Subgroups — #13265367

Order 1792.

270 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈²	C₂	1	1	yes	maximal, characteristic
2	#11727497	C₂	1	2	yes	maximal
3	#11725246	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×#6386	—	2	2	no	—
8	C₇×C₈²	C₄	1	2	yes	—
9	C₇×C₈²	C₂²	1	2	yes	—
10	C₄⋊(C₇⋊C₁₆)	—	2	2	no	—
11	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
12	#12004	—	7	1	no	maximal, Sylow
13	C₂²×C₇×C₈	—	2	1	no	—
14	C₂²×C₇×C₈	C₈	1	2	yes	—
15	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₇×C₈	—	2	1	no	—
19	C₄×C₇×C₈	C₈	1	1	yes	characteristic
20	C₄×C₇×C₈	C₈	1	1	yes	characteristic
21	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇×C₈	D₄	1	1	yes	characteristic
25	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
27	#6386	—	4	4	no	—
28	C₂×C₇⋊C₁₆	—	2	4	no	—
29	#2150	—	7	2	no	—
30	C₂×C₈²	D₇	1	1	yes	characteristic
31	#1370	—	7	4	no	—
32	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
35	C₂×C₇×C₈	—	2	4	no	—
36	C₂×C₇×C₈	—	2	2	no	—
37	C₂×C₇×C₈	—	2	2	no	—
38	C₂×C₇×C₈	—	2	4	no	—
39	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
42	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
43	C₂×C₇×C₈	D₈	1	2	yes	—
44	C₂×C₇×C₈	Q₁₆	1	2	yes	—
45	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₇	M₁₆	1	1	yes	characteristic
50	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₇⋊C₁₆	—	4	4	no	—
52	C₂×M₃₂	—	14	2	no	—
53	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
54	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
56	C₄⋊C₁₆	—	7	4	no	—
57	C₄⋊C₁₆	—	14	2	no	—
58	C₈²	Dic₇	1	2	yes	—
59	C₈²	C₂×D₇	1	2	yes	—
60	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇	—	2	2	no	—
62	C₂×C₄×C₇	—	2	2	no	—
63	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₇	#107	1	2	yes	—
67	C₂×C₄×C₇	#108	1	2	yes	—
68	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
71	C₇×C₈	—	2	4	no	—
72	C₇×C₈	—	2	1	no	—
73	C₇×C₈	—	2	1	no	—
74	C₇×C₈	—	2	2	no	—
75	C₇×C₈	#107	1	2	yes	—
76	C₇×C₈	M₃₂	1	4	yes	complemented
77	C₇×C₈	C₂×D₈	1	1	yes	characteristic
78	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	C₇⋊C₈	1	2	yes	—
81	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
82	C₂²×C₈	D₂₈	1	1	yes	characteristic
83	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
84	M₃₂	—	28	4	no	—
85	C₂×C₁₆	—	14	4	no	—
86	C₄×C₈	—	2	1	no	—
87	C₄×C₈	C₇⋊C₈	1	1	yes	characteristic
88	C₄×C₈	C₇⋊C₈	1	1	yes	characteristic
89	C₄×C₈	Dic₁₄	1	1	yes	characteristic
90	C₄×C₈	D₂₈	1	1	yes	characteristic
91	C₄×C₈	C₂×Dic₇	1	2	yes	—
92	C₄×C₈	C₂×Dic₇	1	2	yes	—
93	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
94	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
95	C₂²×C₇	—	2	1	no	—
96	C₂²×C₇	—	2	1	no	—
97	C₂²×C₇	#335	1	2	yes	—
98	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₇	—	2	1	no	—
100	C₄×C₇	—	2	1	no	—
101	C₄×C₇	#335	1	2	yes	—
102	C₄×C₇	#364	1	2	yes	—
103	C₄×C₇	C₂×#107	1	1	yes	characteristic
104	C₄×C₇	C₂×#108	1	1	yes	characteristic
105	C₄×C₇	C₂×M₃₂	1	2	yes	—
106	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
107	C₂²×C₄	#1136	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	D₅₆	1	2	yes	—
114	C₂×C₈	Dic₂₈	1	2	yes	—
115	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
116	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
117	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
118	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
119	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
120	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
121	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
122	C₄²	#1136	1	1	yes	characteristic
123	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
124	C₄²	C₂²×Dic₇	1	1	yes	characteristic
125	C₁₆	—	28	4	no	—
126	C₂×C₇	—	2	2	no	—
127	C₂×C₇	C₂×#335	1	1	yes	characteristic
128	C₂×C₇	#2150	1	1	yes	characteristic
129	C₂×C₇	C₂×#364	1	1	yes	characteristic
130	C₂³	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
134	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
135	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
136	C₂×C₄	#6392	1	2	yes	—
137	C₂×C₄	#6393	1	2	yes	—
138	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
139	C₂×C₄	C₂×#1136	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	#6386	1	4	yes	complemented
146	C₈	#6392	1	2	yes	—
147	C₈	C₂×D₅₆	1	1	yes	characteristic
148	C₈	C₂×Dic₂₈	1	1	yes	characteristic
149	C₇	#12004	1	1	yes	characteristic, complemented, Sylow
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#89253	1	2	yes	—
153	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#89253	1	2	yes	—
157	C₄	#89258	1	2	yes	—
158	C₄	C₂×#6386	1	2	yes	—
159	C₄	C₂×#6392	1	1	yes	characteristic
160	C₄	C₂×#6393	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#89253	1	1	yes	characteristic
163	C₂	#11727497	1	1	yes	characteristic
164	C₂	C₂×#89258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×C₈²	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×C₈²	D₇
hypocenter	C₇	#12004
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #11725246	C₂	semidirect
76. C₇×C₈	M₃₂	semidirect
145. C₈	#6386	semidirect
149. C₇	#12004	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈²	2
#11727497	2
#11725246	2
#12004	7