Subgroups — #13265164

Order 1792.

292 conjugacy classes of subgroups.

The subgroup lattice has 204 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	C₂×#89258	C₂	1	1	yes	maximal, characteristic
3	C₂×#89258	C₂	1	1	yes	maximal, characteristic
4	#11725606	C₂	1	4	yes	maximal, complemented
5	C₇×#533	C₄	1	2	yes	—
6	C₇×#500	—	2	1	no	—
7	C₇×#499	C₄	1	2	yes	—
8	C₇×#499	C₂²	1	2	yes	—
9	C₇×#498	—	2	1	no	—
10	C₇×#427	—	2	1	no	—
11	C₇×#427	—	2	1	no	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6386	—	2	1	no	—
14	C₂×#6386	—	2	1	no	—
15	#89258	—	2	1	no	—
16	#89258	—	2	1	no	—
17	#89258	C₂²	1	2	yes	—
18	#89258	C₂²	1	2	yes	—
19	#11956	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
22	C₇×C₄⋊Q₈	—	2	2	no	—
23	C₇×#125	C₂×C₄	1	4	yes	—
24	C₇×C₂²⋊Q₈	—	4	2	no	—
25	C₄×C₇×Q₈	—	4	2	no	—
26	C₇×#117	—	2	2	no	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₇×#107	—	2	2	no	—
29	C₇×#107	—	2	2	no	—
30	C₇×#106	—	2	2	no	—
31	C₇×#106	—	2	2	no	—
32	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₇×C₈	D₄	1	1	yes	characteristic
34	C₄×C₇×C₈	D₄	1	1	yes	characteristic
35	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
36	#6386	—	2	2	no	—
37	#6386	—	2	2	no	—
38	#6386	—	4	1	no	—
39	#6386	—	4	1	no	—
40	C₂×C₇⋊C₁₆	—	4	1	no	—
41	C₂×C₇⋊C₁₆	—	4	1	no	—
42	#3160	D₇	1	1	yes	characteristic
43	C₂×#364	—	7	1	no	—
44	C₂×#364	—	7	1	no	—
45	#1366	—	7	4	no	—
46	C₂×C₇×Q₈	—	4	2	no	—
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
54	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
55	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
56	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
57	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
58	C₂×C₇×C₈	Q₁₆	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
61	C₇×C₄⋊C₄	—	2	4	no	—
62	C₇×C₄⋊C₄	—	4	4	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₂²⋊C₄	—	4	2	no	—
65	C₄²×C₇	—	2	1	no	—
66	C₄²×C₇	—	4	2	no	—
67	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₇⋊C₁₆	—	4	2	no	—
70	C₇⋊C₁₆	—	4	2	no	—
71	#533	Dic₇	1	2	yes	—
72	C₂×M₃₂	—	14	1	no	—
73	C₂×M₃₂	—	14	1	no	—
74	#500	—	2	1	no	—
75	#499	Dic₇	1	2	yes	—
76	#499	C₂×D₇	1	2	yes	—
77	#498	—	2	1	no	—
78	#427	—	2	1	no	—
79	#427	—	2	1	no	—
80	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
81	#364	—	7	2	no	—
82	#364	—	7	2	no	—
83	#364	—	14	1	no	—
84	#364	—	14	1	no	—
85	C₂³×C₇	#100	1	1	yes	characteristic
86	C₇×Q₈	—	8	4	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	4	2	no	—
90	C₂×C₄×C₇	—	4	4	no	—
91	C₂×C₄×C₇	#100	1	1	yes	characteristic
92	C₂×C₄×C₇	#103	1	1	yes	characteristic
93	C₂×C₄×C₇	#103	1	1	yes	characteristic
94	C₂×C₄×C₇	#103	1	1	yes	characteristic
95	C₂×C₄×C₇	#103	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₇×C₈	#103	1	1	yes	characteristic
98	C₇×C₈	#103	1	1	yes	characteristic
99	C₇×C₈	#103	1	1	yes	characteristic
100	C₇×C₈	#103	1	1	yes	characteristic
101	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
102	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
103	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
104	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
