Subgroups — #13265366

Order 1792.

270 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2150	C₂	1	1	yes	maximal, characteristic
2	#11727497	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11725361	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×M₃₂	—	2	1	no	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6386	—	2	1	no	—
8	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	—	2	1	no	—
11	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
14	C₈×C₇⋊C₈	C₄	1	2	yes	—
15	C₈×C₇⋊C₈	C₂²	1	2	yes	—
16	#12004	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇⋊C₈	—	2	1	no	—
21	C₂²×C₇⋊C₈	C₈	1	2	yes	—
22	C₇×M₃₂	—	4	2	no	—
23	C₂×C₇×C₁₆	—	2	2	no	—
24	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₇×C₈	D₄	1	1	yes	characteristic
26	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
28	#6386	—	4	2	no	—
29	C₂×C₇⋊C₁₆	—	2	2	no	—
30	C₄×C₇⋊C₈	—	2	1	no	—
31	C₄×C₇⋊C₈	C₈	1	1	yes	characteristic
32	C₄×C₇⋊C₈	C₈	1	1	yes	characteristic
33	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×C₇⋊C₈	C₂×C₄	1	2	yes	—
35	#2150	—	7	1	no	—
36	#2150	D₇	1	1	yes	characteristic
37	C₂×C₈²	—	7	1	no	—
38	#1370	—	7	4	no	—
39	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇×C₈	D₈	1	2	yes	—
46	C₂×C₇×C₈	Q₁₆	1	2	yes	—
47	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₇	M₁₆	1	1	yes	characteristic
52	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₇⋊C₈	—	2	4	no	—
54	C₂×C₇⋊C₈	—	2	2	no	—
55	C₂×C₇⋊C₈	—	2	4	no	—
56	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
57	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
58	C₇×C₁₆	—	4	2	no	—
59	C₇⋊C₁₆	—	4	2	no	—
60	C₂×M₃₂	—	2	1	no	—
61	C₂×M₃₂	—	14	1	no	—
62	C₂×C₄×C₈	—	7	1	no	—
63	C₂×C₄×C₈	—	7	1	no	—
64	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	1	no	—
66	C₄⋊C₁₆	—	7	2	no	—
67	C₄⋊C₁₆	—	14	1	no	—
68	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
69	C₈²	—	7	2	no	—
70	C₈²	—	7	2	no	—
71	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇	#107	1	2	yes	—
78	C₂×C₄×C₇	#108	1	2	yes	—
79	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
82	C₇×C₈	—	2	1	no	—
83	C₇×C₈	—	2	1	no	—
84	C₇×C₈	#107	1	2	yes	—
85	C₇×C₈	C₂×D₈	1	1	yes	characteristic
86	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
87	C₇⋊C₈	—	2	4	no	—
88	C₇⋊C₈	—	2	2	no	—
89	C₇⋊C₈	M₃₂	1	4	yes	complemented
90	C₂²×C₈	—	7	2	no	—
91	C₂²×C₈	—	14	1	no	—
92	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
93	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
95	M₃₂	—	4	2	no	—
96	M₃₂	—	28	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	14	2	no	—
99	C₄×C₈	—	7	1	no	—
100	C₄×C₈	—	7	1	no	—
101	C₄×C₈	—	7	2	no	—
102	C₄×C₈	—	7	2	no	—
103	C₄×C₈	—	14	1	no	—
104	C₄×C₈	Dic₁₄	1	1	yes	characteristic
105	C₄×C₈	C₄×D₇	1	1	yes	characteristic
106	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
107	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
108	C₂²×C₇	—	2	1	no	—
109	C₂²×C₇	—	2	1	no	—
110	C₂²×C₇	#335	1	2	yes	—
111	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₇	—	2	1	no	—
113	C₄×C₇	—	2	1	no	—
114	C₄×C₇	#335	1	2	yes	—
115	C₄×C₇	#364	1	2	yes	—
116	C₄×C₇	C₂×#107	1	1	yes	characteristic
117	C₄×C₇	C₂×#108	1	1	yes	characteristic
118	C₄×C₇	C₂×M₃₂	1	2	yes	—
119	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
120	C₂²×C₄	#1131	1	1	yes	characteristic
121	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	7	2	no	—
124	C₂×C₈	—	7	2	no	—
125	C₂×C₈	—	14	4	no	—
126	C₂×C₈	—	14	4	no	—
127	C₂×C₈	—	14	2	no	—
128	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#1141	1	2	yes	—
131	C₂×C₈	#1144	1	2	yes	—
132	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
133	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
134	C₄²	C₈×D₇	1	1	yes	characteristic
135	C₄²	#1131	1	1	yes	characteristic
136	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
138	C₁₆	—	4	2	no	—
139	C₁₆	—	28	2	no	—
140	C₂×C₇	—	2	2	no	—
141	C₂×C₇	C₂×#335	1	1	yes	characteristic
142	C₂×C₇	#2150	1	1	yes	characteristic
143	C₂×C₇	C₂×#364	1	1	yes	characteristic
144	C₂³	Dic₇⋊C₈	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	#6381	1	2	yes	—
148	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
149	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
150	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
151	C₂×C₄	#6396	1	2	yes	—
152	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
153	C₂×C₄	C₂×#1131	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	7	4	no	—
158	C₈	—	14	4	no	—
159	C₈	—	14	2	no	—
160	C₈	#6381	1	2	yes	—
161	C₈	C₂×#1141	1	1	yes	characteristic
162	C₈	C₂×#1144	1	1	yes	characteristic
163	C₇	#12004	1	1	yes	characteristic, complemented, Sylow
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#89274	1	2	yes	—
167	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#89274	1	2	yes	—
171	C₄	#89306	1	2	yes	—
172	C₄	D₇×M₃₂	1	2	yes	—
173	C₄	C₂×#6381	1	1	yes	characteristic
174	C₄	C₂×#6396	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	C₂×#89274	1	1	yes	characteristic
177	C₂	#11729174	1	1	yes	characteristic
178	C₂	C₂×#89306	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2150	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#2150	D₇
hypocenter	C₇	#12004
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
4. #11725361	C₂	semidirect
89. C₇⋊C₈	M₃₂	semidirect
163. C₇	#12004	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2150	2
#11727497	2
C₂×C₈×C₇⋊C₈	2
#11725361	2
#12004	7