Subgroups — #13323636

Order 1792.

614 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11737408	C₂	1	2	yes	maximal
2	C₄×D₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728706	C₂	1	2	yes	maximal, complemented
5	#11728659	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726321	C₂	1	1	yes	maximal, characteristic
7	#11726318	C₂	1	1	yes	maximal, characteristic, complemented
8	#11725985	C₂	1	2	yes	maximal
9	#11725981	C₂	1	2	yes	maximal, complemented
10	#11725810	C₂	1	1	yes	maximal, characteristic, complemented
11	#11725801	C₂	1	1	yes	maximal, characteristic
12	C₂×D₇×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₇	C₂²	1	2	yes	—
14	C₇×#446	C₂²	1	2	yes	—
15	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×Dic₇	C₄	1	1	yes	characteristic
17	C₄×#1144	C₂²	1	2	yes	—
18	#89796	C₂²	1	2	yes	—
19	D₇×#107	C₄	1	1	yes	characteristic
20	#89645	C₄	1	1	yes	characteristic
21	#89607	—	2	2	no	—
22	#89605	C₂²	1	2	yes	—
23	#89603	C₂²	1	2	yes	—
24	#89600	—	2	2	no	—
25	D₇×#103	C₄	1	2	yes	—
26	#89562	C₄	1	2	yes	—
27	#89477	—	2	2	no	—
28	#89473	—	2	2	no	—
29	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
30	#89457	C₂²	1	1	yes	characteristic
31	#89457	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
33	#89451	C₂²	1	2	yes	—
34	C₇×#335	C₂²	1	1	yes	characteristic
35	C₇×Q₈⋊C₈	C₂²	1	2	yes	—
36	C₇×C₈²	C₂²	1	1	yes	characteristic
37	Q₈⋊(C₇⋊C₈)	C₂²	1	2	yes	—
38	#89279	C₂²	1	2	yes	—
39	#89275	C₂²	1	2	yes	—
40	#89253	C₂²	1	1	yes	characteristic
41	#89251	C₂²	1	2	yes	—
42	#89249	C₂²	1	1	yes	characteristic
43	#89249	C₂²	1	1	yes	characteristic
44	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
45	#17897	—	7	1	no	maximal, Sylow
46	C₂×Q₈×D₇	C₂×C₄	1	2	yes	—
47	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×Q₈	C₂³	1	2	yes	—
49	Q₈×Dic₇	C₂×C₄	1	2	yes	—
50	C₂×#1144	C₂×C₄	1	2	yes	—
51	D₇×Q₁₆	—	2	4	no	—
52	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
55	D₇×C₄⋊C₄	—	2	2	no	—
56	D₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	#6450	—	2	2	no	—
58	#6450	C₂×C₄	1	2	yes	—
59	C₄²×D₇	—	2	2	no	—
60	C₄²×D₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₄	—	2	2	no	—
62	C₄×Dic₁₄	C₂³	1	2	yes	—
63	C₇×#107	C₂×C₄	1	1	yes	characteristic
64	C₇×C₄⋊C₈	—	2	2	no	—
65	C₇×C₄⋊C₈	C₂³	1	2	yes	—
66	C₇×#103	C₂×C₄	1	2	yes	—
67	C₇×#97	—	2	2	no	—
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	#6409	C₂×C₄	1	2	yes	—
72	(C₂×D₇)⋊C₈	—	2	2	no	—
73	(C₂×D₇)⋊C₈	—	2	1	no	—
74	(C₂×D₇)⋊C₈	D₄	1	1	yes	characteristic
75	(C₂×D₇)⋊C₈	D₄	1	1	yes	characteristic
76	#6392	C₂×C₄	1	1	yes	characteristic
77	#6390	C₂×C₄	1	2	yes	—
78	#6389	—	2	2	no	—
79	#6389	—	2	2	no	—
80	Dic₇⋊C₈	—	2	2	no	—
81	Dic₇⋊C₈	—	2	1	no	—
82	Dic₇⋊C₈	—	2	2	no	—
83	Dic₇⋊C₈	D₄	1	1	yes	characteristic
84	Dic₇⋊C₈	D₄	1	1	yes	characteristic
