Subgroups — #13323115

Order 1792.

594 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2965	C₂	1	1	yes	maximal, characteristic
2	#11739967	C₂	1	2	yes	maximal
3	#11728506	C₂	1	2	yes	maximal, complemented
4	#11728505	C₂	1	2	yes	maximal, complemented
5	#11728503	C₂	1	2	yes	maximal, complemented
6	#11727605	C₂	1	2	yes	maximal, complemented
7	#11727604	C₂	1	2	yes	maximal, complemented
8	#11727486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#89252	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₇×#41	C₂²	1	1	yes	characteristic
11	C₇×#446	C₄	1	1	yes	characteristic
12	C₇×#446	C₄	1	1	yes	characteristic
13	C₇×#436	C₄	1	1	yes	characteristic
14	C₇×#436	C₄	1	1	yes	characteristic
15	C₇×#435	C₄	1	2	yes	—
16	C₇×#435	C₄	1	2	yes	—
17	C₇×#431	C₂²	1	2	yes	—
18	C₇×#431	C₂²	1	1	yes	characteristic
19	C₄×C₇×M₁₆	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	#89796	—	2	2	no	—
22	#89783	—	2	2	no	—
23	#89783	—	2	2	no	—
24	D₄×C₇⋊C₈	—	2	2	no	—
25	#89698	C₂²	1	2	yes	—
26	#89698	C₂²	1	2	yes	—
27	#89697	C₂²	1	2	yes	—
28	#89696	C₂²	1	2	yes	—
29	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
30	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
31	#89253	C₂²	1	2	yes	—
32	#89253	C₂²	1	2	yes	—
33	#89253	C₂²	1	2	yes	—
34	#89252	C₂²	1	1	yes	characteristic
35	#89252	C₂²	1	2	yes	—
36	#89252	C₂²	1	2	yes	—
37	#89252	C₂²	1	2	yes	—
38	#89252	C₂²	1	1	yes	characteristic
39	#89252	C₂²	1	1	yes	characteristic
40	#89252	C₂²	1	1	yes	characteristic
41	#17838	—	7	1	no	maximal, Sylow
42	C₂×C₇×#41	Q₈	1	1	yes	characteristic
43	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
49	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
50	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
51	C₄×C₇×D₄	D₄	1	2	yes	—
52	C₄×C₇×D₄	D₄	1	1	yes	characteristic
53	C₇×#117	Q₈	1	1	yes	characteristic
54	C₇×#117	Q₈	1	2	yes	—
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇	C₂³	1	2	yes	—
57	C₂×#1136	—	2	2	no	—
58	C₂²×C₇⋊C₈	—	2	2	no	—
59	C₂²×C₇⋊C₈	—	2	2	no	—
60	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₇×C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	C₇×#97	C₂×C₄	1	2	yes	—
68	C₇×#97	C₂×C₄	1	2	yes	—
69	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇×C₈	C₂³	1	2	yes	—
74	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
75	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
76	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
77	C₄⋊(C₇⋊C₈)	—	2	2	no	—
78	C₄⋊(C₇⋊C₈)	—	2	2	no	—
79	C₄⋊(C₇⋊C₈)	—	2	2	no	—
80	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
81	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
82	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
83	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
84	#6377	—	2	2	no	—
85	#6377	—	2	2	no	—
86	C₄×C₇⋊C₈	—	2	2	no	—
87	#2990	—	7	2	no	—
88	#2965	D₇	1	1	yes	characteristic
89	#1573	—	7	2	no	—
90	#1572	—	7	2	no	—
91	#1570	—	7	2	no	—
92	#1569	—	7	2	no	—
93	#1568	—	7	1	no	—
94	#1565	—	7	2	no	—
95	C₂×#334	—	7	1	no	—
96	C₇×#41	—	4	2	no	—
97	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
99	C₂×C₇×D₄	C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
105	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
106	C₇×M₁₆	—	2	4	no	—
107	C₇×M₁₆	—	4	2	no	—
108	C₂×C₇×C₈	—	2	2	no	—
109	C₂×C₇×C₈	—	4	1	no	—
110	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
117	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
118	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
119	C₇×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
120	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
121	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
