Subgroups — #13324650

Order 1792.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736735	C₂	1	1	yes	maximal, characteristic
2	#11736711	C₂	1	1	yes	maximal, characteristic, complemented
3	#11735913	C₂	1	1	yes	maximal, characteristic
4	C₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
5	#11728525	C₂	1	1	yes	maximal, characteristic, complemented
6	#11728519	C₂	1	1	yes	maximal, characteristic
7	#11727486	C₂	1	1	yes	maximal, characteristic, complemented
8	#11725842	C₂	1	2	yes	maximal, complemented
9	#11725839	C₂	1	2	yes	maximal, complemented
10	#11725834	C₂	1	2	yes	maximal, complemented
11	#11725833	C₂	1	2	yes	maximal, complemented
12	#90168	C₂²	1	1	yes	characteristic
13	#90161	C₂²	1	1	yes	characteristic
14	C₇×#431	C₂²	1	1	yes	characteristic
15	C₇×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#89886	C₄	1	2	yes	—
18	#89886	C₄	1	2	yes	—
19	#89698	C₂²	1	1	yes	characteristic
20	#89698	C₂²	1	1	yes	characteristic
21	#89608	—	2	2	no	—
22	#89600	—	2	2	no	—
23	#89499	—	2	2	no	—
24	#89492	—	2	2	no	—
25	#89466	C₄	1	1	yes	characteristic
26	#89465	C₄	1	1	yes	characteristic
27	C₄×#1132	C₂²	1	2	yes	—
28	C₄×#1132	C₂²	1	2	yes	—
29	#89460	C₂²	1	2	yes	—
30	C₈×D₂₈	C₂²	1	2	yes	—
31	#89453	C₄	1	1	yes	characteristic
32	#89452	C₄	1	1	yes	characteristic
33	#89451	C₂²	1	2	yes	—
34	C₈×Dic₁₄	C₂²	1	2	yes	—
35	C₇×#322	C₂²	1	1	yes	characteristic
36	C₇×#322	C₂²	1	1	yes	characteristic
37	C₇×#322	C₂²	1	1	yes	characteristic
38	C₇×#322	C₂²	1	1	yes	characteristic
39	#89254	C₂²	1	2	yes	—
40	#89254	C₂²	1	2	yes	—
41	#89253	C₂²	1	2	yes	—
42	#89252	C₂²	1	2	yes	—
43	#89251	C₂²	1	2	yes	—
44	#89251	C₂²	1	2	yes	—
45	#17990	—	7	1	no	maximal, Sylow
46	C₂×#1157	—	2	1	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	—	2	1	no	—
51	#6493	C₂×C₄	1	2	yes	—
52	#6489	—	2	1	no	—
53	#6487	—	2	1	no	—
54	#6487	C₂×C₄	1	2	yes	—
55	C₂×#1132	C₂×C₄	1	2	yes	—
56	C₂×#1132	C₂×C₄	1	2	yes	—
57	C₂×#1131	—	2	2	no	—
58	C₂×C₈×D₇	—	2	2	no	—
59	#6438	—	2	1	no	—
60	#6437	C₂×C₄	1	1	yes	characteristic
61	C₄×D₂₈	—	2	1	no	—
62	C₄×D₂₈	C₂³	1	2	yes	—
63	#6433	—	2	1	no	—
64	#6432	C₂×C₄	1	1	yes	characteristic
65	#6432	C₂×C₄	1	1	yes	characteristic
66	#6432	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₁₄	—	2	1	no	—
68	C₄×Dic₁₄	C₂³	1	2	yes	—
69	C₇×C₄⋊C₈	—	2	1	no	—
70	C₇×C₄⋊C₈	D₄	1	2	yes	—
71	C₇×C₂²⋊C₈	—	2	1	no	—
72	C₇×C₂²⋊C₈	D₄	1	2	yes	—
73	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
74	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
82	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
83	#6395	C₂×C₄	1	2	yes	—
84	#6395	C₂×C₄	1	2	yes	—
85	(C₂×D₇)⋊C₈	—	2	2	no	—
86	(C₂×D₇)⋊C₈	—	2	2	no	—
87	#6391	C₂×C₄	1	2	yes	—
88	#6391	C₂×C₄	1	2	yes	—
89	#6390	C₂×C₄	1	2	yes	—
90	#6390	C₂×C₄	1	2	yes	—
91	#6389	—	2	2	no	—
92	Dic₇⋊C₈	—	2	2	no	—