107	C₄⋊Q₈	—	2	2	no	—
108	#125	C₂×Dic₇	1	4	yes	—
109	C₂²⋊Q₈	—	4	2	no	—
110	C₄×Q₈	—	4	2	no	—
111	#117	—	2	2	no	—
112	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
113	M₃₂	—	14	2	no	—
114	M₃₂	—	14	2	no	—
115	M₃₂	—	28	1	no	—
116	M₃₂	—	28	1	no	—
117	C₂×C₁₆	—	28	1	no	—
118	C₂×C₁₆	—	28	1	no	—
119	#107	—	2	2	no	—
120	#107	—	2	2	no	—
121	#106	—	2	2	no	—
122	#106	—	2	2	no	—
123	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
125	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
126	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
127	C₂²×C₇	—	4	1	no	—
128	C₂²×C₇	#333	1	1	yes	characteristic
129	C₂²×C₇	#333	1	1	yes	characteristic
130	C₂²×C₇	C₂×#100	1	1	yes	characteristic
131	C₄×C₇	—	2	2	no	—
132	C₄×C₇	—	4	1	no	—
133	C₄×C₇	—	8	4	no	—
134	C₄×C₇	#333	1	1	yes	characteristic
135	C₄×C₇	#333	1	1	yes	characteristic
136	C₄×C₇	C₂×#103	1	1	yes	characteristic
137	C₄×C₇	C₂×#103	1	1	yes	characteristic
138	C₂×Q₈	—	4	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
141	C₂×C₈	#1143	1	1	yes	characteristic
142	C₂×C₈	#1143	1	1	yes	characteristic
143	C₂×C₈	#1143	1	1	yes	characteristic
144	C₂×C₈	#1143	1	1	yes	characteristic
145	C₂×C₈	#1144	1	1	yes	characteristic
146	C₂×C₈	#1144	1	1	yes	characteristic
147	C₂×C₈	#1144	1	1	yes	characteristic
148	C₂×C₈	#1144	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
150	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
160	C₄²	C₂²×Dic₇	1	1	yes	characteristic
161	C₁₆	—	28	2	no	—
162	C₁₆	—	28	2	no	—
163	C₂×C₇	—	2	1	no	—
164	C₂×C₇	—	2	1	no	—
165	C₂×C₇	—	2	1	no	—
166	C₂×C₇	C₂×#333	1	1	yes	characteristic
167	C₂³	#6407	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	#6407	1	1	yes	characteristic
174	C₂×C₄	#6409	1	1	yes	characteristic
175	C₂×C₄	#6409	1	1	yes	characteristic
176	C₂×C₄	#6409	1	1	yes	characteristic
177	C₂×C₄	#6409	1	1	yes	characteristic
178	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
179	C₈	#6409	1	1	yes	characteristic
180	C₈	#6409	1	1	yes	characteristic
181	C₈	#6409	1	1	yes	characteristic
182	C₈	#6409	1	1	yes	characteristic
183	C₈	C₂×#1143	1	1	yes	characteristic
184	C₈	C₂×#1143	1	1	yes	characteristic
185	C₈	C₂×#1144	1	1	yes	characteristic
186	C₈	C₂×#1144	1	1	yes	characteristic
187	C₇	#11956	1	1	yes	characteristic, complemented, Sylow
188	C₂²	—	4	1	no	—
189	C₂²	#89341	1	1	yes	characteristic
190	C₂²	#89341	1	1	yes	characteristic
191	C₂²	C₂×#6407	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	4	1	no	—
194	C₄	—	8	4	no	—
195	C₄	#89341	1	1	yes	characteristic
196	C₄	#89341	1	1	yes	characteristic
197	C₄	C₂×#6409	1	1	yes	characteristic
198	C₄	C₂×#6409	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	C₂×#89341	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#89341
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#3160	C₂
socle	C₂×C₇	C₂×#333
prime radical	C₂³×C₇	#100
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#3160	D₇
hypocenter	C₇	#11956
LCS	G ▷ C₄²×C₇ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
4. #11725606	C₂	semidirect
187. C₇	#11956	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
C₂×#89258	2
#11725606	2
#11956	7