Subgroups — #13324475

Order 1792.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11737410	C₂	1	2	yes	maximal
2	#11737170	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	Q₁₆×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726324	C₂	1	1	yes	maximal, characteristic
7	#11726311	C₂	1	1	yes	maximal, characteristic, complemented
8	#11725997	C₂	1	2	yes	maximal
9	#11725969	C₂	1	2	yes	maximal, complemented
10	#11725810	C₂	1	1	yes	maximal, characteristic, complemented
11	#11725801	C₂	1	1	yes	maximal, characteristic
12	#90269	C₂²	1	2	yes	—
13	C₇×#446	C₂²	1	2	yes	—
14	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	#89960	C₄	1	1	yes	characteristic
16	#89953	C₄	1	1	yes	characteristic
17	C₄×#1144	C₂²	1	2	yes	—
18	Q₈×C₇⋊C₈	C₂²	1	2	yes	—
19	#89659	C₄	1	1	yes	characteristic
20	#89646	C₄	1	1	yes	characteristic
21	#89612	—	2	2	no	—
22	#89607	C₂²	1	2	yes	—
23	#89603	—	2	2	no	—
24	#89602	C₂²	1	2	yes	—
25	#89585	C₄	1	2	yes	—
26	#89564	C₄	1	2	yes	—
27	#89481	—	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	#17972	—	7	1	no	maximal, Sylow
46	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×Q₈	C₂³	1	2	yes	—
48	#6509	C₂×C₄	1	2	yes	—
49	#6507	C₂×C₄	1	2	yes	—
50	C₂×#1144	C₂×C₄	1	2	yes	—
51	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
54	#6460	C₂×C₄	1	2	yes	—
55	#6459	—	2	2	no	—
56	#6452	—	2	2	no	—
57	#6451	C₂×C₄	1	2	yes	—
58	#6435	—	2	2	no	—
59	C₄²×D₇	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₄	—	2	2	no	—
61	C₄×Dic₁₄	C₂³	1	2	yes	—
62	C₇×#107	C₂×C₄	1	1	yes	characteristic
63	C₇×C₄⋊C₈	—	2	2	no	—
64	C₇×C₄⋊C₈	C₂³	1	2	yes	—
65	C₇×#103	C₂×C₄	1	2	yes	—
66	C₇×#97	—	2	2	no	—
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	#6409	C₂×C₄	1	2	yes	—
71	(C₂×D₇)⋊C₈	—	2	2	no	—
72	(C₂×D₇)⋊C₈	—	2	1	no	—
73	(C₂×D₇)⋊C₈	D₄	1	1	yes	characteristic
74	(C₂×D₇)⋊C₈	D₄	1	1	yes	characteristic
75	#6392	C₂×C₄	1	1	yes	characteristic
76	#6390	C₂×C₄	1	2	yes	—
77	#6389	—	2	2	no	—
78	#6389	—	2	2	no	—
79	Dic₇⋊C₈	—	2	2	no	—
80	Dic₇⋊C₈	—	2	1	no	—
81	Dic₇⋊C₈	—	2	2	no	—
82	Dic₇⋊C₈	D₄	1	1	yes	characteristic
83	Dic₇⋊C₈	D₄	1	1	yes	characteristic
84	C₈×Dic₇	—	2	2	no	—
85	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
86	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
87	#6384	C₂×C₄	1	2	yes	—
88	#6381	C₂×C₄	1	2	yes	—
89	C₄⋊(C₇⋊C₈)	—	2	2	no	—
90	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
91	C₄×C₇⋊C₈	—	2	2	no	—
92	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
