Subgroups — #13324494

Order 1792.

592 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	#11736498	C₂	1	2	yes	maximal, complemented
2	#11735914	C₂	1	1	yes	maximal, characteristic
3	C₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728253	C₂	1	2	yes	maximal, complemented
5	#11727478	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726279	C₂	1	1	yes	maximal, characteristic, complemented
7	#11726278	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726277	C₂	1	2	yes	maximal, complemented
9	#11726272	C₂	1	1	yes	maximal, characteristic, complemented
10	#11726271	C₂	1	1	yes	maximal, characteristic, complemented
11	#11726270	C₂	1	2	yes	maximal, complemented
12	#90170	C₂²	1	2	yes	—
13	C₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89900	—	2	2	no	—
16	#89899	—	2	2	no	—
17	#89888	C₄	1	1	yes	characteristic
18	#89887	C₄	1	1	yes	characteristic
19	#89886	C₄	1	2	yes	—
20	#89697	C₂²	1	1	yes	characteristic
21	#89697	C₂²	1	1	yes	characteristic
22	#89612	—	2	2	no	—
23	#89611	—	2	2	no	—
24	#89610	C₂²	1	2	yes	—
25	#89606	C₂²	1	2	yes	—
26	#89603	C₂²	1	2	yes	—
27	#89602	C₂²	1	2	yes	—
28	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
29	#89469	C₄	1	2	yes	—
30	#89468	C₄	1	1	yes	characteristic
31	C₄×D₅₆	C₂²	1	1	yes	characteristic
32	C₄×#1132	C₂²	1	2	yes	—
33	#89455	C₄	1	1	yes	characteristic
34	C₇×#334	C₂²	1	2	yes	—
35	C₇×#334	C₂²	1	1	yes	characteristic
36	C₇×#334	C₂²	1	1	yes	characteristic
37	#89279	C₂²	1	2	yes	—
38	#89279	C₂²	1	2	yes	—
39	#89278	C₂²	1	2	yes	—
40	#89278	C₂²	1	2	yes	—
41	#89250	C₂²	1	1	yes	characteristic
42	#89250	C₂²	1	1	yes	characteristic
43	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
44	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
45	#17974	—	7	1	no	maximal, Sylow
46	C₂×C₇×M₁₆	—	2	2	no	—
47	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	#6493	C₂×C₄	1	2	yes	—
51	#6492	—	2	2	no	—
52	C₄×C₂²⋊D₇	—	2	2	no	—
53	#6487	C₂×C₄	1	2	yes	—
54	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
55	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
56	C₂×#1132	C₂×C₄	1	2	yes	—
57	C₂×#1131	—	2	2	no	—
58	C₂×C₈×D₇	—	2	2	no	—
59	#6439	—	2	2	no	—
60	#6438	C₂×C₄	1	2	yes	—
61	C₄×D₂₈	C₂³	1	2	yes	—
62	#6435	—	2	2	no	—
63	#6433	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₄	C₂³	1	2	yes	—
65	C₇×C₄⋊C₈	—	2	2	no	—
66	C₇×C₄⋊C₈	C₂³	1	2	yes	—
67	C₇×C₄⋊C₈	C₂³	1	2	yes	—
68	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
73	C₂²⋊(C₇⋊C₈)	D₄	1	2	yes	—
74	#6395	C₂×C₄	1	2	yes	—
75	#6395	C₂×C₄	1	2	yes	—
76	(C₂×D₇)⋊C₈	—	2	2	no	—
77	(C₂×D₇)⋊C₈	—	2	2	no	—
78	#6392	C₂×C₄	1	1	yes	characteristic
79	#6392	C₂×C₄	1	1	yes	characteristic
80	#6391	C₂×C₄	1	2	yes	—