93	Dic₇⋊C₈	—	2	2	no	—
94	C₈×Dic₇	—	2	2	no	—
95	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
96	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
97	#6377	—	2	1	no	—
98	#6377	—	2	1	no	—
99	#3167	—	7	1	no	—
100	#2985	—	7	1	no	—
101	#2981	—	7	1	no	—
102	#1588	—	7	2	no	—
103	#1585	—	7	2	no	—
104	#1583	—	7	2	no	—
105	#1582	—	7	2	no	—
106	#1568	—	7	1	no	—
107	#1497	—	7	1	no	—
108	#1496	—	7	1	no	—
109	#1452	D₇	1	1	yes	characteristic, complemented
110	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇	#41	1	1	yes	characteristic
113	C₂×C₂²⋊D₇	—	2	2	no	—
114	#1157	—	8	2	no	—
115	C₂×D₂₈	—	2	1	no	—
116	C₂×D₂₈	—	2	1	no	—
117	C₂×D₂₈	C₂²×C₄	1	2	yes	—
118	C₂×C₄×D₇	—	2	2	no	—
119	C₂×Dic₁₄	—	2	1	no	—
120	C₂×Dic₁₄	—	2	1	no	—
121	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
122	C₂×C₇×C₈	—	2	1	no	—
123	C₂×C₇×C₈	—	2	1	no	—
124	C₂×C₇×C₈	—	4	1	no	—
125	C₂×C₇×C₈	—	4	1	no	—
126	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
131	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
132	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₇	#41	1	1	yes	characteristic
135	C₄²×C₇	C₂⁴	1	1	yes	characteristic
136	C₂²⋊Dic₇	—	2	2	no	—
137	(C₂×D₇)⋊C₄	—	2	2	no	—
138	(C₂×D₇)⋊C₄	—	2	2	no	—
139	C₄⋊Dic₇	—	2	1	no	—
140	C₄⋊Dic₇	—	2	1	no	—
141	C₄⋊Dic₇	—	2	1	no	—
142	C₄⋊Dic₇	—	2	1	no	—
143	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
144	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
145	Dic₇⋊C₄	—	2	2	no	—
146	Dic₇⋊C₄	—	2	2	no	—
147	C₄×Dic₇	—	2	2	no	—
148	C₂×C₇⋊C₈	—	2	2	no	—
149	C₂×C₇⋊C₈	—	2	2	no	—
150	#1132	—	4	2	no	—
151	#1132	—	4	2	no	—
152	#1131	—	8	2	no	—
153	C₈×D₇	—	8	2	no	—
154	#533	—	7	1	no	—
155	#532	—	7	1	no	—
156	#498	—	7	1	no	—
157	#492	—	7	1	no	—
158	#487	—	7	1	no	—
159	#486	—	7	1	no	—
160	#465	—	7	2	no	—
161	#465	—	7	2	no	—
162	#446	—	7	2	no	—
163	#446	—	14	2	no	—
164	C₈×Q₈	—	7	2	no	—
165	C₄×SD₁₆	—	7	2	no	—
166	C₄×SD₁₆	—	7	2	no	—
167	#436	—	7	2	no	—
168	#435	—	14	2	no	—
169	C₈×D₄	—	7	2	no	—
170	#433	—	14	2	no	—
171	#432	—	7	1	no	—
172	#432	—	7	1	no	—
173	#431	C₂×D₇	1	1	yes	characteristic
174	#431	C₂×D₇	1	1	yes	characteristic
175	#408	—	14	2	no	—
176	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
177	#335	—	7	2	no	—
178	#334	—	7	2	no	—
179	Q₈⋊C₈	—	7	2	no	—
180	Q₈⋊C₈	—	7	2	no	—
181	D₄⋊C₈	—	7	2	no	—
182	D₄⋊C₈	—	7	2	no	—
183	#322	C₂×D₇	1	1	yes	characteristic
184	#322	C₂×D₇	1	1	yes	characteristic
185	#322	C₂×D₇	1	1	yes	characteristic
186	#322	C₂×D₇	1	1	yes	characteristic
187	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
188	C₂²×D₇	—	2	2	no	—
189	C₂×C₄×C₇	—	2	1	no	—
190	C₂×C₄×C₇	—	2	1	no	—
191	C₂×C₄×C₇	—	2	2	no	—
192	C₂×C₄×C₇	—	4	1	no	—
193	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
199	C₂²⋊D₇	—	8	2	no	—
200	C₂×Dic₇	—	2	2	no	—
