Group #22558499 — subgroups

Subgroups — #22558499

Order 8576.

208 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#16863118	C₂	1	2	yes	maximal, complemented
4	#16863117	C₂	1	2	yes	maximal, complemented
5	#431×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₄	1	2	yes	—
7	C₂²×C₆₇⋊C₈	C₄	1	2	yes	—
8	#14523954	C₄	1	2	yes	—
9	C₄×C₆₇⋊C₈	C₄	1	2	yes	—
10	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₇	C₄	1	2	yes	—
15	C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
16	C₄×C₈×C₆₇	C₂²	1	2	yes	—
17	#11761612	—	2	4	no	—
18	C₂×C₆₇⋊C₈	—	2	4	no	—
19	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
26	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₆₇	C₂³	1	1	yes	characteristic
30	C₂³×C₆₇	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₆₇	—	2	2	no	—
32	C₂×C₄×C₆₇	—	2	2	no	—
33	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₆₇	M₁₆	1	4	yes	—
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₆₇	—	2	4	no	—
41	C₆₇⋊C₈	—	2	2	no	—
42	C₆₇⋊C₈	—	2	4	no	—
43	C₆₇⋊C₈	M₁₆	1	4	yes	complemented
44	C₂²×C₆₇	—	2	1	no	—
45	C₂²×C₆₇	—	2	1	no	—
46	C₂²×C₆₇	#97	1	2	yes	—
47	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₆₇	—	2	1	no	—
49	C₄×C₆₇	—	2	1	no	—
50	C₄×C₆₇	#97	1	2	yes	—
51	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
52	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
53	C₄×C₆₇	#131	1	2	yes	—
54	C₂×C₆₇	—	2	2	no	—
55	C₂×C₆₇	C₂×#97	1	1	yes	characteristic
56	C₂×C₆₇	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₆₇	#405	1	1	yes	characteristic
58	#1451	—	67	1	no	maximal, Sylow
59	C₆₇	#1451	1	1	yes	characteristic, complemented, Sylow
60	#431	D₆₇	1	1	yes	characteristic
61	C₄×M₁₆	—	67	1	no	—
62	C₂×C₄×C₈	—	67	1	no	—
63	#322	—	67	2	no	—
64	#322	—	67	2	no	—
65	C₂×M₁₆	—	67	2	no	—
66	C₂²×C₈	—	67	2	no	—
67	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
68	C₄⋊C₈	Dic₆₇	1	2	yes	—
69	C₂²⋊C₈	Dic₆₇	1	2	yes	—
70	#97	—	67	2	no	—
71	C₄×C₈	—	67	2	no	—
72	C₄×C₈	—	67	1	no	—
73	C₄×C₈	—	67	1	no	—
74	C₄×C₈	—	67	2	no	—
75	C₄×C₈	C₂×D₆₇	1	2	yes	—
76	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
79	M₁₆	—	134	4	no	—
80	C₂×C₈	—	67	2	no	—
81	C₂×C₈	—	67	2	no	—
82	C₂×C₈	—	67	4	no	—
83	C₂×C₈	—	134	4	no	—
84	C₂×C₈	C₂×Dic₆₇	1	4	yes	—
85	C₄²	C₄×D₆₇	1	1	yes	characteristic
86	C₄²	C₄×D₆₇	1	1	yes	characteristic
87	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
88	C₄²	C₂²×D₆₇	1	1	yes	characteristic
89	C₂³	C₄×Dic₆₇	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
95	C₂×C₄	#11761611	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
99	C₈	—	2	4	no	—
100	C₈	—	67	4	no	—
101	C₈	—	134	2	no	—
102	C₈	—	134	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#14523953	1	2	yes	—
106	C₂²	C₂×C₄×Dic₆₇	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#14523953	1	2	yes	—
110	C₄	M₁₆×D₆₇	1	2	yes	—
111	C₄	C₂×#11761611	1	2	yes	—
112	C₄	#14524050	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#14523953	1	1	yes	characteristic
115	C₂	M₁₆×Dic₆₇	1	1	yes	characteristic
116	C₂	#16863352	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11761612

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂

yes

maximal, characteristic

#16863118

C₂

yes

maximal, complemented

#16863117

C₂

yes

maximal, complemented

#431×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₄

yes

—

C₂²×C₆₇⋊C₈

C₄

yes

—

#14523954

C₄

yes

—

C₄×C₆₇⋊C₈

C₄

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₄

yes

—

C₂²⋊C₈×C₆₇

C₄

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

M₁₆

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#97

yes

—

C₂²×C₆₇

C₂×C₄²

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#97

yes

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

#131

yes

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#97

yes

characteristic

C₂×C₆₇

C₄×M₁₆

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#431

D₆₇

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

#322

—

#322

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

Dic₆₇

yes

—

C₂²⋊C₈

Dic₆₇

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂×Dic₆₇

yes

—

C₄²

C₄×D₆₇

yes

characteristic

C₄²

C₄×D₆₇

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

C₄×Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₆₇

yes

characteristic

C₂×C₄

C₄×Dic₆₇

yes

characteristic

C₂×C₄

C₄×Dic₆₇

yes

characteristic

C₂×C₄

#11761611

yes

—

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

134

—

102

C₈

—

134

—

103

C₂²

—

104

C₂²

—

105

C₂²

#14523953

yes

—

106

C₂²

C₂×C₄×Dic₆₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#14523953

yes

—

110

C₄

M₁₆×D₆₇

yes

—

111

C₄

C₂×#11761611

yes

—

112

C₄

#14524050

yes

—

113

C₂

—

114

C₂

C₂×#14523953

yes

characteristic

115

C₂

M₁₆×Dic₆₇

yes

characteristic

116

C₂

#16863352

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄²
Fitting	#431×C₆₇	C₂
hypercenter	#431	D₆₇
hypocenter	C₆₇	#1451
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂×C₄²

Fitting

#431×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #16863118	C₂	semidirect
4. #16863117	C₂	semidirect
43. C₆₇⋊C₈	M₁₆	semidirect
59. C₆₇	#1451	semidirect

H ≅ G/N

type

3. #16863118

C₂

semidirect

4. #16863117

C₂

semidirect

43. C₆₇⋊C₈

M₁₆

semidirect

59. C₆₇

#1451

semidirect

Subgroups — #22558499

Conjugacy classes of subgroups

Complements