Subgroups — #13258342

Order 1792.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11728222	C₂	1	1	yes	maximal, characteristic, complemented
2	#11728211	C₂	1	1	yes	maximal, characteristic
3	C₂×#89280	C₂	1	1	yes	maximal, characteristic
4	C₂×#89274	C₂	1	1	yes	maximal, characteristic, complemented
5	#11727334	C₂	1	1	yes	maximal, characteristic
6	#11727321	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6437	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₇×#384	C₂²	1	1	yes	characteristic
12	C₂×#6395	—	2	1	no	—
13	C₂×#6395	—	2	1	no	—
14	C₂×Dic₇⋊C₈	—	2	1	no	—
15	C₂×#6383	—	2	1	no	—
16	C₂×#6383	—	2	1	no	—
17	C₂×#6381	—	2	1	no	—
18	C₄×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	#89345	—	2	1	no	—
22	#89324	—	2	1	no	—
23	#89280	—	2	2	no	—
24	#89274	—	2	2	no	—
25	#10591	—	7	1	no	maximal, Sylow
26	C₂²×D₂₈	—	2	1	no	—
27	C₂²×D₂₈	—	2	1	no	—
28	C₂²×D₂₈	—	2	1	no	—
29	C₂²×C₇×C₈	—	2	1	no	—
30	C₂²×C₇×C₈	—	2	1	no	—
31	C₂×C₇×C₄⋊C₄	—	2	1	no	—
32	C₂×C₇×C₄⋊C₄	—	2	1	no	—
33	C₂×C₇×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
36	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇	—	2	1	no	—
38	C₂²×C₇⋊C₈	—	2	1	no	—
39	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
41	#6437	—	2	1	no	—
42	#6437	—	2	1	no	—
43	#6437	—	2	1	no	—
44	#6437	Q₈	1	2	yes	—
45	C₇×C₄⋊C₈	—	2	1	no	—
46	C₇×C₄⋊C₈	—	2	1	no	—
47	C₇×C₄⋊C₈	—	2	1	no	—
48	C₇×C₄⋊C₈	—	2	1	no	—
49	C₇⋊#95	—	2	1	no	—
50	#6395	—	4	2	no	—
51	#6395	—	4	2	no	—
52	Dic₇⋊C₈	—	4	2	no	—
53	#6383	—	4	2	no	—
54	#6383	—	4	2	no	—
55	#6381	—	4	2	no	—
56	C₄⋊(C₇⋊C₈)	—	2	1	no	—
57	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
58	C₄×C₇⋊C₈	—	2	1	no	—
59	C₄×C₇⋊C₈	—	2	1	no	—
60	#1982	—	7	1	no	—
61	#1957	—	7	1	no	—
62	#1947	—	7	1	no	—
63	#1939	—	7	1	no	—
64	C₂×#335	—	7	1	no	—
65	C₂×#331	—	7	1	no	—
66	#1296	D₇	1	1	yes	characteristic, complemented
67	C₂³×D₇	—	4	1	no	—
68	C₂²×C₄×C₇	—	4	1	no	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	#41	1	1	yes	characteristic
72	C₂²×Dic₇	—	2	1	no	—
73	C₂²×Dic₇	—	2	1	no	—
74	C₂×D₂₈	—	2	2	no	—
75	C₂×D₂₈	—	2	2	no	—
76	C₂×D₂₈	—	2	2	no	—
77	C₂×D₂₈	—	4	1	no	—
78	C₂×D₂₈	—	4	1	no	—
79	C₂×D₂₈	—	4	1	no	—
80	C₂×D₂₈	—	4	1	no	—
81	C₂×D₂₈	—	4	1	no	—
82	C₂×D₂₈	—	4	1	no	—
83	C₂×D₂₈	—	4	1	no	—
84	C₂×D₂₈	—	4	1	no	—
85	C₂×D₂₈	—	4	1	no	—
86	C₂×D₂₈	—	4	1	no	—
87	C₂×D₂₈	—	4	1	no	—
88	C₂×D₂₈	—	8	1	no	—
89	C₂×D₂₈	—	8	1	no	—
90	C₂×C₇×C₈	—	4	1	no	—
91	C₂×C₇×C₈	—	4	1	no	—
92	C₂×C₇×C₈	—	4	1	no	—
93	C₂×C₇×C₈	—	4	1	no	—
94	C₂×C₇×C₈	—	4	1	no	—
95	C₂×C₇×C₈	—	4	1	no	—
96	C₇×C₄⋊C₄	—	4	1	no	—
97	C₇×C₄⋊C₄	—	4	1	no	—
98	C₇×C₄⋊C₄	—	4	1	no	—
99	C₇×C₄⋊C₄	—	4	1	no	—