81	#6390	C₂×C₄	1	2	yes	—
82	#6390	C₂×C₄	1	2	yes	—
83	#6389	—	2	2	no	—
84	Dic₇⋊C₈	—	2	2	no	—
85	Dic₇⋊C₈	—	2	2	no	—
86	C₈×Dic₇	—	2	2	no	—
87	C₄⋊(C₇⋊C₈)	—	2	1	no	—
88	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
89	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
90	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
91	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
92	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
93	#3168	—	7	1	no	—
94	#2993	—	7	2	no	—
95	#1594	—	7	1	no	—
96	#1593	—	7	1	no	—
97	#1590	—	7	2	no	—
98	#1580	—	7	1	no	—
99	#1579	—	7	1	no	—
100	C₈×SD₁₆	—	7	2	no	—
101	#1566	D₇	1	1	yes	characteristic, complemented
102	#1495	—	7	2	no	—
103	#1450	—	7	1	no	—
104	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇	#41	1	1	yes	characteristic
107	C₂×C₂²⋊D₇	—	2	2	no	—
108	C₂×D₂₈	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₇	—	2	2	no	—
110	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
111	C₇×M₁₆	—	8	2	no	—
112	C₂×C₇×C₈	—	2	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	2	no	—
115	C₂×C₇×C₈	—	2	2	no	—
116	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇	#41	1	1	yes	characteristic
123	C₄²×C₇	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₇	—	2	2	no	—
125	(C₂×D₇)⋊C₄	—	2	2	no	—
126	(C₂×D₇)⋊C₄	—	2	2	no	—
127	(C₂×D₇)⋊C₄	—	2	2	no	—
128	(C₂×D₇)⋊C₄	—	2	2	no	—
129	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
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	Dic₇⋊C₄	—	2	2	no	—
135	C₄×Dic₇	—	2	2	no	—
136	C₂×C₇⋊C₈	—	2	1	no	—
137	C₂×C₇⋊C₈	—	2	1	no	—
138	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
139	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
140	Dic₂₈	—	2	2	no	—
141	D₅₆	—	2	2	no	—
142	#1132	—	2	4	no	—
143	#1131	—	8	2	no	—
144	C₈×D₇	—	8	2	no	—
145	#529	—	7	2	no	—
146	#499	—	7	1	no	—
147	#495	—	7	1	no	—
148	#488	—	7	2	no	—
149	#467	—	7	1	no	—
150	#466	—	7	1	no	—
151	#465	—	7	2	no	—
152	#446	—	7	2	no	—
153	C₈×Q₈	—	7	2	no	—
154	C₄×Q₁₆	—	7	1	no	—
155	C₄×SD₁₆	—	7	2	no	—
156	C₄×D₈	—	7	1	no	—
157	#436	—	7	2	no	—
158	#436	—	14	2	no	—
159	#435	—	14	2	no	—
160	C₈×D₄	—	7	2	no	—
161	#433	—	14	2	no	—
162	#431	—	7	1	no	—
163	#431	—	7	1	no	—
164	#424	—	14	2	no	—
165	#423	C₂×D₇	1	2	yes	—
166	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
167	#334	C₂×D₇	1	2	yes	—
168	#334	C₂×D₇	1	1	yes	characteristic
169	#334	C₂×D₇	1	1	yes	characteristic
170	Q₈⋊C₈	—	7	2	no	—
171	Q₈⋊C₈	—	7	2	no	—
172	D₄⋊C₈	—	7	2	no	—
173	D₄⋊C₈	—	7	2	no	—
174	#322	—	7	1	no	—
175	#322	—	7	1	no	—
176	C₈²	—	7	1	no	—
177	C₈²	—	7	1	no	—
178	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
179	C₂²×D₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	1	no	—