201	C₂×Dic₇	—	2	2	no	—
202	C₂×Dic₇	—	2	2	no	—
203	D₂₈	—	2	2	no	—
204	D₂₈	—	2	2	no	—
205	D₂₈	—	8	1	no	—
206	D₂₈	—	8	1	no	—
207	C₄×D₇	—	4	2	no	—
208	C₄×D₇	—	4	2	no	—
209	Dic₁₄	—	2	2	no	—
210	Dic₁₄	—	2	2	no	—
211	Dic₁₄	—	8	1	no	—
212	Dic₁₄	—	8	1	no	—
213	C₇×C₈	—	2	1	no	—
214	C₇×C₈	—	2	1	no	—
215	C₇×C₈	—	2	1	no	—
216	C₇×C₈	—	2	1	no	—
217	C₇×C₈	—	2	2	no	—
218	C₇×C₈	—	2	2	no	—
219	C₇×C₈	—	4	1	no	—
220	C₇×C₈	—	4	1	no	—
221	C₇⋊C₈	—	4	2	no	—
222	C₇⋊C₈	—	4	2	no	—
223	C₂×#41	—	14	1	no	—
224	C₂×SD₁₆	—	7	2	no	—
225	C₂×SD₁₆	—	7	2	no	—
226	C₂×M₁₆	—	14	2	no	—
227	C₂²×C₈	—	14	2	no	—
228	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
230	C₄⋊Q₈	—	7	1	no	—
231	C₄⋊Q₈	—	7	1	no	—
232	C₄⋊Q₈	—	7	1	no	—
233	#127	—	7	1	no	—
234	#125	—	14	1	no	—
235	#124	—	14	1	no	—
236	C₂²⋊Q₈	—	7	2	no	—
237	C₂²⋊Q₈	—	14	1	no	—
238	#121	—	7	2	no	—
239	#121	—	14	1	no	—
240	C₄×Q₈	—	7	2	no	—
241	C₄×Q₈	—	14	1	no	—
242	C₄×D₄	—	7	2	no	—
243	C₄×D₄	—	14	1	no	—
244	#117	—	14	1	no	—
245	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
246	#106	—	7	2	no	—
247	#106	—	7	2	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	7	2	no	—
250	C₄⋊C₈	—	7	2	no	—
251	C₄⋊C₈	—	14	2	no	—
252	C₄⋊C₈	—	14	2	no	—
253	C₄⋊C₈	D₂₈	1	2	yes	—
254	#103	—	7	2	no	—
255	#103	—	7	2	no	—
256	#102	—	7	2	no	—
257	#102	—	7	2	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	14	1	no	—
260	C₂²⋊C₈	—	14	1	no	—
261	C₂²⋊C₈	—	14	2	no	—
262	C₂²⋊C₈	—	14	2	no	—
263	C₂²⋊C₈	D₂₈	1	2	yes	—
264	#97	—	14	2	no	—
265	#97	—	14	1	no	—
266	#97	—	14	1	no	—
267	C₄×C₈	—	14	2	no	—
268	C₄×C₈	C₄×D₇	1	1	yes	characteristic
269	C₄×C₈	C₄×D₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
276	C₂²×C₇	—	2	1	no	—
277	C₂²×C₇	—	2	1	no	—
278	C₂²×C₇	—	2	1	no	—
279	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
280	C₂×D₇	—	4	2	no	—
281	C₂×D₇	—	4	2	no	—
282	C₂×D₇	—	4	2	no	—
283	C₄×C₇	—	2	1	no	—
284	C₄×C₇	—	2	1	no	—
285	C₄×C₇	—	2	1	no	—
286	C₄×C₇	—	2	1	no	—
287	C₄×C₇	#568	1	1	yes	characteristic
288	C₄×C₇	#568	1	1	yes	characteristic
289	C₄×C₇	#577	1	2	yes	—
290	Dic₇	—	4	2	no	—
291	Dic₇	—	4	2	no	—
292	Dic₇	—	4	2	no	—
293	#41	—	56	2	no	—
294	C₂×Q₈	—	7	2	no	—
295	C₂×Q₈	—	14	1	no	—
296	C₂×Q₈	—	14	1	no	—
297	C₂×D₄	—	7	2	no	—
298	C₂×D₄	—	14	2	no	—
299	C₂×D₄	—	14	1	no	—
300	C₂×D₄	—	14	1	no	—
301	C₂²×C₄	—	14	2	no	—
302	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
303	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
304	C₂²×C₄	#1157	1	1	yes	characteristic
305	SD₁₆	—	28	2	no	—
306	SD₁₆	—	28	2	no	—
307	M₁₆	—	56	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	4	1	no	—
311	C₂×C₈	—	4	1	no	—
312	C₂×C₈	—	14	2	no	—
313	C₂×C₈	—	14	2	no	—
314	C₂×C₈	—	56	2	no	—
315	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
317	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
