278	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
279	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
280	C₁₆	—	2	2	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	7	2	no	—
283	C₁₆	—	7	2	no	—
284	C₁₆	—	14	1	no	—
285	C₁₆	—	14	1	no	—
286	C₂×C₇	—	2	2	no	—
287	C₂×C₇	C₈×#41	1	1	yes	characteristic
288	C₂×C₇	C₂×#504	1	1	yes	characteristic
289	C₂×C₇	C₂×#504	1	1	yes	characteristic
290	D₇	—	4	2	no	—
291	C₂³	—	7	2	no	—
292	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
293	Q₈	—	14	2	no	—
294	D₄	—	14	4	no	—
295	D₄	—	14	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	7	2	no	—
301	C₂×C₄	—	7	2	no	—
302	C₂×C₄	—	7	2	no	—
303	C₂×C₄	—	14	2	no	—
304	C₂×C₄	—	14	2	no	—
305	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	14	2	no	—
314	C₈	—	14	2	no	—
315	C₈	#6470	1	2	yes	—
316	C₈	D₇×#41	1	1	yes	characteristic
317	C₈	D₇×#41	1	1	yes	characteristic
318	C₇	#33480	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	14	2	no	—
322	C₂²	—	14	2	no	—
323	C₂²	—	14	2	no	—
324	C₂²	#89679	1	2	yes	—
325	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	14	2	no	—
331	C₄	—	14	2	no	—
332	C₄	—	14	2	no	—
333	C₄	#89679	1	2	yes	—
334	C₄	#90213	1	1	yes	characteristic
335	C₄	C₂×#6470	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	28	2	no	—
338	C₂	#11736721	1	1	yes	characteristic
339	C₂	C₂×#89679	1	1	yes	characteristic
340	C₂	C₂×#89679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×#2163	C₂
prime radical	#6493	C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	#2163	D₇
hypocenter	C₇	#33480
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. C₇×#2163	C₂	semidirect
3. C₂×C₄×C₇⋊C₁₆	C₂	semidirect
4. #11727035	C₂	semidirect
5. #11727034	C₂	semidirect
6. #11727033	C₂	semidirect
7. #11727031	C₂	semidirect
8. #11726856	C₂	semidirect
9. (C₂²⋊D₇)⋊C₁₆	C₂	semidirect
10. #11726828	C₂	semidirect
11. #11726826	C₂	semidirect
89. #2163	D₇	semidirect
318. C₇	#33480	semidirect

Maximal subgroups

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

subgroup	index
#11735893	2
C₇×#2163	2
C₂×C₄×C₇⋊C₁₆	2
#11727035	2
#11727034	2
#11727033	2
#11727031	2
#11726856	2
(C₂²⋊D₇)⋊C₁₆	2
#11726828	2
#11726826	2
#33480	7