Subgroups — #13265096

Order 1792.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	#11727497	C₂	1	1	yes	maximal, characteristic
3	C₂×#89253	C₂	1	1	yes	maximal, characteristic
4	#11725608	C₂	1	2	yes	maximal, complemented
5	#11725607	C₂	1	2	yes	maximal, complemented
6	C₇×#533	—	2	1	no	—
7	C₇×#500	C₂²	1	1	yes	characteristic
8	C₇×#500	C₂²	1	1	yes	characteristic
9	C₇×#499	—	2	1	no	—
10	C₇×#499	C₄	1	2	yes	—
11	C₇×#498	—	2	1	no	—
12	C₇×#427	—	2	1	no	—
13	C₇×#427	C₄	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6386	—	2	1	no	—
16	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
17	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
18	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
19	#89253	—	2	1	no	—
20	#89253	C₂²	1	2	yes	—
21	#11942	—	7	1	no	maximal, Sylow
22	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₇×C₄⋊Q₈	—	2	1	no	—
25	C₇×C₄⋊Q₈	—	2	1	no	—
26	C₇×#125	—	2	2	no	—
27	C₇×C₂²⋊Q₈	—	4	1	no	—
28	C₇×C₂²⋊Q₈	—	4	1	no	—
29	C₄×C₇×Q₈	—	4	1	no	—
30	C₄×C₇×Q₈	—	4	1	no	—
31	C₇×#117	—	2	2	no	—
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂²×C₇⋊C₈	—	4	1	no	—
34	C₇×#107	—	2	1	no	—
35	C₇×#107	—	2	1	no	—
36	C₇×#107	C₂×C₄	1	2	yes	—
37	C₇×#107	C₂×C₄	1	2	yes	—
38	C₇×#106	—	2	2	no	—
39	C₇×#106	—	2	2	no	—
40	C₄×C₇×C₈	C₂×C₄	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	C₂³	1	1	yes	characteristic
44	#6386	—	4	2	no	—
45	C₂×C₇⋊C₁₆	—	2	2	no	—
46	C₄⋊(C₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	—	4	1	no	—
48	#3160	D₇	1	1	yes	characteristic
49	#2150	—	7	1	no	—
50	C₂×#335	—	7	1	no	—
51	#1365	—	7	2	no	—
52	#1364	—	7	2	no	—
53	C₂×C₇×Q₈	—	4	1	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
58	C₂×C₇×C₈	—	2	2	no	—
59	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	SD₁₆	1	2	yes	—
62	C₂×C₇×C₈	Q₁₆	1	2	yes	—
63	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
65	C₇×C₄⋊C₄	—	2	2	no	—
66	C₇×C₄⋊C₄	—	2	2	no	—
67	C₇×C₄⋊C₄	—	4	2	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₄⋊C₄	—	4	1	no	—
70	C₇×C₄⋊C₄	—	4	1	no	—
71	C₇×C₂²⋊C₄	—	4	2	no	—
72	C₄²×C₇	—	4	2	no	—
73	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₇	D₈	1	1	yes	characteristic
75	C₄²×C₇	SD₁₆	1	1	yes	characteristic
76	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇⋊C₈	—	4	2	no	—
78	C₂×C₇⋊C₈	—	8	2	no	—
79	C₇⋊C₁₆	—	4	2	no	—
80	#533	—	2	1	no	—
81	C₂×M₃₂	—	14	1	no	—
82	#500	C₂×D₇	1	1	yes	characteristic
83	#500	C₂×D₇	1	1	yes	characteristic
84	#499	—	2	1	no	—
85	#499	Dic₇	1	2	yes	—
86	#498	—	2	1	no	—
87	#427	—	2	1	no	—
88	#427	Dic₇	1	2	yes	—
89	C₂×C₄⋊C₈	—	14	1	no	—
90	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
91	C₄⋊C₁₆	—	7	2	no	—
92	C₄⋊C₁₆	—	14	1	no	—
93	#335	—	7	2	no	—
94	#335	—	14	1	no	—
95	C₂³×C₇	#102	1	1	yes	characteristic
96	C₇×Q₈	—	8	2	no	—
97	C₇×Q₈	—	8	1	no	—
98	C₇×Q₈	—	8	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	1	no	—
101	C₂×C₄×C₇	—	4	1	no	—
102	C₂×C₄×C₇	—	4	2	no	—
103	C₂×C₄×C₇	—	4	2	no	—
104	C₂×C₄×C₇	#101	1	2	yes	—
105	C₂×C₄×C₇	#102	1	1	yes	characteristic
106	C₂×C₄×C₇	#102	1	1	yes	characteristic
107	C₂×C₄×C₇	#102	1	1	yes	characteristic
108	C₂×C₄×C₇	#103	1	2	yes	—
109	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
112	C₇×C₈	—	2	1	no	—
113	C₇×C₈	—	2	1	no	—
114	C₇×C₈	#103	1	2	yes	—
115	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