100	C₇×C₄⋊C₄	—	8	1	no	—
101	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
103	C₄²×C₇	#41	1	1	yes	characteristic
104	C₄²×C₇	#41	1	1	yes	characteristic
105	C₄⋊Dic₇	—	2	1	no	—
106	C₄⋊Dic₇	—	2	1	no	—
107	C₄⋊Dic₇	—	2	1	no	—
108	C₄⋊Dic₇	D₈	1	2	yes	—
109	C₄×Dic₇	—	4	1	no	—
110	C₄×Dic₇	—	4	1	no	—
111	C₂×C₇⋊C₈	—	2	1	no	—
112	C₂×C₇⋊C₈	—	2	1	no	—
113	C₂×C₇⋊C₈	—	2	2	no	—
114	C₂×C₇⋊C₈	—	2	1	no	—
115	C₂×C₇⋊C₈	—	2	1	no	—
116	C₂×C₇⋊C₈	—	2	1	no	—
117	C₂×C₇⋊C₈	—	4	1	no	—
118	C₂×C₇⋊C₈	—	4	1	no	—
119	C₂×C₇⋊C₈	D₈	1	2	yes	—
120	C₂×#127	—	7	1	no	—
121	C₂×#107	—	14	1	no	—
122	C₂×C₄⋊C₈	—	7	1	no	—
123	C₂×C₄⋊C₈	—	14	1	no	—
124	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
126	C₂×#102	—	14	1	no	—
127	C₂×#102	—	14	1	no	—
128	C₂×#102	—	14	1	no	—
129	C₂×#102	—	14	1	no	—
130	C₂×C₄×C₈	—	7	1	no	—
131	#384	C₂×D₇	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	7	1	no	—
133	#340	—	14	1	no	—
134	#336	—	14	1	no	—
135	#335	—	14	2	no	—
136	#331	—	14	2	no	—
137	C₂³×C₇	#121	1	1	yes	characteristic
138	C₂²×D₇	—	4	2	no	—
139	C₂²×D₇	—	8	1	no	—
140	C₂²×D₇	—	8	1	no	—
141	C₂²×D₇	—	8	1	no	—
142	C₂²×D₇	—	8	1	no	—
143	C₂²×D₇	—	8	1	no	—
144	C₂²×D₇	—	8	1	no	—
145	C₂×C₄×C₇	—	2	1	no	—
146	C₂×C₄×C₇	—	2	1	no	—
147	C₂×C₄×C₇	—	8	1	no	—
148	C₂×C₄×C₇	—	8	1	no	—
149	C₂×C₄×C₇	—	8	1	no	—
150	C₂×C₄×C₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₇	#126	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₇	#135	1	1	yes	characteristic
159	C₂×C₄×C₇	#135	1	1	yes	characteristic
160	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄×C₇	#137	1	1	yes	characteristic
163	C₂×C₄×C₇	#137	1	1	yes	characteristic
164	C₂×Dic₇	—	2	2	no	—
165	C₂×Dic₇	—	4	1	no	—
166	C₂×Dic₇	—	4	1	no	—
167	C₂×Dic₇	—	4	1	no	—
168	C₂×Dic₇	—	4	1	no	—
169	C₂×Dic₇	—	4	1	no	—
170	D₂₈	—	4	2	no	—
171	D₂₈	—	4	2	no	—
172	D₂₈	—	4	2	no	—
173	D₂₈	—	4	2	no	—
174	D₂₈	—	8	2	no	—
175	D₂₈	—	8	2	no	—
176	D₂₈	—	8	1	no	—
177	D₂₈	—	8	1	no	—
178	D₂₈	—	8	1	no	—
179	D₂₈	—	8	1	no	—
180	C₇×C₈	—	8	1	no	—
181	C₇×C₈	—	8	1	no	—
182	C₇⋊C₈	—	2	2	no	—
183	C₇⋊C₈	—	4	2	no	—
184	C₇⋊C₈	—	4	1	no	—
185	C₂²×D₄	—	14	1	no	—
186	C₂²×D₄	—	14	1	no	—
187	C₂²×D₄	—	14	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	7	1	no	—
191	C₂²×C₈	—	7	1	no	—
192	C₂²×C₈	—	14	1	no	—
193	#127	—	7	2	no	—
194	#127	—	14	1	no	—
195	#127	—	14	1	no	—
196	#127	—	14	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	7	1	no	—
201	C₂×C₄⋊C₄	—	7	1	no	—
202	C₂×C₄²	—	14	1	no	—
203	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
204	#107	—	28	2	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	7	2	no	—
210	C₄⋊C₈	—	14	1	no	—
211	C₄⋊C₈	—	28	2	no	—
