Subgroups — #13257652

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89614	C₂	1	1	yes	maximal, characteristic
2	#11728224	C₂	1	1	yes	maximal, characteristic
3	C₂×#89276	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89274	C₂	1	1	yes	maximal, characteristic, complemented
5	#11727338	C₂	1	1	yes	maximal, characteristic
6	#11727336	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₈	C₂²	1	1	yes	characteristic
9	C₂×#6432	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₇×#384	C₂²	1	1	yes	characteristic
13	C₂×#6395	—	2	1	no	—
14	C₂×(C₂×D₇)⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₇)⋊C₈	—	2	1	no	—
16	C₂×#6391	—	2	1	no	—
17	C₂×#6383	—	2	1	no	—
18	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	#89614	—	2	2	no	—
21	#89614	—	2	2	no	—
22	#89324	—	2	1	no	—
23	#89324	—	2	1	no	—
24	#89324	—	2	1	no	—
25	#89276	—	2	2	no	—
26	#89274	—	2	2	no	—
27	#10481	—	7	1	no	maximal, Sylow
28	C₂²×D₂₈	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₇	—	2	1	no	—
30	C₂²×Dic₁₄	—	2	1	no	—
31	C₂²×C₇×C₈	—	2	1	no	—
32	C₂²×C₇×C₈	—	2	1	no	—
33	C₂×C₇×C₄⋊C₄	—	2	1	no	—
34	C₂×C₇×C₄⋊C₄	—	2	1	no	—
35	C₂×C₇×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₂×(C₂×D₇)⋊C₄	—	2	1	no	—
38	C₂×C₄⋊Dic₇	—	2	1	no	—
39	C₂×C₄⋊Dic₇	—	2	1	no	—
40	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
41	C₂²×C₇⋊C₈	—	2	1	no	—
42	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
44	C₄×D₂₈	—	2	1	no	—
45	C₄×D₂₈	—	2	1	no	—
46	C₄×D₂₈	—	2	1	no	—
47	C₄×D₂₈	—	2	1	no	—
48	#6432	—	2	1	no	—
49	#6432	—	2	1	no	—
50	#6432	—	2	1	no	—
51	#6432	D₄	1	2	yes	—
52	C₇×C₄⋊C₈	—	2	1	no	—
53	C₇×C₄⋊C₈	—	2	1	no	—
54	C₇×C₄⋊C₈	—	2	1	no	—
55	C₇×C₄⋊C₈	D₄	1	2	yes	—
56	#6395	—	2	4	no	—
57	(C₂×D₇)⋊C₈	—	4	2	no	—
58	(C₂×D₇)⋊C₈	—	4	2	no	—
59	#6391	—	2	4	no	—
60	#6383	—	4	2	no	—
61	C₄⋊(C₇⋊C₈)	—	2	1	no	—
62	C₄⋊(C₇⋊C₈)	—	2	1	no	—
63	C₄×C₇⋊C₈	—	2	1	no	—
64	C₄×C₇⋊C₈	—	2	1	no	—
65	C₂×#476	—	7	1	no	—
66	#1990	—	7	1	no	—
67	#1954	—	7	1	no	—
68	#1802	—	7	1	no	—
69	C₂×#335	—	7	1	no	—
70	C₂×#333	—	7	1	no	—
71	#1296	D₇	1	1	yes	characteristic, complemented
72	C₂³×D₇	—	2	1	no	—
73	C₂²×C₄×C₇	—	4	1	no	—
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	#41	1	1	yes	characteristic
77	C₂²×Dic₇	—	2	1	no	—
78	C₂²×Dic₇	—	4	1	no	—