85	C₈×Dic₇	—	2	2	no	—
86	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
88	#6384	C₂×C₄	1	2	yes	—
89	#6381	C₂×C₄	1	2	yes	—
90	C₄⋊(C₇⋊C₈)	—	2	2	no	—
91	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
92	#6377	—	2	2	no	—
93	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
94	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
95	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
96	#2983	—	7	2	no	—
97	C₂×C₄×Q₁₆	—	7	1	no	—
98	#1593	—	7	2	no	—
99	#1593	—	7	2	no	—
100	#1591	—	7	1	no	—
101	#1591	—	7	1	no	—
102	#1591	—	7	1	no	—
103	#1591	D₇	1	1	yes	characteristic, complemented
104	#1564	—	7	1	no	—
105	#1479	—	7	2	no	—
106	#1450	—	7	1	no	—
107	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
108	Q₈×D₇	—	2	4	no	—
109	Q₈×D₇	—	2	4	no	—
110	C₂×C₄×D₇	—	2	2	no	—
111	C₂×C₄×D₇	—	2	2	no	—
112	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×D₇	#41	1	1	yes	characteristic
115	C₂×Dic₁₄	—	2	2	no	—
116	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
117	C₇×Q₁₆	—	2	2	no	—
118	C₂×C₇×C₈	—	2	2	no	—
119	C₂×C₇×C₈	—	2	1	no	—
120	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
124	C₇×C₄⋊C₄	—	2	2	no	—
125	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
126	C₄²×C₇	—	2	2	no	—
127	C₄²×C₇	C₂⁴	1	1	yes	characteristic
128	#1144	—	2	4	no	—
129	C₄⋊Dic₇	—	2	2	no	—
130	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
131	Dic₇⋊C₄	—	2	2	no	—
132	Dic₇⋊C₄	—	2	2	no	—
133	Dic₇⋊C₄	—	2	2	no	—
134	C₄×Dic₇	—	2	2	no	—
135	C₄×Dic₇	—	2	2	no	—
136	C₄×Dic₇	—	2	2	no	—
137	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
138	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
139	C₄×Dic₇	#41	1	1	yes	characteristic
140	C₂×C₇⋊C₈	—	2	2	no	—
141	C₂×C₇⋊C₈	—	2	1	no	—
142	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
146	Dic₂₈	—	2	2	no	—
147	C₈×D₇	—	2	2	no	—
148	C₈×D₇	M₁₆	1	4	yes	—
149	C₂²×Q₁₆	—	7	1	no	—
150	C₂×C₄×Q₈	—	7	2	no	—
151	#446	—	7	2	no	—
152	#446	—	7	2	no	—
153	#446	—	7	2	no	—
154	#446	—	14	2	no	—
155	#446	—	14	2	no	—
156	#446	C₂×D₇	1	2	yes	—
157	C₄×Q₁₆	—	7	1	no	—
158	C₄×Q₁₆	—	7	2	no	—
159	C₄×Q₁₆	—	7	1	no	—
160	C₄×Q₁₆	—	7	2	no	—
161	C₄×Q₁₆	—	7	1	no	—
162	C₄×Q₁₆	C₂×D₇	1	1	yes	characteristic
163	#432	—	14	2	no	—
164	#432	—	14	2	no	—
165	#431	—	7	1	no	—
166	#431	—	7	1	no	—
167	#431	—	7	2	no	—
168	C₂×#107	—	7	1	no	—
169	C₂×#103	—	7	2	no	—
170	C₂×C₄×C₈	—	7	1	no	—
171	#335	—	7	1	no	—
172	#335	C₂×D₇	1	1	yes	characteristic
173	#334	—	7	2	no	—
174	Q₈⋊C₈	—	7	2	no	—
175	Q₈⋊C₈	—	7	2	no	—
176	Q₈⋊C₈	—	7	2	no	—
177	Q₈⋊C₈	C₂×D₇	1	2	yes	—
178	#322	—	7	1	no	—
179	#322	—	7	1	no	—
180	C₈²	—	7	1	no	—
181	C₈²	C₂×D₇	1	1	yes	characteristic
182	C₂²×D₇	C₄×D₄	1	1	yes	characteristic
183	C₇×Q₈	—	2	2	no	—
184	C₇×Q₈	—	2	2	no	—
185	C₂×C₄×C₇	—	2	2	no	—
186	C₂×C₄×C₇	—	2	2	no	—