122	C₇×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
123	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇	C₂²×C₄	1	2	yes	—
125	C₄²×C₇	C₂×D₄	1	2	yes	—
126	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
128	C₄²×C₇	C₂×Q₈	1	2	yes	—
129	C₄²×C₇	C₂⁴	1	1	yes	characteristic
130	#1136	—	8	2	no	—
131	C₂×C₇⋊C₈	—	2	2	no	—
132	C₂×C₇⋊C₈	—	2	2	no	—
133	C₂×C₇⋊C₈	—	2	2	no	—
134	C₂×C₇⋊C₈	—	2	2	no	—
135	C₂×C₇⋊C₈	—	8	2	no	—
136	C₂×C₇⋊C₈	—	8	2	no	—
137	C₄×#41	C₂×D₇	1	1	yes	characteristic
138	#446	—	14	2	no	—
139	#446	Dic₇	1	1	yes	characteristic
140	#446	Dic₇	1	1	yes	characteristic
141	#436	Dic₇	1	1	yes	characteristic
142	#436	Dic₇	1	1	yes	characteristic
143	#435	—	14	2	no	—
144	#435	—	14	2	no	—
145	#435	Dic₇	1	2	yes	—
146	#435	Dic₇	1	2	yes	—
147	C₈×D₄	—	14	2	no	—
148	#432	—	7	2	no	—
149	#432	—	7	2	no	—
150	#431	—	7	2	no	—
151	#431	C₂×D₇	1	2	yes	—
152	#431	C₂×D₇	1	1	yes	characteristic
153	#423	—	7	2	no	—
154	C₂×C₄⋊C₈	—	7	2	no	—
155	C₂×C₄⋊C₈	—	7	2	no	—
156	C₄×M₁₆	C₂×D₇	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
158	#335	—	7	2	no	—
159	#335	—	7	2	no	—
160	#335	—	7	2	no	—
161	#334	—	7	1	no	—
162	#334	—	7	1	no	—
163	#334	—	7	2	no	—
164	#334	—	7	2	no	—
165	#334	—	7	1	no	—
166	#334	—	7	1	no	—
167	#334	—	7	2	no	—
168	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
169	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₇×Q₈	—	4	1	no	—
171	C₇×D₄	—	2	4	no	—
172	C₇×D₄	—	4	1	no	—
173	C₂×C₄×C₇	—	2	2	no	—
174	C₂×C₄×C₇	—	2	1	no	—
175	C₂×C₄×C₇	—	2	2	no	—
176	C₂×C₄×C₇	—	2	1	no	—
177	C₂×C₄×C₇	—	2	2	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	4	2	no	—
182	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
184	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
186	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
187	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
191	C₇×C₈	—	2	1	no	—
192	C₇×C₈	—	2	2	no	—
193	C₇×C₈	—	2	1	no	—
194	C₇×C₈	—	2	1	no	—
195	C₇×C₈	—	2	2	no	—
196	C₇×C₈	#131	1	2	yes	—
197	C₇⋊C₈	—	4	2	no	—
198	C₇⋊C₈	—	4	2	no	—
199	C₇⋊C₈	—	4	2	no	—
200	C₇⋊C₈	—	4	2	no	—
201	C₂×#41	Dic₁₄	1	1	yes	characteristic
202	C₂×M₁₆	—	14	2	no	—
203	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
204	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
205	C₂²×C₈	—	14	2	no	—
206	C₂²×C₈	—	14	2	no	—
207	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
208	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
209	C₄×Q₈	D₂₈	1	1	yes	characteristic
210	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
211	C₄×D₄	D₂₈	1	2	yes	—
212	C₄×D₄	D₂₈	1	1	yes	characteristic
213	C₄×D₄	C₂×Dic₇	1	2	yes	—
214	C₄×D₄	C₂×Dic₇	1	1	yes	characteristic
215	#117	Dic₁₄	1	1	yes	characteristic
216	#117	Dic₁₄	1	2	yes	—
217	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₇	1	2	yes	—
219	C₄⋊C₈	—	7	2	no	—
220	C₄⋊C₈	—	7	2	no	—
221	C₄⋊C₈	—	7	2	no	—
222	C₄⋊C₈	—	7	2	no	—
223	C₄⋊C₈	—	14	2	no	—
224	C₄⋊C₈	—	14	2	no	—
225	C₄⋊C₈	—	14	2	no	—
226	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
227	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
228	C₄⋊C₈	C₂×Dic₇	1	1	yes	characteristic
229	C₄⋊C₈	C₂×Dic₇	1	1	yes	characteristic
230	C₂²⋊C₈	—	14	2	no	—
231	C₂²⋊C₈	—	14	2	no	—
232	C₂²⋊C₈	—	14	2	no	—
233	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
234	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
235	C₂²⋊C₈	C₂×Dic₇	1	2	yes	—
236	#97	—	14	2	no	—
237	#97	—	14	2	no	—
238	#97	C₂×Dic₇	1	2	yes	—
239	#97	C₂×Dic₇	1	2	yes	—
240	C₄×C₈	—	14	2	no	—
241	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
242	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₇	1	2	yes	—
246	C₂²×C₇	—	2	1	no	—
