Subgroups — #13253957

Order 1792.

438 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2859	C₂	1	1	yes	maximal, characteristic
2	#11727148	C₂	1	1	yes	maximal, characteristic
3	#11727145	C₂	1	1	yes	maximal, characteristic
4	#11725440	C₂	1	2	yes	maximal
5	#11725440	C₂	1	2	yes	maximal
6	C₂×C₇×#125	C₄	1	2	yes	—
7	C₇×#389	—	2	1	no	—
8	C₇×#389	—	2	1	no	—
9	C₇×#389	—	2	1	no	—
10	C₇×#389	—	2	1	no	—
11	C₇×#384	C₂²	1	1	yes	characteristic
12	C₇×#384	C₂²	1	1	yes	characteristic
13	C₇×#382	C₄	1	2	yes	—
14	C₄³×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
16	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
17	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×#6377	—	2	1	no	—
20	#89319	—	2	1	no	—
21	#9906	—	7	1	no	maximal, Sylow
22	C₇×#125	—	4	4	no	—
23	C₂×C₇×C₄⋊C₄	—	2	2	no	—
24	C₂×C₇×C₄⋊C₄	—	2	2	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	—	2	1	no	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	D₄	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	C₂³	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	—	2	2	no	—
39	C₂²×C₇⋊C₈	—	2	2	no	—
40	C₇×#95	—	2	2	no	—
41	C₇×#95	—	2	2	no	—
42	C₄⋊(C₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₇⋊C₈)	—	2	4	no	—
44	C₄⋊(C₇⋊C₈)	—	4	2	no	—
45	C₄⋊(C₇⋊C₈)	—	4	2	no	—
46	C₄⋊(C₇⋊C₈)	Q₈	1	4	yes	—
47	#6377	—	4	2	no	—
48	#2859	D₇	1	1	yes	characteristic
49	#1842	—	7	1	no	—
50	#1767	—	7	1	no	—
51	#1296	—	7	2	no	—
52	#1296	—	7	2	no	—
53	C₂²×C₄×C₇	—	2	2	no	—
54	C₂²×C₄×C₇	—	2	2	no	—
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₄⋊C₄	—	4	4	no	—
65	C₇×C₄⋊C₄	—	4	4	no	—
66	C₇×C₄⋊C₄	—	4	2	no	—
67	C₇×C₄⋊C₄	—	4	1	no	—
68	C₇×C₄⋊C₄	—	4	1	no	—
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	C₄²×C₇	—	2	2	no	—
74	C₄²×C₇	—	2	2	no	—
75	C₄²×C₇	—	2	2	no	—
76	C₄²×C₇	—	2	1	no	—
77	C₄²×C₇	—	2	1	no	—
78	C₄²×C₇	—	2	1	no	—
79	C₄²×C₇	—	2	1	no	—
80	C₄²×C₇	—	2	1	no	—
81	C₄²×C₇	—	2	1	no	—
82	C₄²×C₇	C₂×Q₈	1	2	yes	—
83	C₄²×C₇	#41	1	2	yes	—
84	C₂×C₇⋊C₈	—	2	4	no	—
85	C₂×C₇⋊C₈	—	4	2	no	—
86	C₂×C₇⋊C₈	—	4	2	no	—
87	C₂×C₇⋊C₈	—	4	2	no	—
88	C₂×C₇⋊C₈	—	4	2	no	—
89	C₂×C₇⋊C₈	—	4	2	no	—
90	C₂×#125	Dic₇	1	2	yes	—
91	C₂×C₄⋊C₈	—	7	2	no	—
92	C₂×C₄⋊C₈	—	7	2	no	—
93	C₂×C₄⋊C₈	—	14	1	no	—
94	C₂×C₄⋊C₈	—	14	1	no	—
95	C₂×#97	—	14	1	no	—
96	#389	—	2	1	no	—
97	#389	—	2	1	no	—
98	#389	—	2	1	no	—
99	#389	—	2	1	no	—
100	#384	C₂×D₇	1	1	yes	characteristic
101	#384	C₂×D₇	1	1	yes	characteristic
102	#382	Dic₇	1	2	yes	—
103	C₄³	C₂×D₇	1	1	yes	characteristic
104	#336	—	14	1	no	—
105	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
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	—	4	2	no	—
119	C₂×C₄×C₇	—	4	2	no	—
120	C₂×C₄×C₇	—	4	2	no	—
121	C₂×C₄×C₇	—	4	2	no	—
122	C₂×C₄×C₇	—	4	2	no	—
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	#100	1	2	yes	—
125	C₂×C₄×C₇	#101	1	2	yes	—
126	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₇	#135	1	2	yes	—
131	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₇	#137	1	1	yes	characteristic
135	C₂×C₄×C₇	#137	1	1	yes	characteristic
136	C₂×C₄×C₇	#137	1	1	yes	characteristic
137	C₇⋊C₈	—	4	4	no	—
138	C₇⋊C₈	—	8	2	no	—
139	C₂²×C₈	—	14	2	no	—
140	C₂²×C₈	—	14	2	no	—
141	#125	—	4	4	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
150	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
155	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