319	C₂×C₈	C₂×D₂₈	1	2	yes	—
320	C₂×C₈	C₂×D₂₈	1	2	yes	—
321	C₄⋊C₄	—	7	2	no	—
322	C₄⋊C₄	—	7	2	no	—
323	C₄⋊C₄	—	14	2	no	—
324	C₄⋊C₄	—	14	1	no	—
325	C₄⋊C₄	—	14	1	no	—
326	C₄⋊C₄	—	14	1	no	—
327	C₄⋊C₄	—	14	1	no	—
328	C₄⋊C₄	—	14	2	no	—
329	C₂²⋊C₄	—	14	2	no	—
330	C₂²⋊C₄	—	14	2	no	—
331	C₂²⋊C₄	—	14	2	no	—
332	C₄²	—	14	2	no	—
333	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
334	C₄²	C₂×D₂₈	1	1	yes	characteristic
335	C₄²	#1157	1	1	yes	characteristic
336	C₄²	C₂³×D₇	1	1	yes	characteristic
337	C₂×C₇	—	4	1	no	—
338	C₂×C₇	#2932	1	1	yes	characteristic
339	C₂×C₇	#2949	1	1	yes	characteristic
340	C₂×C₇	#2957	1	1	yes	characteristic
341	D₇	—	8	2	no	—
342	C₂³	—	14	2	no	—
343	C₂³	C₄×D₂₈	1	1	yes	characteristic
344	Q₈	—	14	2	no	—
345	Q₈	—	14	2	no	—
346	Q₈	—	56	1	no	—
347	Q₈	—	56	1	no	—
348	D₄	—	14	2	no	—
349	D₄	—	14	2	no	—
350	D₄	—	56	1	no	—
351	D₄	—	56	1	no	—
352	D₄	—	56	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	14	2	no	—
358	C₂×C₄	—	14	2	no	—
359	C₂×C₄	—	14	2	no	—
360	C₂×C₄	—	28	2	no	—
361	C₂×C₄	—	28	2	no	—
362	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
363	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
364	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
365	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
367	C₂×C₄	C₂×#1157	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	2	no	—
373	C₈	—	2	2	no	—
374	C₈	—	4	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	28	2	no	—
377	C₈	—	28	2	no	—
378	C₇	#17990	1	1	yes	characteristic, complemented, Sylow
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	28	2	no	—
383	C₂²	—	28	2	no	—
384	C₂²	—	28	2	no	—
385	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	28	2	no	—
391	C₄	—	28	2	no	—
392	C₄	—	28	2	no	—
393	C₄	#90436	1	1	yes	characteristic
394	C₄	#90436	1	1	yes	characteristic
395	C₄	#90442	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	56	2	no	—
398	C₂	#11735936	1	1	yes	characteristic
399	C₂	#11736017	1	1	yes	characteristic
400	C₂	#11736025	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1452	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	#90168	C₂²
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1452	D₇
hypocenter	C₇	#17990
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #11736711	C₂	semidirect
4. C₇×#1452	C₂	semidirect
5. #11728525	C₂	semidirect
7. #11727486	C₂	semidirect
8. #11725842	C₂	semidirect
9. #11725839	C₂	semidirect
10. #11725834	C₂	semidirect
11. #11725833	C₂	semidirect
109. #1452	D₇	semidirect
378. C₇	#17990	semidirect

Maximal subgroups

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

subgroup	index
#11736735	2
#11736711	2
#11735913	2
C₇×#1452	2
#11728525	2
#11728519	2
#11727486	2
#11725842	2
#11725839	2
#11725834	2
#11725833	2
#17990	7