93	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
94	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
95	#3170	—	7	1	no	—
96	#2985	—	7	2	no	—
97	#1593	—	7	2	no	—
98	#1591	—	7	1	no	—
99	#1591	D₇	1	1	yes	characteristic, complemented
100	#1579	—	7	2	no	—
101	C₈×Q₁₆	—	7	1	no	—
102	C₈×Q₁₆	—	7	1	no	—
103	#1568	—	7	1	no	—
104	#1497	—	7	2	no	—
105	#1450	—	7	1	no	—
106	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
107	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₇	#41	1	1	yes	characteristic
110	C₂×Dic₁₄	—	2	2	no	—
111	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
112	C₇×Q₁₆	—	2	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	1	no	—
115	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
119	C₇×C₄⋊C₄	—	2	2	no	—
120	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
121	C₄²×C₇	—	2	2	no	—
122	C₄²×C₇	C₂⁴	1	1	yes	characteristic
123	#1144	—	2	4	no	—
124	(C₂×D₇)⋊C₄	—	2	2	no	—
125	(C₂×D₇)⋊C₄	—	2	2	no	—
126	C₄⋊Dic₇	—	2	2	no	—
127	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
128	Dic₇⋊C₄	—	2	2	no	—
129	Dic₇⋊C₄	—	2	2	no	—
130	Dic₇⋊C₄	—	2	2	no	—
131	Dic₇⋊C₄	—	2	2	no	—
132	Dic₇⋊C₄	—	2	2	no	—
133	C₄×Dic₇	—	2	2	no	—
134	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
135	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₇	#41	1	1	yes	characteristic
137	C₂×C₇⋊C₈	—	2	2	no	—
138	C₂×C₇⋊C₈	—	2	1	no	—
139	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
142	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
143	Dic₂₈	—	2	2	no	—
144	C₈×D₇	—	2	2	no	—
145	C₈×D₇	—	2	2	no	—
146	#533	—	7	2	no	—
147	#500	—	7	1	no	—
148	#494	—	7	1	no	—
149	#487	—	7	2	no	—
150	#467	—	7	1	no	—
151	#467	—	7	1	no	—
152	#467	—	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	C₈×Q₈	—	7	2	no	—
158	C₈×Q₈	—	7	2	no	—
159	C₄×Q₁₆	—	7	2	no	—
160	C₄×Q₁₆	—	7	1	no	—
161	C₄×Q₁₆	C₂×D₇	1	1	yes	characteristic
162	#433	—	14	2	no	—
163	#433	—	14	2	no	—
164	#432	—	7	2	no	—
165	#431	—	7	1	no	—
166	#431	—	7	1	no	—
167	C₂×C₄×C₈	—	7	1	no	—
168	#335	—	7	1	no	—
169	#335	C₂×D₇	1	1	yes	characteristic
170	#334	—	7	2	no	—
171	Q₈⋊C₈	—	7	2	no	—
172	Q₈⋊C₈	—	7	2	no	—
173	Q₈⋊C₈	—	7	2	no	—
174	Q₈⋊C₈	C₂×D₇	1	2	yes	—
175	#322	—	7	1	no	—
176	#322	—	7	1	no	—
177	C₈²	—	7	1	no	—
178	C₈²	C₂×D₇	1	1	yes	characteristic
179	C₂²×D₇	C₄×D₄	1	1	yes	characteristic
180	C₇×Q₈	—	2	2	no	—
181	C₇×Q₈	—	2	2	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	2	2	no	—
184	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