212	#102	—	28	2	no	—
213	#102	—	28	2	no	—
214	#102	—	28	2	no	—
215	#102	—	28	2	no	—
216	C₄×C₈	—	14	1	no	—
217	C₄×C₈	—	14	1	no	—
218	#95	—	14	1	no	—
219	C₂²×C₇	#396	1	1	yes	characteristic
220	C₂²×C₇	#459	1	1	yes	characteristic
221	C₂²×C₇	#464	1	1	yes	characteristic
222	C₂²×C₇	#466	1	1	yes	characteristic
223	C₂²×C₇	#468	1	1	yes	characteristic
224	C₂²×C₇	#471	1	1	yes	characteristic
225	C₂²×C₇	#472	1	1	yes	characteristic
226	C₂×D₇	—	8	2	no	—
227	C₂×D₇	—	8	2	no	—
228	C₂×D₇	—	8	2	no	—
229	C₂×D₇	—	16	1	no	—
230	C₂×D₇	—	16	1	no	—
231	C₂×D₇	—	16	1	no	—
232	C₂×D₇	—	16	1	no	—
233	C₄×C₇	—	2	1	no	—
234	C₄×C₇	—	2	1	no	—
235	C₄×C₇	—	16	1	no	—
236	C₄×C₇	#459	1	1	yes	characteristic
237	C₄×C₇	#464	1	1	yes	characteristic
238	C₄×C₇	#466	1	1	yes	characteristic
239	C₄×C₇	#468	1	1	yes	characteristic
240	C₄×C₇	#474	1	1	yes	characteristic
241	C₄×C₇	#474	1	1	yes	characteristic
242	C₄×C₇	#478	1	1	yes	characteristic
243	C₄×C₇	#478	1	1	yes	characteristic
244	Dic₇	—	4	2	no	—
245	Dic₇	—	8	1	no	—
246	C₂⁴	—	28	1	no	—
247	C₂×D₄	—	14	2	no	—
248	C₂×D₄	—	14	2	no	—
249	C₂×D₄	—	14	2	no	—
250	C₂×D₄	—	28	1	no	—
251	C₂×D₄	—	28	1	no	—
252	C₂×D₄	—	28	1	no	—
253	C₂×D₄	—	28	1	no	—
254	C₂×D₄	—	28	1	no	—
255	C₂×D₄	—	28	1	no	—
256	C₂×D₄	—	28	1	no	—
257	C₂×D₄	—	28	1	no	—
258	C₂×D₄	—	28	1	no	—
259	C₂×D₄	—	28	1	no	—
260	C₂×D₄	—	28	1	no	—
261	C₂×D₄	—	56	1	no	—
262	C₂×D₄	—	56	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	14	1	no	—
265	C₂²×C₄	—	14	1	no	—
266	C₂²×C₄	#1157	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	7	2	no	—
276	C₂×C₈	—	14	1	no	—
277	C₂×C₈	—	14	1	no	—
278	C₂×C₈	—	14	1	no	—
279	C₂×C₈	—	14	2	no	—
280	C₂×C₈	—	14	1	no	—
281	C₂×C₈	—	14	1	no	—
282	C₂×C₈	—	28	1	no	—
283	C₂×C₈	—	28	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	7	2	no	—
289	C₄⋊C₄	—	8	1	no	—
290	C₄⋊C₄	—	14	1	no	—
291	C₄⋊C₄	—	14	1	no	—
292	C₄⋊C₄	—	14	1	no	—
293	C₄²	—	28	1	no	—
294	C₄²	—	28	1	no	—
295	C₄²	#1157	1	1	yes	characteristic
296	C₄²	#1157	1	1	yes	characteristic
297	C₄²	D₄×D₇	1	1	yes	characteristic
298	C₄²	Q₈×D₇	1	1	yes	characteristic
299	C₂×C₇	#1667	1	1	yes	characteristic
300	C₂×C₇	#1682	1	1	yes	characteristic
301	C₂×C₇	#1685	1	1	yes	characteristic
302	C₂×C₇	#1698	1	1	yes	characteristic
303	C₂×C₇	#2063	1	1	yes	characteristic
304	C₂×C₇	#2067	1	1	yes	characteristic
305	C₂×C₇	#2067	1	1	yes	characteristic
306	D₇	—	16	2	no	—
307	C₂³	—	28	2	no	—
308	C₂³	—	56	1	no	—
309	C₂³	—	56	1	no	—
310	C₂³	—	56	1	no	—
311	C₂³	—	56	1	no	—
312	C₂³	—	56	1	no	—
313	C₂³	—	56	1	no	—
314	C₂³	#6447	1	1	yes	characteristic
315	D₄	—	28	2	no	—
316	D₄	—	28	2	no	—
317	D₄	—	28	2	no	—