187	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
190	C₂×Dic₇	—	2	2	no	—
191	C₂×Dic₇	—	2	2	no	—
192	C₂×Dic₇	C₄×D₄	1	1	yes	characteristic
193	C₂×Dic₇	C₄×D₄	1	1	yes	characteristic
194	C₂×Dic₇	C₄×D₄	1	1	yes	characteristic
195	C₄×D₇	—	2	1	no	—
196	C₄×D₇	—	2	1	no	—
197	C₄×D₇	—	4	1	no	—
198	C₄×D₇	—	4	4	no	—
199	C₄×D₇	—	8	2	no	—
200	C₄×D₇	C₂×M₁₆	1	2	yes	—
201	C₄×D₇	#131	1	2	yes	—
202	Dic₁₄	—	2	2	no	—
203	Dic₁₄	—	2	2	no	—
204	Dic₁₄	—	4	4	no	—
205	C₇×C₈	—	2	1	no	—
206	C₇×C₈	—	2	1	no	—
207	C₇×C₈	—	2	1	no	—
208	C₇×C₈	—	4	1	no	—
209	C₇×C₈	—	4	2	no	—
210	C₇×C₈	C₂×M₁₆	1	2	yes	—
211	C₇⋊C₈	—	2	1	no	—
212	C₇⋊C₈	—	2	1	no	—
213	C₇⋊C₈	—	2	1	no	—
214	C₇⋊C₈	—	4	1	no	—
215	C₇⋊C₈	—	4	2	no	—
216	C₇⋊C₈	C₂×M₁₆	1	2	yes	—
217	C₂²×Q₈	—	7	2	no	—
218	C₂×Q₁₆	—	7	2	no	—
219	C₂×Q₁₆	—	7	1	no	—
220	C₂×Q₁₆	—	14	4	no	—
221	C₂×Q₁₆	C₄×D₇	1	1	yes	characteristic
222	C₂²×C₈	—	7	1	no	—
223	C₂²×C₈	—	7	1	no	—
224	C₄×Q₈	—	7	2	no	—
225	C₄×Q₈	—	7	2	no	—
226	C₄×Q₈	—	7	2	no	—
227	C₄×Q₈	—	14	2	no	—
228	C₄×Q₈	—	14	2	no	—
229	C₄×Q₈	C₂²×D₇	1	2	yes	—
230	C₂×C₄⋊C₄	—	7	2	no	—
231	C₂×C₄⋊C₄	—	14	2	no	—
232	C₂×C₄²	—	7	1	no	—
233	C₂×C₄²	—	14	2	no	—
234	#107	—	7	1	no	—
235	#107	—	7	2	no	—
236	#107	C₄×D₇	1	1	yes	characteristic
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	7	1	no	—
239	C₄⋊C₈	—	7	1	no	—
240	C₄⋊C₈	—	7	2	no	—
241	C₄⋊C₈	—	14	2	no	—
242	C₄⋊C₈	—	14	1	no	—
243	C₄⋊C₈	—	14	2	no	—
244	C₄⋊C₈	—	14	2	no	—
245	C₄⋊C₈	C₂²×D₇	1	2	yes	—
246	#103	—	7	2	no	—
247	#103	—	7	2	no	—
248	#103	—	7	2	no	—
249	#103	C₄×D₇	1	2	yes	—
250	C₂²⋊C₈	—	7	1	no	—
251	C₂²⋊C₈	—	7	1	no	—
252	C₂²⋊C₈	—	14	2	no	—
253	C₂²⋊C₈	—	14	1	no	—
254	#97	—	2	2	no	—
255	#97	—	14	2	no	—
256	#97	—	14	2	no	—
257	#97	—	14	2	no	—
258	C₄×C₈	—	7	1	no	—
259	C₄×C₈	—	7	1	no	—
260	C₄×C₈	—	7	1	no	—
261	C₄×C₈	—	7	1	no	—
262	C₄×C₈	—	7	1	no	—
263	C₄×C₈	—	14	2	no	—
264	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
267	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂×D₇	—	2	1	no	—
269	C₂×D₇	—	2	1	no	—
270	C₂×D₇	#436	1	2	yes	—
271	C₄×C₇	—	2	1	no	—
272	C₄×C₇	—	4	2	no	—
273	C₄×C₇	—	4	2	no	—
274	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
275	C₄×C₇	C₂×#131	1	1	yes	characteristic
276	C₄×C₇	#578	1	1	yes	characteristic
277	C₄×C₇	#578	1	1	yes	characteristic
278	Dic₇	—	2	1	no	—
279	Dic₇	—	2	1	no	—
280	Dic₇	—	4	2	no	—
281	Dic₇	—	4	2	no	—
282	Dic₇	#436	1	2	yes	—
283	C₂×Q₈	—	7	2	no	—
284	C₂×Q₈	—	14	2	no	—
285	C₂×Q₈	—	14	4	no	—
286	C₂×Q₈	—	14	4	no	—
287	C₂×Q₈	C₂×C₄×D₇	1	2	yes	—
288	C₂²×C₄	—	7	1	no	—
289	C₂²×C₄	—	7	1	no	—
290	C₂²×C₄	—	7	1	no	—
291	C₂²×C₄	—	14	2	no	—
292	C₂²×C₄	—	14	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	14	4	no	—
295	Q₁₆	—	14	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	7	1	no	—
299	C₂×C₈	—	7	4	no	—