187	C₂×Dic₇	—	2	2	no	—
188	C₂×Dic₇	—	2	2	no	—
189	C₂×Dic₇	C₄×D₄	1	1	yes	characteristic
190	C₂×Dic₇	C₄×D₄	1	1	yes	characteristic
191	C₂×Dic₇	C₄×D₄	1	1	yes	characteristic
192	C₄×D₇	—	2	1	no	—
193	C₄×D₇	—	2	1	no	—
194	C₄×D₇	—	2	1	no	—
195	C₄×D₇	—	2	1	no	—
196	C₄×D₇	—	4	1	no	—
197	Dic₁₄	—	2	2	no	—
198	Dic₁₄	—	2	2	no	—
199	Dic₁₄	—	8	2	no	—
200	C₇×C₈	—	2	1	no	—
201	C₇×C₈	—	2	1	no	—
202	C₇×C₈	—	2	1	no	—
203	C₇×C₈	—	4	1	no	—
204	C₇×C₈	—	4	2	no	—
205	C₇×C₈	#131	1	2	yes	—
206	C₇⋊C₈	—	2	1	no	—
207	C₇⋊C₈	—	2	1	no	—
208	C₇⋊C₈	—	2	1	no	—
209	C₇⋊C₈	—	4	1	no	—
210	C₇⋊C₈	—	4	2	no	—
211	C₇⋊C₈	#131	1	2	yes	—
212	C₂×Q₁₆	—	7	2	no	—
213	C₂×Q₁₆	—	7	1	no	—
214	C₂×Q₁₆	C₄×D₇	1	1	yes	characteristic
215	C₂²×C₈	—	7	1	no	—
216	C₂²×C₈	—	7	1	no	—
217	C₄⋊Q₈	—	7	2	no	—
218	C₄⋊Q₈	—	7	2	no	—
219	#125	—	14	2	no	—
220	C₂²⋊Q₈	—	7	2	no	—
221	C₂²⋊Q₈	—	7	2	no	—
222	C₂²⋊Q₈	—	14	2	no	—
223	C₄×Q₈	—	7	2	no	—
224	C₄×Q₈	—	14	2	no	—
225	C₄×Q₈	C₂²×D₇	1	2	yes	—
226	#117	—	14	2	no	—
227	C₂×C₄²	—	7	1	no	—
228	#107	—	7	1	no	—
229	#107	—	7	2	no	—
230	#107	C₄×D₇	1	1	yes	characteristic
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	—	7	1	no	—
233	C₄⋊C₈	—	7	1	no	—
234	C₄⋊C₈	—	7	2	no	—
235	C₄⋊C₈	—	14	2	no	—
236	C₄⋊C₈	—	14	1	no	—
237	C₄⋊C₈	—	14	2	no	—
238	C₄⋊C₈	—	14	2	no	—
239	C₄⋊C₈	C₂²×D₇	1	2	yes	—
240	#103	—	7	2	no	—
241	#103	—	7	2	no	—
242	#103	—	7	2	no	—
243	#103	C₄×D₇	1	2	yes	—
244	C₂²⋊C₈	—	7	1	no	—
245	C₂²⋊C₈	—	7	1	no	—
246	C₂²⋊C₈	—	14	1	no	—
247	C₂²⋊C₈	—	14	2	no	—
248	#97	—	2	2	no	—
249	#97	—	14	2	no	—
250	#97	—	14	2	no	—
251	C₄×C₈	—	7	1	no	—
252	C₄×C₈	—	7	1	no	—
253	C₄×C₈	—	7	1	no	—
254	C₄×C₈	—	7	1	no	—
255	C₄×C₈	—	7	1	no	—
256	C₄×C₈	—	14	2	no	—
257	C₄×C₈	—	14	2	no	—
258	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
261	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
262	C₂×D₇	—	2	1	no	—
263	C₂×D₇	—	2	1	no	—
264	C₂×D₇	—	2	1	no	—
265	C₄×C₇	—	2	1	no	—
266	C₄×C₇	—	4	2	no	—
267	C₄×C₇	—	4	2	no	—
268	C₄×C₇	C₂×#131	1	1	yes	characteristic
269	C₄×C₇	C₂×#131	1	1	yes	characteristic
270	C₄×C₇	#578	1	1	yes	characteristic
271	C₄×C₇	#578	1	1	yes	characteristic
272	Dic₇	—	2	1	no	—
273	Dic₇	—	2	1	no	—
274	Dic₇	—	2	1	no	—
275	Dic₇	—	4	2	no	—
276	Dic₇	—	4	2	no	—
277	C₂×Q₈	—	7	2	no	—
278	C₂×Q₈	—	14	2	no	—
279	C₂×Q₈	C₂×C₄×D₇	1	2	yes	—
280	C₂²×C₄	—	7	1	no	—
281	C₂²×C₄	—	7	1	no	—
282	C₂²×C₄	—	7	1	no	—
283	Q₁₆	—	2	2	no	—
284	Q₁₆	—	14	4	no	—
285	Q₁₆	—	14	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	7	1	no	—
289	C₂×C₈	—	7	1	no	—