318	D₄	—	28	2	no	—
319	D₄	—	56	2	no	—
320	D₄	—	56	2	no	—
321	D₄	—	56	1	no	—
322	D₄	—	56	1	no	—
323	D₄	—	56	1	no	—
324	D₄	—	56	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	14	2	no	—
331	C₂×C₄	—	28	1	no	—
332	C₂×C₄	—	28	1	no	—
333	C₂×C₄	—	28	1	no	—
334	C₂×C₄	—	28	1	no	—
335	C₂×C₄	—	28	1	no	—
336	C₂×C₄	#6439	1	1	yes	characteristic
337	C₂×C₄	#6447	1	1	yes	characteristic
338	C₂×C₄	#6447	1	1	yes	characteristic
339	C₂×C₄	#6459	1	1	yes	characteristic
340	C₂×C₄	#6459	1	1	yes	characteristic
341	C₂×C₄	#6459	1	1	yes	characteristic
342	C₂×C₄	D₇×D₈	1	1	yes	characteristic
343	C₂×C₄	D₇×D₈	1	1	yes	characteristic
344	C₂×C₄	#6477	1	1	yes	characteristic
345	C₂×C₄	#6477	1	1	yes	characteristic
346	C₂×C₄	#6479	1	1	yes	characteristic
347	C₂×C₄	#6479	1	1	yes	characteristic
348	C₂×C₄	#6481	1	1	yes	characteristic
349	C₂×C₄	#6481	1	1	yes	characteristic
350	C₈	—	8	1	no	—
351	C₈	—	8	1	no	—
352	C₈	—	14	2	no	—
353	C₈	—	28	2	no	—
354	C₈	—	28	1	no	—
355	C₇	#10591	1	1	yes	characteristic, complemented, Sylow
356	C₂²	—	56	2	no	—
357	C₂²	—	56	2	no	—
358	C₂²	—	56	2	no	—
359	C₂²	—	112	1	no	—
360	C₂²	—	112	1	no	—
361	C₂²	—	112	1	no	—
362	C₂²	—	112	1	no	—
363	C₂²	#89448	1	1	yes	characteristic
364	C₂²	#89521	1	1	yes	characteristic
365	C₂²	#89526	1	1	yes	characteristic
366	C₂²	#89547	1	1	yes	characteristic
367	C₂²	#89558	1	1	yes	characteristic
368	C₂²	#89570	1	1	yes	characteristic
369	C₂²	#89584	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	16	1	no	—
373	C₄	—	28	2	no	—
374	C₄	—	56	1	no	—
375	C₄	#89547	1	1	yes	characteristic
376	C₄	#89558	1	1	yes	characteristic
377	C₄	#89570	1	1	yes	characteristic
378	C₄	#89584	1	1	yes	characteristic
379	C₄	#89643	1	1	yes	characteristic
380	C₄	#89643	1	1	yes	characteristic
381	C₄	#89662	1	1	yes	characteristic
382	C₄	#89662	1	1	yes	characteristic
383	C₂	—	112	2	no	—
384	C₂	#11726149	1	1	yes	characteristic
385	C₂	#11726202	1	1	yes	characteristic
386	C₂	#11726224	1	1	yes	characteristic
387	C₂	#11726268	1	1	yes	characteristic
388	C₂	#11726655	1	1	yes	characteristic
389	C₂	#11726655	1	1	yes	characteristic
390	C₂	#11726750	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	C₂×#6437	C₂²
hypercenter	#1296	D₇
hypocenter	C₇	#10591
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
1. #11728222	C₂	semidirect
4. C₂×#89274	C₂	semidirect
6. #11727321	C₂	semidirect
7. C₇×#1296	C₂	semidirect
66. #1296	D₇	semidirect
355. C₇	#10591	semidirect

Maximal subgroups

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

subgroup	index
#11728222	2
#11728211	2
C₂×#89280	2
C₂×#89274	2
#11727334	2
#11727321	2
C₇×#1296	2
#10591	7