79	C₂×D₂₈	—	2	1	no	—
80	C₂×D₂₈	—	2	1	no	—
81	C₂×D₂₈	—	2	1	no	—
82	C₂×D₂₈	—	2	1	no	—
83	C₂×D₂₈	—	2	1	no	—
84	C₂×D₂₈	Q₁₆	1	2	yes	—
85	C₂×C₄×D₇	—	4	1	no	—
86	C₂×C₄×D₇	—	4	1	no	—
87	C₂×C₄×D₇	—	4	1	no	—
88	C₂×C₄×D₇	—	4	1	no	—
89	C₂×C₄×D₇	—	4	1	no	—
90	C₂×C₄×D₇	—	4	1	no	—
91	C₂×Dic₁₄	—	2	2	no	—
92	C₂×Dic₁₄	—	4	1	no	—
93	C₂×Dic₁₄	—	8	1	no	—
94	C₂×Dic₁₄	—	8	1	no	—
95	C₂×C₇×C₈	—	2	2	no	—
96	C₂×C₇×C₈	—	4	1	no	—
97	C₂×C₇×C₈	—	4	1	no	—
98	C₂×C₇×C₈	—	4	1	no	—
99	C₂×C₇×C₈	—	4	1	no	—
100	C₂×C₇×C₈	—	4	1	no	—
101	C₇×C₄⋊C₄	—	4	1	no	—
102	C₇×C₄⋊C₄	—	4	1	no	—
103	C₇×C₄⋊C₄	—	4	1	no	—
104	C₇×C₄⋊C₄	—	4	1	no	—
105	C₇×C₄⋊C₄	—	8	1	no	—
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	#41	1	1	yes	characteristic
109	C₄²×C₇	#41	1	1	yes	characteristic
110	(C₂×D₇)⋊C₄	—	4	1	no	—
111	(C₂×D₇)⋊C₄	—	4	1	no	—
112	C₄⋊Dic₇	—	2	2	no	—
113	C₄⋊Dic₇	—	2	1	no	—
114	C₄⋊Dic₇	—	2	2	no	—
115	C₄⋊Dic₇	—	2	1	no	—
116	C₄⋊Dic₇	—	4	1	no	—
117	C₄⋊Dic₇	—	4	1	no	—
118	C₂×C₇⋊C₈	—	2	1	no	—
119	C₂×C₇⋊C₈	—	2	1	no	—
120	C₂×C₇⋊C₈	—	2	1	no	—
121	C₂×C₇⋊C₈	—	2	1	no	—
122	C₂×C₇⋊C₈	—	2	1	no	—
123	C₂×C₇⋊C₈	—	4	1	no	—
124	C₂×C₇⋊C₈	—	4	1	no	—
125	C₂×C₇⋊C₈	—	4	1	no	—
126	C₂×C₇⋊C₈	SD₁₆	1	2	yes	—
127	C₂×C₄⋊Q₈	—	7	1	no	—
128	C₂×C₄×D₄	—	7	1	no	—
129	#476	—	14	2	no	—
130	#476	—	14	2	no	—
131	C₂×#106	—	14	1	no	—
132	C₂×C₄⋊C₈	—	7	1	no	—
133	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
135	C₂×#102	—	14	1	no	—
136	C₂×#102	—	14	1	no	—
137	C₂×C₂²⋊C₈	—	14	1	no	—
138	C₂×C₂²⋊C₈	—	14	1	no	—
139	C₂×C₄×C₈	—	7	1	no	—
140	#384	C₂×D₇	1	1	yes	characteristic
141	#340	—	14	1	no	—
142	#340	—	14	1	no	—
143	#340	—	14	1	no	—
144	#335	—	14	2	no	—
145	#333	—	14	2	no	—
146	C₂³×C₇	#121	1	1	yes	characteristic
147	C₂²×D₇	—	2	2	no	—
148	C₂²×D₇	—	4	1	no	—
149	C₂²×D₇	—	4	1	no	—
150	C₂²×D₇	—	4	1	no	—
151	C₂²×D₇	—	4	1	no	—
152	C₂²×D₇	—	4	1	no	—
153	C₂²×D₇	—	4	1	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	1	no	—
156	C₂×C₄×C₇	—	8	1	no	—
157	C₂×C₄×C₇	—	8	1	no	—
158	C₂×C₄×C₇	—	8	1	no	—
159	C₂×C₄×C₇	#120	1	1	yes	characteristic
160	C₂×C₄×C₇	#121	1	1	yes	characteristic
161	C₂×C₄×C₇	#121	1	1	yes	characteristic
162	C₂×C₄×C₇	#123	1	1	yes	characteristic
163	C₂×C₄×C₇	#123	1	1	yes	characteristic
164	C₂×C₄×C₇	#124	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