290	C₂×C₈	—	7	1	no	—
291	C₂×C₈	—	7	1	no	—
292	C₂×C₈	—	14	2	no	—
293	C₂×C₈	—	14	2	no	—
294	C₂×C₈	—	14	2	no	—
295	C₂×C₈	—	14	1	no	—
296	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
298	C₂×C₈	D₄×D₇	1	1	yes	characteristic
299	C₂×C₈	D₄×D₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	7	2	no	—
302	C₄⋊C₄	—	14	2	no	—
303	C₄⋊C₄	—	14	2	no	—
304	C₄⋊C₄	—	14	2	no	—
305	C₄⋊C₄	—	14	2	no	—
306	C₄⋊C₄	—	14	2	no	—
307	C₄⋊C₄	—	14	2	no	—
308	C₄⋊C₄	C₂×C₄×D₇	1	2	yes	—
309	C₂²⋊C₄	—	14	2	no	—
310	C₂²⋊C₄	—	14	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	7	1	no	—
313	C₄²	—	7	1	no	—
314	C₄²	—	7	1	no	—
315	C₄²	—	14	2	no	—
316	C₄²	C₂³×D₇	1	1	yes	characteristic
317	C₂×C₇	#2931	1	1	yes	characteristic
318	C₂×C₇	#2950	1	1	yes	characteristic
319	C₂×C₇	#2957	1	1	yes	characteristic
320	D₇	—	4	1	no	—
321	C₂³	—	7	1	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	14	2	no	—
325	Q₈	—	14	2	no	—
326	Q₈	—	56	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	7	1	no	—
330	C₂×C₄	—	7	1	no	—
331	C₂×C₄	—	7	1	no	—
332	C₂×C₄	—	14	1	no	—
333	C₂×C₄	—	14	1	no	—
334	C₂×C₄	—	14	1	no	—
335	C₂×C₄	—	14	1	no	—
336	C₂×C₄	—	14	2	no	—
337	C₂×C₄	—	14	2	no	—
338	C₂×C₄	—	28	1	no	—
339	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
341	C₂×C₄	D₇×#41	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	1	no	—
346	C₈	—	4	2	no	—
347	C₈	—	7	2	no	—
348	C₈	—	14	1	no	—
349	C₈	—	14	1	no	—
350	C₈	—	14	1	no	—
351	C₈	—	28	1	no	—
352	C₈	—	28	2	no	—
353	C₈	#6470	1	2	yes	—
354	C₇	#17972	1	1	yes	characteristic, complemented, Sylow
355	C₂²	—	14	1	no	—
356	C₂²	—	14	1	no	—
357	C₂²	—	14	1	no	—
358	C₂²	C₄×D₄×D₇	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	14	1	no	—
363	C₄	—	14	1	no	—
364	C₄	—	14	1	no	—
365	C₄	—	28	2	no	—
366	C₄	—	28	2	no	—
367	C₄	C₂×#6470	1	1	yes	characteristic
368	C₄	D₇×#131	1	1	yes	characteristic
369	C₄	#90457	1	1	yes	characteristic
370	C₄	#90462	1	1	yes	characteristic
371	C₂	—	28	1	no	—
372	C₂	#11736983	1	1	yes	characteristic
373	C₂	#11737165	1	1	yes	characteristic
374	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	#17972
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. Q₁₆×C₇⋊C₈	C₂	semidirect
7. #11726311	C₂	semidirect
9. #11725969	C₂	semidirect
10. #11725810	C₂	semidirect
99. #1591	D₇	semidirect
354. C₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#11737410	2
#11737170	2
C₇×#1591	2
#11728706	2
Q₁₆×C₇⋊C₈	2
#11726324	2
#11726311	2
#11725997	2
#11725969	2
#11725810	2
#11725801	2
#17972	7