300	C₂×C₈	—	7	1	no	—
301	C₂×C₈	—	7	1	no	—
302	C₂×C₈	—	7	1	no	—
303	C₂×C₈	—	14	2	no	—
304	C₂×C₈	—	14	2	no	—
305	C₂×C₈	—	14	1	no	—
306	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
308	C₂×C₈	D₄×D₇	1	1	yes	characteristic
309	C₂×C₈	D₄×D₇	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	7	2	no	—
312	C₄⋊C₄	—	14	2	no	—
313	C₄⋊C₄	—	14	2	no	—
314	C₄⋊C₄	—	14	2	no	—
315	C₄⋊C₄	—	14	2	no	—
316	C₄⋊C₄	C₂×C₄×D₇	1	2	yes	—
317	C₄²	—	2	2	no	—
318	C₄²	—	7	1	no	—
319	C₄²	—	7	1	no	—
320	C₄²	—	7	1	no	—
321	C₄²	—	14	2	no	—
322	C₄²	—	14	2	no	—
323	C₄²	—	14	2	no	—
324	C₄²	C₂³×D₇	1	1	yes	characteristic
325	C₂×C₇	C₂×#436	1	1	yes	characteristic
326	C₂×C₇	#2950	1	1	yes	characteristic
327	C₂×C₇	#2957	1	1	yes	characteristic
328	D₇	—	2	2	no	—
329	C₂³	—	7	1	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	14	2	no	—
333	Q₈	—	14	2	no	—
334	Q₈	—	28	4	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	7	1	no	—
338	C₂×C₄	—	7	2	no	—
339	C₂×C₄	—	7	2	no	—
340	C₂×C₄	—	7	1	no	—
341	C₂×C₄	—	7	1	no	—
342	C₂×C₄	—	14	1	no	—
343	C₂×C₄	—	14	1	no	—
344	C₂×C₄	—	14	2	no	—
345	C₂×C₄	—	14	2	no	—
346	C₂×C₄	—	28	1	no	—
347	C₂×C₄	—	28	4	no	—
348	C₂×C₄	—	56	2	no	—
349	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
350	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
351	C₂×C₄	D₇×#41	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	7	2	no	—
358	C₈	—	14	1	no	—
359	C₈	—	14	1	no	—
360	C₈	—	14	1	no	—
361	C₈	—	28	1	no	—
362	C₈	—	28	2	no	—
363	C₈	D₇×M₁₆	1	2	yes	—
364	C₇	#17897	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	7	2	no	—
366	C₂²	—	14	1	no	—
367	C₂²	—	14	1	no	—
368	C₂²	C₄×D₄×D₇	1	1	yes	characteristic
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	7	2	no	—
373	C₄	—	14	1	no	—
374	C₄	—	14	1	no	—
375	C₄	—	28	2	no	—
376	C₄	—	28	2	no	—
377	C₄	C₂×D₇×M₁₆	1	1	yes	characteristic
378	C₄	D₇×#131	1	1	yes	characteristic
379	C₄	#90457	1	1	yes	characteristic
380	C₄	#90462	1	1	yes	characteristic
381	C₂	—	14	2	no	—
382	C₂	D₇×#436	1	1	yes	characteristic
383	C₂	#11737165	1	1	yes	characteristic
384	C₂	#11737335	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1591	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂²×D₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1591	D₇
hypocenter	C₇	#17897
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. C₇×#1591	C₂	semidirect
4. #11728706	C₂	semidirect
5. #11728659	C₂	semidirect
7. #11726318	C₂	semidirect
9. #11725981	C₂	semidirect
10. #11725810	C₂	semidirect
103. #1591	D₇	semidirect
364. C₇	#17897	semidirect

Maximal subgroups

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

subgroup	index
#11737408	2
C₄×D₇×Q₁₆	2
C₇×#1591	2
#11728706	2
#11728659	2
#11726321	2
#11726318	2
#11725985	2
#11725981	2
#11725810	2
#11725801	2
#17897	7