167	C₂×C₄×C₇	#135	1	1	yes	characteristic
168	C₂×C₄×C₇	#135	1	1	yes	characteristic
169	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₄×C₇	#137	1	1	yes	characteristic
172	C₂×C₄×C₇	#137	1	1	yes	characteristic
173	C₂×Dic₇	—	4	2	no	—
174	C₂×Dic₇	—	4	1	no	—
175	C₂×Dic₇	—	4	1	no	—
176	C₂×Dic₇	—	4	1	no	—
177	C₂×Dic₇	—	8	1	no	—
178	C₂×Dic₇	—	8	1	no	—
179	D₂₈	—	2	2	no	—
180	D₂₈	—	2	2	no	—
181	D₂₈	—	4	1	no	—
182	D₂₈	—	4	1	no	—
183	C₄×D₇	—	8	1	no	—
184	C₄×D₇	—	8	1	no	—
185	C₄×D₇	—	8	1	no	—
186	C₄×D₇	—	8	1	no	—
187	Dic₁₄	—	8	2	no	—
188	Dic₁₄	—	8	2	no	—
189	C₇×C₈	—	4	2	no	—
190	C₇×C₈	—	8	1	no	—
191	C₇⋊C₈	—	2	2	no	—
192	C₇⋊C₈	—	4	1	no	—
193	C₇⋊C₈	—	8	1	no	—
194	C₂²×Q₈	—	14	1	no	—
195	C₂²×D₄	—	7	1	no	—
196	C₂³×C₄	—	14	1	no	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	7	1	no	—
200	C₂²×C₈	—	7	1	no	—
201	C₂²×C₈	—	14	1	no	—
202	C₄⋊Q₈	—	7	2	no	—
203	C₄⋊Q₈	—	14	1	no	—
204	C₄⋊Q₈	—	14	1	no	—
205	C₄⋊Q₈	—	14	1	no	—
206	C₄×D₄	—	14	1	no	—
207	C₄×D₄	—	14	1	no	—
208	C₄×D₄	—	14	1	no	—
209	C₄×D₄	—	14	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	7	1	no	—
214	C₂×C₄⋊C₄	—	14	1	no	—
215	C₂×C₄⋊C₄	—	14	1	no	—
216	C₂×C₂²⋊C₄	—	14	1	no	—
217	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
218	#106	—	14	4	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	—	14	1	no	—
223	C₄⋊C₈	—	14	1	no	—
224	C₄⋊C₈	D₂₈	1	2	yes	—
225	#102	—	14	4	no	—
226	#102	—	28	2	no	—
227	C₂²⋊C₈	—	28	2	no	—
228	C₂²⋊C₈	—	28	2	no	—
229	C₄×C₈	—	14	1	no	—
230	C₄×C₈	—	14	1	no	—
231	C₂²×C₇	#394	1	1	yes	characteristic
232	C₂²×C₇	#461	1	1	yes	characteristic
233	C₂²×C₇	#463	1	1	yes	characteristic
234	C₂²×C₇	#467	1	1	yes	characteristic
235	C₂²×C₇	#468	1	1	yes	characteristic
236	C₂²×C₇	#471	1	1	yes	characteristic
237	C₂²×C₇	#472	1	1	yes	characteristic
238	C₂×D₇	—	4	2	no	—
239	C₂×D₇	—	4	2	no	—
240	C₂×D₇	—	4	2	no	—
241	C₂×D₇	—	8	1	no	—
242	C₂×D₇	—	8	1	no	—
243	C₂×D₇	—	8	1	no	—
244	C₂×D₇	—	8	1	no	—
245	C₄×C₇	—	2	1	no	—
246	C₄×C₇	—	2	1	no	—
247	C₄×C₇	—	16	1	no	—
248	C₄×C₇	#449	1	1	yes	characteristic
249	C₄×C₇	#451	1	1	yes	characteristic
250	C₄×C₇	#460	1	1	yes	characteristic
251	C₄×C₇	#464	1	1	yes	characteristic
252	C₄×C₇	#465	1	1	yes	characteristic
253	C₄×C₇	#468	1	1	yes	characteristic