116	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
117	C₇⋊C₈	—	8	2	no	—
118	C₂²×C₈	—	28	1	no	—
119	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
120	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
121	C₄⋊Q₈	—	2	1	no	—
122	C₄⋊Q₈	—	2	1	no	—
123	#125	—	2	2	no	—
124	C₂²⋊Q₈	—	4	1	no	—
125	C₂²⋊Q₈	—	4	1	no	—
126	C₄×Q₈	—	4	1	no	—
127	C₄×Q₈	—	4	1	no	—
128	#117	—	2	2	no	—
129	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
130	M₃₂	—	28	2	no	—
131	C₂×C₁₆	—	14	2	no	—
132	#107	—	2	1	no	—
133	#107	—	2	1	no	—
134	#107	C₂×Dic₇	1	2	yes	—
135	#107	C₂×Dic₇	1	2	yes	—
136	#106	—	2	2	no	—
137	#106	—	2	2	no	—
138	C₄⋊C₈	—	14	2	no	—
139	C₄⋊C₈	—	28	1	no	—
140	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
143	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
144	C₂²×C₇	—	2	1	no	—
145	C₂²×C₇	—	2	1	no	—
146	C₂²×C₇	#332	1	2	yes	—
147	C₂²×C₇	C₂×#102	1	1	yes	characteristic
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	—	2	1	no	—
150	C₄×C₇	—	2	1	no	—
151	C₄×C₇	—	8	2	no	—
152	C₄×C₇	—	8	2	no	—
153	C₄×C₇	#332	1	2	yes	—
154	C₄×C₇	C₂×#101	1	1	yes	characteristic
155	C₄×C₇	C₂×#103	1	1	yes	characteristic
156	C₄×C₇	#509	1	1	yes	characteristic
157	C₄×C₇	#510	1	1	yes	characteristic
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂²×C₄	#1141	1	1	yes	characteristic
161	C₂²×C₄	#1142	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	28	2	no	—
165	C₂×C₈	—	56	2	no	—
166	C₂×C₈	#1143	1	2	yes	—
167	C₂×C₈	#1144	1	2	yes	—
168	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	#1141	1	1	yes	characteristic
181	C₄²	#1142	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
184	C₁₆	—	28	2	no	—
185	C₂×C₇	—	2	2	no	—
186	C₂×C₇	C₂×#332	1	1	yes	characteristic
187	C₂×C₇	#2139	1	1	yes	characteristic
188	C₂×C₇	#2148	1	1	yes	characteristic
189	C₂³	#6406	1	1	yes	characteristic
190	Q₈	—	8	2	no	—
191	Q₈	—	8	1	no	—
192	Q₈	—	8	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#6406	1	1	yes	characteristic
199	C₂×C₄	#6406	1	1	yes	characteristic
200	C₂×C₄	#6406	1	1	yes	characteristic
201	C₂×C₄	#6409	1	2	yes	—
202	C₂×C₄	#6410	1	2	yes	—
203	C₂×C₄	C₂×#1141	1	1	yes	characteristic
204	C₂×C₄	C₂×#1142	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	56	2	no	—
209	C₈	#6409	1	2	yes	—
210	C₈	C₂×#1143	1	1	yes	characteristic
211	C₈	C₂×#1144	1	1	yes	characteristic
212	C₇	#11942	1	1	yes	characteristic, complemented, Sylow
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#89342	1	2	yes	—
216	C₂²	C₂×#6406	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	8	2	no	—
221	C₄	—	8	2	no	—
222	C₄	#89342	1	2	yes	—
223	C₄	#89965	1	1	yes	characteristic
224	C₄	#89967	1	1	yes	characteristic
225	C₄	C₂×#6409	1	1	yes	characteristic
226	C₄	C₂×#6410	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#89342	1	1	yes	characteristic
229	C₂	#11729832	1	1	yes	characteristic
230	C₂	#11729835	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂×#102
prime radical	C₂³×C₇	#102
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#3160	D₇
hypocenter	C₇	#11942
LCS	G ▷ C₂×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
4. #11725608	C₂	semidirect
5. #11725607	C₂	semidirect
212. C₇	#11942	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
#11727497	2
C₂×#89253	2
#11725608	2
#11725607	2
#11942	7