Subgroups — #13263129

Order 1792.

360 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2931	C₂	1	1	yes	maximal, characteristic
2	#11727499	C₂	1	1	yes	maximal, characteristic, complemented
3	#11727498	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725495	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#131	C₄	1	2	yes	—
6	C₇×#532	C₄	1	1	yes	characteristic
7	C₇×#436	—	2	1	no	—
8	C₇×#436	—	2	1	no	—
9	C₇×C₈×D₄	—	2	1	no	—
10	C₇×C₈×D₄	—	2	1	no	—
11	C₇×#423	C₄	1	1	yes	characteristic
12	C₇×#408	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₂²⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
17	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
18	#89256	—	2	1	no	—
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	#11492	—	7	1	no	maximal, Sylow
21	C₂×C₇×#41	C₂×C₄	1	2	yes	—
22	C₇×#131	—	4	4	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₇×#127	C₈	1	1	yes	characteristic
30	C₇×#124	C₈	1	2	yes	—
31	C₇×#121	—	2	1	no	—
32	C₇×#121	—	2	1	no	—
33	C₄×C₇×D₄	—	2	1	no	—
34	C₄×C₇×D₄	—	2	1	no	—
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₇×C₄⋊C₈	—	2	1	no	—
37	C₇×C₄⋊C₈	—	2	1	no	—
38	C₇×C₂²⋊C₈	—	2	2	no	—
39	C₇×C₂²⋊C₈	—	2	2	no	—
40	C₄×C₇×C₈	D₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₂×C₇⋊C₁₆	—	2	2	no	—
45	C₂×C₇⋊C₁₆	—	2	2	no	—
46	#2931	D₇	1	1	yes	characteristic
47	#2163	—	7	1	no	—
48	#2162	—	7	1	no	—
49	#1315	—	7	4	no	—
50	C₇×#41	—	4	2	no	—
51	C₇×#41	—	4	2	no	—
52	C₂×C₇×Q₈	C₂×C₈	1	2	yes	—
53	C₂×C₇×D₄	—	2	2	no	—
54	C₂×C₇×D₄	—	2	1	no	—
55	C₂×C₇×D₄	—	2	1	no	—
56	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇	—	2	2	no	—
58	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₇×M₁₆	—	4	4	no	—
62	C₇×M₁₆	—	4	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	4	1	no	—
66	C₂×C₇×C₈	—	4	4	no	—
67	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	2	1	no	—
72	C₇×C₄⋊C₄	—	2	1	no	—
73	C₇×C₂²⋊C₄	—	2	2	no	—
74	C₇×C₂²⋊C₄	—	2	2	no	—
75	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
79	C₇⋊C₁₆	—	4	2	no	—
80	C₇⋊C₁₆	—	4	2	no	—
81	C₂×#131	Dic₇	1	2	yes	—
82	#532	Dic₇	1	1	yes	characteristic
83	#436	—	2	1	no	—
84	#436	—	2	1	no	—
85	C₈×D₄	—	2	1	no	—
86	C₈×D₄	—	2	1	no	—
87	#423	Dic₇	1	1	yes	characteristic
88	#408	C₂×D₇	1	2	yes	—
89	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
90	C₄⋊C₁₆	—	14	1	no	—
91	C₄⋊C₁₆	—	14	1	no	—
92	C₂²⋊C₁₆	—	7	2	no	—
93	C₂²⋊C₁₆	—	7	2	no	—
94	#346	—	14	1	no	—
95	C₄×C₁₆	—	14	1	no	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
98	C₇×Q₈	—	4	2	no	—
99	C₇×D₄	—	4	2	no	—
100	C₇×D₄	—	4	2	no	—
101	C₇×D₄	—	4	4	no	—
102	C₇×D₄	—	4	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	4	1	no	—
108	C₂×C₄×C₇	—	4	2	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
116	C₇×C₈	—	2	1	no	—
117	C₇×C₈	—	2	1	no	—
118	C₇×C₈	—	2	1	no	—
119	C₇×C₈	—	4	2	no	—
120	C₇×C₈	#100	1	2	yes	—
121	C₂×#41	C₂×Dic₇	1	2	yes	—
122	#131	—	4	4	no	—
123	C₂×M₁₆	—	2	2	no	—
124	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
128	C₄⋊Q₈	C₇⋊C₈	1	1	yes	characteristic
129	#127	C₇⋊C₈	1	1	yes	characteristic
130	#124	C₇⋊C₈	1	2	yes	—
131	#121	—	2	1	no	—
132	#121	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	—	2	1	no	—
135	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
136	C₂×C₁₆	—	14	2	no	—
137	C₂×C₁₆	—	14	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
146	C₂²×C₇	—	2	1	no	—
147	C₂²×C₇	—	2	1	no	—
148	C₂²×C₇	—	2	1	no	—
149	C₂²×C₇	—	4	2	no	—
150	C₂²×C₇	—	4	2	no	—
151	C₂²×C₇	—	4	2	no	—
152	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	2	1	no	—
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	4	2	no	—
158	C₄×C₇	#350	1	2	yes	—
159	C₄×C₇	#410	1	1	yes	characteristic
160	C₄×C₇	C₂×#100	1	1	yes	characteristic
161	#41	—	4	2	no	—
162	#41	—	4	2	no	—
163	C₂×Q₈	C₂×C₇⋊C₈	1	2	yes	—
164	C₂×D₄	—	2	2	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂×C₇⋊C₈	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	#1136	1	1	yes	characteristic
170	C₂²×C₄	#1136	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
172	M₁₆	—	4	4	no	—
173	M₁₆	—	4	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
187	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
190	C₁₆	—	28	2	no	—
191	C₁₆	—	28	2	no	—
192	C₂×C₇	—	4	1	no	—
193	C₂×C₇	—	8	2	no	—
194	C₂×C₇	#1460	1	1	yes	characteristic
195	C₂×C₇	C₂×#350	1	1	yes	characteristic
196	C₂×C₇	#2117	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	C₂×#1136	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
212	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
213	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
216	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	4	2	no	—
221	C₈	#6407	1	2	yes	—
222	C₇	#11492	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	#89356	1	2	yes	—
236	C₄	C₂×#6407	1	1	yes	characteristic
237	C₄	#90012	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	8	2	no	—
240	C₂	#11729355	1	1	yes	characteristic
241	C₂	C₂×#89356	1	1	yes	characteristic
242	C₂	#11729760	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#2931	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₇×#532	C₄
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#2931	D₇
hypocenter	C₇	#11492
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. #11727499	C₂	semidirect
3. #11727498	C₂	semidirect
4. #11725495	C₂	semidirect
222. C₇	#11492	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2931	2
#11727499	2
#11727498	2
#11725495	2
#11492	7