254	C₄×C₇	#484	1	1	yes	characteristic
255	C₄×C₇	#485	1	1	yes	characteristic
256	Dic₇	—	8	2	no	—
257	Dic₇	—	8	1	no	—
258	C₂⁴	—	14	1	no	—
259	C₂×Q₈	—	14	2	no	—
260	C₂×Q₈	—	28	1	no	—
261	C₂×Q₈	—	56	1	no	—
262	C₂×Q₈	—	56	1	no	—
263	C₂×D₄	—	7	2	no	—
264	C₂×D₄	—	14	1	no	—
265	C₂×D₄	—	14	1	no	—
266	C₂×D₄	—	14	1	no	—
267	C₂×D₄	—	14	1	no	—
268	C₂×D₄	—	14	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	14	1	no	—
271	C₂²×C₄	—	28	1	no	—
272	C₂²×C₄	—	28	1	no	—
273	C₂²×C₄	—	28	1	no	—
274	C₂²×C₄	—	28	1	no	—
275	C₂²×C₄	—	28	1	no	—
276	C₂²×C₄	—	28	1	no	—
277	C₂²×C₄	—	28	1	no	—
278	C₂²×C₄	#1157	1	1	yes	characteristic
279	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	7	2	no	—
288	C₂×C₈	—	14	1	no	—
289	C₂×C₈	—	14	1	no	—
290	C₂×C₈	—	14	1	no	—
291	C₂×C₈	—	14	1	no	—
292	C₂×C₈	—	14	1	no	—
293	C₂×C₈	—	28	1	no	—
294	C₂×C₈	—	28	1	no	—
295	C₂×C₈	—	28	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	8	1	no	—
301	C₄⋊C₄	—	14	2	no	—
302	C₄⋊C₄	—	14	1	no	—
303	C₄⋊C₄	—	14	2	no	—
304	C₄⋊C₄	—	14	1	no	—
305	C₄⋊C₄	—	28	1	no	—
306	C₄⋊C₄	—	28	1	no	—
307	C₂²⋊C₄	—	28	1	no	—
308	C₂²⋊C₄	—	28	1	no	—
309	C₄²	C₂×D₂₈	1	1	yes	characteristic
310	C₄²	#1157	1	1	yes	characteristic
311	C₄²	D₄×D₇	1	1	yes	characteristic
312	C₄²	#1161	1	1	yes	characteristic
313	C₂×C₇	#1675	1	1	yes	characteristic
314	C₂×C₇	#1683	1	1	yes	characteristic
315	C₂×C₇	#1691	1	1	yes	characteristic
316	C₂×C₇	#1697	1	1	yes	characteristic
317	C₂×C₇	#2036	1	1	yes	characteristic
318	C₂×C₇	#2051	1	1	yes	characteristic
319	C₂×C₇	#2055	1	1	yes	characteristic
320	D₇	—	8	2	no	—
321	C₂³	—	14	2	no	—
322	C₂³	—	28	1	no	—
323	C₂³	—	28	1	no	—
324	C₂³	—	28	1	no	—
325	C₂³	—	28	1	no	—
326	C₂³	—	28	1	no	—
327	C₂³	—	28	1	no	—
328	C₂³	#6447	1	1	yes	characteristic
329	Q₈	—	56	2	no	—
330	Q₈	—	56	2	no	—
331	D₄	—	14	2	no	—
332	D₄	—	14	2	no	—
333	D₄	—	28	1	no	—
334	D₄	—	28	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	28	1	no	—
341	C₂×C₄	—	28	2	no	—
342	C₂×C₄	—	28	1	no	—
343	C₂×C₄	—	28	1	no	—
344	C₂×C₄	—	56	1	no	—
345	C₂×C₄	—	56	1	no	—
346	C₂×C₄	—	56	1	no	—
347	C₂×C₄	—	56	1	no	—
348	C₂×C₄	—	56	1	no	—
349	C₂×C₄	—	56	1	no	—
350	C₂×C₄	#6438	1	1	yes	characteristic
351	C₂×C₄	#6445	1	1	yes	characteristic
352	C₂×C₄	#6447	1	1	yes	characteristic
