Subgroups — #13253902

Order 1792.

388 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2864	C₂	1	1	yes	maximal, characteristic
2	#11727145	C₂	1	1	yes	maximal, characteristic
3	#11727143	C₂	1	1	yes	maximal, characteristic
4	#11725436	C₂	1	2	yes	maximal
5	#11725436	C₂	1	2	yes	maximal
6	C₇×#389	C₄	1	1	yes	characteristic
7	C₇×#389	C₄	1	1	yes	characteristic
8	C₇×#389	C₄	1	1	yes	characteristic
9	C₇×#389	C₄	1	1	yes	characteristic
10	C₇×#387	—	2	1	no	—
11	C₇×#387	—	2	1	no	—
12	C₇×#387	—	2	1	no	—
13	C₇×#387	—	2	1	no	—
14	C₇×#382	C₂²	1	1	yes	characteristic
15	C₇×#382	C₂²	1	1	yes	characteristic
16	C₄³×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
18	C₂×#6377	C₂²	1	2	yes	—
19	C₂×#6377	C₂²	1	2	yes	—
20	#89319	—	2	1	no	—
21	#89319	—	2	1	no	—
22	#89319	—	2	1	no	—
23	#9893	—	7	1	no	maximal, Sylow
24	C₂×C₇×C₄⋊C₄	—	2	1	no	—
25	C₂×C₇×C₄⋊C₄	—	2	1	no	—
26	C₂×C₇×C₄⋊C₄	—	2	1	no	—
27	C₂×C₇×C₄⋊C₄	—	2	1	no	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₇⋊C₈	—	2	2	no	—
40	C₂²×C₇⋊C₈	—	2	2	no	—
41	C₇×#95	—	2	1	no	—
42	C₇×#95	—	2	1	no	—
43	C₇×#95	—	2	1	no	—
44	C₇×#95	—	2	1	no	—
45	C₇×#95	—	2	1	no	—
46	C₇×#95	—	2	1	no	—
47	C₇×#95	—	2	1	no	—
48	C₇×#95	—	2	1	no	—
49	C₄⋊(C₇⋊C₈)	—	2	4	no	—
50	#6377	—	2	2	no	—
51	#6377	—	2	4	no	—
52	#6377	Q₈	1	4	yes	—
53	#2864	D₇	1	1	yes	characteristic
54	#1767	—	7	1	no	—
55	#1745	—	7	1	no	—
56	#1295	—	7	2	no	—
57	#1295	—	7	2	no	—
58	C₂²×C₄×C₇	—	2	1	no	—
59	C₂²×C₄×C₇	—	2	1	no	—
60	C₂²×C₄×C₇	—	2	1	no	—
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	4	1	no	—
70	C₇×C₄⋊C₄	—	4	1	no	—
71	C₇×C₄⋊C₄	—	4	1	no	—
72	C₇×C₄⋊C₄	—	4	1	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₄⋊C₄	—	4	2	no	—
75	C₇×C₄⋊C₄	—	4	2	no	—
76	C₇×C₄⋊C₄	—	4	2	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	—	4	1	no	—
79	C₄²×C₇	—	4	1	no	—
80	C₄²×C₇	—	4	1	no	—
81	C₄²×C₇	—	4	1	no	—
82	C₄²×C₇	—	4	1	no	—
83	C₄²×C₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₇	#41	1	2	yes	—
85	C₂×C₇⋊C₈	—	2	4	no	—
86	C₂×C₇⋊C₈	—	2	4	no	—
87	C₂×C₇⋊C₈	—	4	2	no	—
88	C₂×C₇⋊C₈	—	4	2	no	—
89	C₂×C₇⋊C₈	—	4	2	no	—
90	C₂×C₇⋊C₈	—	4	2	no	—
91	C₂×C₄⋊C₈	—	14	1	no	—
92	C₂×#97	—	7	2	no	—
93	C₂×#97	—	7	2	no	—
94	#389	Dic₇	1	1	yes	characteristic
95	#389	Dic₇	1	1	yes	characteristic
96	#389	Dic₇	1	1	yes	characteristic
97	#389	Dic₇	1	1	yes	characteristic
98	#387	—	2	1	no	—
99	#387	—	2	1	no	—
100	#387	—	2	1	no	—
101	#387	—	2	1	no	—
102	#382	C₂×D₇	1	1	yes	characteristic
103	#382	C₂×D₇	1	1	yes	characteristic
104	C₄³	C₂×D₇	1	1	yes	characteristic
105	#336	—	14	1	no	—
106	#336	—	14	1	no	—
107	#336	—	14	1	no	—
108	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	1	no	—
114	C₂×C₄×C₇	—	2	1	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	1	no	—
119	C₂×C₄×C₇	—	2	1	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	1	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	4	1	no	—
127	C₂×C₄×C₇	—	4	1	no	—
128	C₂×C₄×C₇	—	4	1	no	—
129	C₂×C₄×C₇	—	4	1	no	—
130	C₂×C₄×C₇	—	4	1	no	—
131	C₂×C₄×C₇	—	4	1	no	—
132	C₂×C₄×C₇	—	4	1	no	—
133	C₂×C₄×C₇	—	4	1	no	—
134	C₂×C₄×C₇	—	4	1	no	—