247	C₂²×C₇	—	2	1	no	—
248	C₂²×C₇	—	2	2	no	—
249	C₂²×C₇	—	2	2	no	—
250	C₂²×C₇	—	2	2	no	—
251	C₂²×C₇	#430	1	2	yes	—
252	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
253	C₄×C₇	—	2	1	no	—
254	C₄×C₇	—	2	2	no	—
255	C₄×C₇	—	2	1	no	—
256	C₄×C₇	—	2	2	no	—
257	C₄×C₇	—	2	1	no	—
258	C₄×C₇	—	2	1	no	—
259	C₄×C₇	—	2	2	no	—
260	C₄×C₇	C₂×#131	1	1	yes	characteristic
261	C₄×C₇	#568	1	1	yes	characteristic
262	C₄×C₇	#577	1	2	yes	—
263	#41	—	4	2	no	—
264	C₂×Q₈	C₄⋊Dic₇	1	1	yes	characteristic
265	C₂×D₄	C₄⋊Dic₇	1	2	yes	—
266	C₂×D₄	C₄⋊Dic₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
268	C₂²×C₄	C₂×Dic₁₄	1	2	yes	—
269	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₂₈	1	2	yes	—
271	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
272	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
273	M₁₆	—	2	4	no	—
274	M₁₆	—	4	2	no	—
275	M₁₆	—	56	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	14	2	no	—
279	C₂×C₈	—	14	2	no	—
280	C₂×C₈	—	14	2	no	—
281	C₂×C₈	—	14	2	no	—
282	C₂×C₈	—	56	2	no	—
283	C₂×C₈	—	56	2	no	—
284	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
285	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
286	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
287	C₂×C₈	C₂²×Dic₇	1	2	yes	—
288	C₂×C₈	C₂²×Dic₇	1	2	yes	—
289	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
290	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
291	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
292	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
293	C₄⋊C₄	C₄⋊Dic₇	1	1	yes	characteristic
294	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
295	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
296	C₂²⋊C₄	C₄⋊Dic₇	1	2	yes	—
297	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
298	C₄²	C₂×Dic₁₄	1	2	yes	—
299	C₄²	C₂×D₂₈	1	2	yes	—
300	C₄²	C₂×D₂₈	1	1	yes	characteristic
301	C₄²	C₂²×Dic₇	1	1	yes	characteristic
302	C₄²	C₂²×Dic₇	1	2	yes	—
303	C₄²	C₂³×D₇	1	1	yes	characteristic
304	C₂×C₇	—	2	2	no	—
305	C₂×C₇	—	4	2	no	—
306	C₂×C₇	#2906	1	1	yes	characteristic
307	C₂×C₇	#2912	1	1	yes	characteristic
308	C₂×C₇	C₂×#430	1	1	yes	characteristic
309	C₂³	C₂×C₄⋊Dic₇	1	2	yes	—
310	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
311	Q₈	—	4	1	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊Dic₇	1	2	yes	—
329	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
331	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	2	no	—
337	C₈	—	28	2	no	—
338	C₈	—	28	2	no	—
339	C₈	—	28	2	no	—
340	C₈	—	28	2	no	—
341	C₈	#6511	1	2	yes	—
342	C₇	#17838	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	#89897	1	2	yes	—
349	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	2	no	—
357	C₄	#90442	1	2	yes	—
358	C₄	C₂×#6511	1	1	yes	characteristic
359	C₄	#90510	1	1	yes	characteristic
360	C₂	—	2	2	no	—
361	C₂	—	4	2	no	—
362	C₂	#11739957	1	1	yes	characteristic
363	C₂	C₂×#89897	1	1	yes	characteristic
364	C₂	#11741229	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#2965	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂×C₇×#41	Q₈
cosocle	C₄²×C₇	C₂⁴
hypercenter	#2965	D₇
hypocenter	C₇	#17838
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2965

Complements

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

N	H ≅ G/N	type
3. #11728506	C₂	semidirect
4. #11728505	C₂	semidirect
5. #11728503	C₂	semidirect
6. #11727605	C₂	semidirect
7. #11727604	C₂	semidirect
8. #11727486	C₂	semidirect
9. C₂×#89252	C₂	semidirect
342. C₇	#17838	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2965	2
#11739967	2
#11728506	2
#11728505	2
#11728503	2
#11727605	2
#11727604	2
#11727486	2
C₂×#89252	2
#17838	7