181	C₂×C₄×C₇	—	2	1	no	—
182	C₂×C₄×C₇	—	2	2	no	—
183	C₂×C₄×C₇	—	4	1	no	—
184	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
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	C₂²⋊D₇	—	8	2	no	—
191	C₂×Dic₇	—	2	2	no	—
192	C₂×Dic₇	—	2	2	no	—
193	C₂×Dic₇	—	2	2	no	—
194	D₂₈	—	2	2	no	—
195	D₂₈	—	2	2	no	—
196	C₄×D₇	—	4	2	no	—
197	C₄×D₇	—	4	2	no	—
198	Dic₁₄	—	2	2	no	—
199	Dic₁₄	—	2	2	no	—
200	C₇×C₈	—	2	1	no	—
201	C₇×C₈	—	2	1	no	—
202	C₇×C₈	—	4	2	no	—
203	C₇×C₈	—	4	2	no	—
204	C₇×C₈	#131	1	2	yes	—
205	C₇×C₈	#131	1	2	yes	—
206	C₇⋊C₈	—	2	2	no	—
207	C₇⋊C₈	—	2	2	no	—
208	C₇⋊C₈	—	4	1	no	—
209	C₇⋊C₈	—	4	1	no	—
210	C₂×Q₁₆	—	7	1	no	—
211	C₂×SD₁₆	—	7	2	no	—
212	C₂×D₈	—	7	1	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	14	2	no	—
215	C₂²×C₈	—	14	2	no	—
216	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
217	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
218	#126	—	14	2	no	—
219	#125	—	7	2	no	—
220	#124	—	7	2	no	—
221	#123	—	14	2	no	—
222	C₂²⋊Q₈	—	7	2	no	—
223	#121	—	7	2	no	—
224	C₄×Q₈	—	7	2	no	—
225	C₄×D₄	—	7	2	no	—
226	C₄×D₄	—	14	2	no	—
227	#117	—	14	2	no	—
228	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
229	#107	—	7	1	no	—
230	#107	—	7	1	no	—
231	#106	—	7	2	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	—	7	2	no	—
234	C₄⋊C₈	—	14	2	no	—
235	C₄⋊C₈	—	14	2	no	—
236	C₄⋊C₈	—	14	1	no	—
237	C₄⋊C₈	C₂²×D₇	1	2	yes	—
238	C₄⋊C₈	C₂²×D₇	1	2	yes	—
239	#103	—	7	2	no	—
240	#103	—	7	2	no	—
241	#102	—	7	2	no	—
242	#102	—	7	2	no	—
243	C₂²⋊C₈	—	7	2	no	—
244	C₂²⋊C₈	—	14	2	no	—
245	C₂²⋊C₈	—	14	2	no	—
246	C₂²⋊C₈	—	14	1	no	—
247	#97	—	14	2	no	—
248	C₄×C₈	—	7	1	no	—
249	C₄×C₈	—	7	1	no	—
250	C₄×C₈	—	7	1	no	—
251	C₄×C₈	—	7	1	no	—
252	C₄×C₈	—	14	2	no	—
253	C₄×C₈	C₄×D₇	1	1	yes	characteristic
254	C₄×C₈	C₄×D₇	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
257	C₂²×C₇	—	2	1	no	—
258	C₂²×C₇	—	2	1	no	—
259	C₂²×C₇	—	2	1	no	—
260	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂×D₇	—	4	2	no	—
262	C₂×D₇	—	4	2	no	—
263	C₂×D₇	—	4	2	no	—
264	C₄×C₇	—	2	1	no	—
265	C₄×C₇	—	2	1	no	—
266	C₄×C₇	—	2	1	no	—
267	C₄×C₇	—	2	1	no	—
268	C₄×C₇	C₂×#131	1	1	yes	characteristic
269	C₄×C₇	C₂×#131	1	1	yes	characteristic
270	C₄×C₇	#577	1	2	yes	—
271	Dic₇	—	4	2	no	—
272	Dic₇	—	4	2	no	—
273	Dic₇	—	4	2	no	—
274	C₂×Q₈	—	7	2	no	—
275	C₂×D₄	—	7	2	no	—
276	C₂×D₄	—	14	2	no	—
277	C₂²×C₄	—	14	2	no	—
278	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
280	C₂²×C₄	#1161	1	1	yes	characteristic
281	Q₁₆	—	14	2	no	—
282	SD₁₆	—	14	4	no	—
283	D₈	—	14	2	no	—
284	M₁₆	—	8	2	no	—
285	M₁₆	—	56	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	7	2	no	—
291	C₂×C₈	—	7	2	no	—
292	C₂×C₈	—	14	1	no	—
293	C₂×C₈	—	14	1	no	—