353	C₂×C₄	#6457	1	1	yes	characteristic
354	C₂×C₄	#6457	1	1	yes	characteristic
355	C₂×C₄	#6458	1	1	yes	characteristic
356	C₂×C₄	#6474	1	1	yes	characteristic
357	C₂×C₄	#6474	1	1	yes	characteristic
358	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
359	C₂×C₄	#6478	1	1	yes	characteristic
360	C₂×C₄	#6479	1	1	yes	characteristic
361	C₂×C₄	#6479	1	1	yes	characteristic
362	C₂×C₄	D₇×Q₁₆	1	1	yes	characteristic
363	C₂×C₄	#6481	1	1	yes	characteristic
364	C₈	—	4	2	no	—
365	C₈	—	8	1	no	—
366	C₈	—	14	2	no	—
367	C₈	—	28	1	no	—
368	C₈	—	56	1	no	—
369	C₇	#10481	1	1	yes	characteristic, complemented, Sylow
370	C₂²	—	28	2	no	—
371	C₂²	—	28	2	no	—
372	C₂²	—	28	2	no	—
373	C₂²	—	56	1	no	—
374	C₂²	—	56	1	no	—
375	C₂²	—	56	1	no	—
376	C₂²	—	56	1	no	—
377	C₂²	#89444	1	1	yes	characteristic
378	C₂²	#89521	1	1	yes	characteristic
379	C₂²	#89526	1	1	yes	characteristic
380	C₂²	#89552	1	1	yes	characteristic
381	C₂²	#89559	1	1	yes	characteristic
382	C₂²	#89585	1	1	yes	characteristic
383	C₂²	#89590	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	16	1	no	—
387	C₄	—	56	1	no	—
388	C₄	—	56	2	no	—
389	C₄	#89534	1	1	yes	characteristic
390	C₄	#89553	1	1	yes	characteristic
391	C₄	#89570	1	1	yes	characteristic
392	C₄	#89580	1	1	yes	characteristic
393	C₄	#89636	1	1	yes	characteristic
394	C₄	#89639	1	1	yes	characteristic
395	C₄	#89656	1	1	yes	characteristic
396	C₄	#89658	1	1	yes	characteristic
397	C₂	—	56	2	no	—
398	C₂	#11726154	1	1	yes	characteristic
399	C₂	#11726201	1	1	yes	characteristic
400	C₂	#11726204	1	1	yes	characteristic
401	C₂	#11726251	1	1	yes	characteristic
402	C₂	#11726579	1	1	yes	characteristic
403	C₂	#11726617	1	1	yes	characteristic
404	C₂	#11726740	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6447
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1296	C₂
socle	C₂³×C₇	#121
prime radical	C₂²×D₂₈	D₄
hypercenter	#1296	D₇
hypocenter	C₇	#10481
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
3. C₂×#89276	C₂	semidirect
4. C₂×#89274	C₂	semidirect
6. #11727336	C₂	semidirect
7. C₇×#1296	C₂	semidirect
71. #1296	D₇	semidirect
369. C₇	#10481	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89614	2
#11728224	2
C₂×#89276	2
C₂×#89274	2
#11727338	2
#11727336	2
C₇×#1296	2
#10481	7