156	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
157	C₄⋊C₈	—	7	4	no	—
158	C₄⋊C₈	—	14	2	no	—
159	C₄⋊C₈	—	14	4	no	—
160	C₄⋊C₈	—	28	2	no	—
161	C₄⋊C₈	—	28	2	no	—
162	#97	—	28	2	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	C₂²×C₇	#391	1	1	yes	characteristic
166	C₂²×C₇	C₂×#100	1	1	yes	characteristic
167	C₂²×C₇	C₂×#101	1	1	yes	characteristic
168	C₂²×C₇	#416	1	1	yes	characteristic
169	C₂²×C₇	#417	1	1	yes	characteristic
170	C₂²×C₇	#417	1	1	yes	characteristic
171	C₂²×C₇	#417	1	1	yes	characteristic
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	4	1	no	—
176	C₄×C₇	—	4	1	no	—
177	C₄×C₇	—	8	2	no	—
178	C₄×C₇	—	8	2	no	—
179	C₄×C₇	#478	1	2	yes	—
180	C₄×C₇	#479	1	2	yes	—
181	C₄×C₇	#490	1	1	yes	characteristic
182	C₄×C₇	#491	1	1	yes	characteristic
183	C₄×C₇	#501	1	1	yes	characteristic
184	C₄×C₇	#501	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
194	C₂×C₈	—	14	4	no	—
195	C₂×C₈	—	28	2	no	—
196	C₂×C₈	—	28	2	no	—
197	C₂×C₈	—	28	2	no	—
198	C₂×C₈	—	28	2	no	—
199	C₂×C₈	—	28	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	Q₈×D₇	1	2	yes	—
221	C₄²	#1161	1	2	yes	—
222	C₂×C₇	#1551	1	1	yes	characteristic
223	C₂×C₇	#1552	1	1	yes	characteristic
224	C₂×C₇	#1560	1	1	yes	characteristic
225	C₂×C₇	#1561	1	1	yes	characteristic
226	C₂×C₇	#1989	1	1	yes	characteristic
227	C₂×C₇	#1992	1	1	yes	characteristic
228	C₂×C₇	#1997	1	1	yes	characteristic
229	C₂³	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
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	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	#6407	1	2	yes	—
249	C₂×C₄	#6410	1	2	yes	—
250	C₂×C₄	#6507	1	1	yes	characteristic
251	C₂×C₄	Q₈×Dic₇	1	2	yes	—
252	C₂×C₄	#6509	1	2	yes	—
253	C₂×C₄	#6510	1	1	yes	characteristic
254	C₂×C₄	#6512	1	1	yes	characteristic
255	C₂×C₄	#6512	1	1	yes	characteristic
256	C₂×C₄	#6512	1	1	yes	characteristic
257	C₂×C₄	#6513	1	2	yes	—
258	C₂×C₄	#6514	1	1	yes	characteristic
259	C₂×C₄	#6514	1	1	yes	characteristic
260	C₂×C₄	#6514	1	1	yes	characteristic
261	C₈	—	28	4	no	—
262	C₈	—	56	2	no	—
263	C₇	#9906	1	1	yes	characteristic, complemented, Sylow
264	C₂²	C₂×#6407	1	1	yes	characteristic
265	C₂²	C₂×#6410	1	1	yes	characteristic
266	C₂²	#90002	1	1	yes	characteristic
267	C₂²	#90004	1	1	yes	characteristic
268	C₂²	#90004	1	1	yes	characteristic
269	C₂²	#90004	1	1	yes	characteristic
270	C₂²	#90005	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	#89835	1	2	yes	—
279	C₄	#89837	1	1	yes	characteristic
280	C₄	#89837	1	1	yes	characteristic
281	C₄	#89838	1	2	yes	—
282	C₄	#89839	1	1	yes	characteristic
283	C₄	#89841	1	1	yes	characteristic
284	C₂	#11729492	1	1	yes	characteristic
285	C₂	#11729495	1	1	yes	characteristic
286	C₂	#11729544	1	1	yes	characteristic
287	C₂	#11729561	1	1	yes	characteristic
288	C₂	#11729564	1	1	yes	characteristic
289	C₂	#11729585	1	1	yes	characteristic
290	C₂	#11729588	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×#2859	C₂
socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2859	D₇
hypocenter	C₇	#9906
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

Complements

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

N	H ≅ G/N	type
263. C₇	#9906	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2859	2
#11727148	2
#11727145	2
#11725440	2
#9906	7