294	C₂×C₈	—	56	2	no	—
295	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
299	C₄⋊C₄	—	7	2	no	—
300	C₄⋊C₄	—	7	2	no	—
301	C₄⋊C₄	—	14	2	no	—
302	C₄⋊C₄	—	14	2	no	—
303	C₄⋊C₄	—	14	2	no	—
304	C₄⋊C₄	—	14	2	no	—
305	C₂²⋊C₄	—	14	2	no	—
306	C₂²⋊C₄	—	14	2	no	—
307	C₂²⋊C₄	—	14	2	no	—
308	C₂²⋊C₄	—	14	2	no	—
309	C₂²⋊C₄	—	14	2	no	—
310	C₄²	—	14	2	no	—
311	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
312	C₄²	D₄×D₇	1	1	yes	characteristic
313	C₄²	#1161	1	1	yes	characteristic
314	C₄²	C₂³×D₇	1	1	yes	characteristic
315	C₂×C₇	—	4	1	no	—
316	C₂×C₇	#2931	1	1	yes	characteristic
317	C₂×C₇	#2952	1	1	yes	characteristic
318	C₂×C₇	#2957	1	1	yes	characteristic
319	D₇	—	8	2	no	—
320	C₂³	—	14	2	no	—
321	C₂³	#6456	1	1	yes	characteristic
322	Q₈	—	14	2	no	—
323	Q₈	—	14	2	no	—
324	D₄	—	14	2	no	—
325	D₄	—	14	2	no	—
326	D₄	—	56	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	14	2	no	—
332	C₂×C₄	—	14	2	no	—
333	C₂×C₄	—	14	2	no	—
334	C₂×C₄	—	28	2	no	—
335	C₂×C₄	—	28	2	no	—
336	C₂×C₄	#6456	1	1	yes	characteristic
337	C₂×C₄	#6456	1	1	yes	characteristic
338	C₂×C₄	#6456	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#1161	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	2	no	—
346	C₈	—	14	2	no	—
347	C₈	—	14	2	no	—
348	C₈	—	28	1	no	—
349	C₈	—	28	1	no	—
350	C₈	#6470	1	2	yes	—
351	C₈	#6470	1	2	yes	—
352	C₇	#17974	1	1	yes	characteristic, complemented, Sylow
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	28	2	no	—
357	C₂²	—	28	2	no	—
358	C₂²	—	28	2	no	—
359	C₂²	C₂×#6456	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	28	2	no	—
365	C₄	—	28	2	no	—
366	C₄	—	28	2	no	—
367	C₄	C₂×#6470	1	1	yes	characteristic
368	C₄	C₂×#6470	1	1	yes	characteristic
369	C₄	#90453	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	56	2	no	—
372	C₂	#11736486	1	1	yes	characteristic
373	C₂	#11736551	1	1	yes	characteristic
374	C₂	#11736694	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×#1566	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#89887	C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1566	D₇
hypocenter	C₇	#17974
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
1. #11736498	C₂	semidirect
3. C₇×#1566	C₂	semidirect
4. #11728253	C₂	semidirect
5. #11727478	C₂	semidirect
6. #11726279	C₂	semidirect
7. #11726278	C₂	semidirect
8. #11726277	C₂	semidirect
9. #11726272	C₂	semidirect
10. #11726271	C₂	semidirect
11. #11726270	C₂	semidirect
101. #1566	D₇	semidirect
352. C₇	#17974	semidirect

Maximal subgroups

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

subgroup	index
#11736498	2
#11735914	2
C₇×#1566	2
#11728253	2
#11727478	2
#11726279	2
#11726278	2
#11726277	2
#11726272	2
#11726271	2
#11726270	2
#17974	7