135	C₂×C₄×C₇	—	4	1	no	—
136	C₂×C₄×C₇	—	4	1	no	—
137	C₂×C₄×C₇	—	4	1	no	—
138	C₂×C₄×C₇	#104	1	2	yes	—
139	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₇	#124	1	1	yes	characteristic
142	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
143	C₇⋊C₈	—	4	4	no	—
144	C₇⋊C₈	—	4	4	no	—
145	C₂²×C₈	—	14	2	no	—
146	C₂²×C₈	—	14	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
162	C₄⋊C₈	—	14	4	no	—
163	#97	—	7	4	no	—
164	#97	—	14	2	no	—
165	#97	—	14	4	no	—
166	#95	—	2	1	no	—
167	#95	—	2	1	no	—
168	#95	—	2	1	no	—
169	#95	—	2	1	no	—
170	#95	—	2	1	no	—
171	#95	—	2	1	no	—
172	#95	—	2	1	no	—
173	#95	—	2	1	no	—
174	C₂²×C₇	#391	1	1	yes	characteristic
175	C₂²×C₇	#413	1	1	yes	characteristic
176	C₂²×C₇	#413	1	1	yes	characteristic
177	C₂²×C₇	C₂×#104	1	1	yes	characteristic
178	C₂²×C₇	#421	1	1	yes	characteristic
179	C₂²×C₇	#421	1	1	yes	characteristic
180	C₂²×C₇	#421	1	1	yes	characteristic
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	2	2	no	—
186	C₄×C₇	—	4	1	no	—
187	C₄×C₇	—	4	1	no	—
188	C₄×C₇	—	4	1	no	—
189	C₄×C₇	—	8	1	no	—
190	C₄×C₇	—	8	1	no	—
191	C₄×C₇	—	8	1	no	—
192	C₄×C₇	—	8	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
204	C₂×C₈	—	14	4	no	—
205	C₂×C₈	—	14	4	no	—
206	C₂×C₈	—	28	2	no	—
207	C₂×C₈	—	28	2	no	—
208	C₂×C₈	—	28	2	no	—
209	C₂×C₈	—	28	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₄²	—	4	1	no	—
224	C₄²	Q₈×D₇	1	2	yes	—
225	C₄²	#1161	1	2	yes	—
226	C₂×C₇	#1529	1	1	yes	characteristic
227	C₂×C₇	#1532	1	1	yes	characteristic
228	C₂×C₇	#1535	1	1	yes	characteristic
229	C₂×C₇	#1535	1	1	yes	characteristic
230	C₂×C₇	#1986	1	1	yes	characteristic
231	C₂×C₇	#1995	1	1	yes	characteristic
232	C₂×C₇	#1998	1	1	yes	characteristic
233	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	#6411	1	2	yes	—
264	C₂×C₄	#6507	1	1	yes	characteristic
265	C₂×C₄	Q₈×Dic₇	1	2	yes	—
266	C₂×C₄	#6509	1	2	yes	—
267	C₂×C₄	#6510	1	1	yes	characteristic
268	C₈	—	28	4	no	—
269	C₈	—	28	4	no	—
270	C₇	#9893	1	1	yes	characteristic, complemented, Sylow
271	C₂²	#90002	1	1	yes	characteristic
272	C₂²	C₂×#6411	1	1	yes	characteristic
273	C₂²	#90008	1	1	yes	characteristic
274	C₂²	#90008	1	1	yes	characteristic
275	C₂²	#90008	1	1	yes	characteristic
276	C₂²	#90009	1	1	yes	characteristic
277	C₂²	#90009	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	8	1	no	—
287	C₄	—	8	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₂	#11729496	1	1	yes	characteristic
291	C₂	#11729501	1	1	yes	characteristic
292	C₂	#11729502	1	1	yes	characteristic
293	C₂	#11729503	1	1	yes	characteristic
294	C₂	#11729503	1	1	yes	characteristic
295	C₂	#11729594	1	1	yes	characteristic
296	C₂	#11729595	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2864	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2864	D₇
hypocenter	C₇	#9893
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2864

Complements

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

N	H ≅ G/N	type
270. C₇	#9893	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2864	2
#11727145	2
#11727143	2
#11725436	2
#9893	7