Group #22566155 — subgroups

Subgroups — #22566155

Order 8576.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#14524040	C₂	1	1	yes	maximal, characteristic
2	#16863313	C₂	1	8	yes	maximal, complemented
3	#16863297	C₂	1	4	yes	maximal, complemented
4	#16863242	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	2	yes	—
7	C₂²×D₂₆₈	C₂²	1	4	yes	—
8	#14524040	—	2	4	no	—
9	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	16	yes	—
10	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
11	C₆₇⋊#95	C₂²	1	4	yes	—
12	#125×C₆₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₆₇	C₂³	1	4	yes	—
17	C₂×D₂₆₈	—	2	8	no	—
18	C₂×D₂₆₈	—	2	4	no	—
19	C₂×D₂₆₈	—	2	8	no	—
20	C₂×D₂₆₈	—	4	8	no	—
21	C₂²×Dic₆₇	C₂³	1	4	yes	—
22	(C₂×D₆₇)⋊C₄	—	4	16	no	—
23	C₄×Dic₆₇	—	2	4	no	—
24	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₆₇	—	2	8	no	—
28	C₄⋊C₄×C₆₇	—	2	4	no	—
29	C₄²×C₆₇	D₄	1	4	yes	—
30	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₂²×D₆₇	—	2	4	no	—
32	C₂²×D₆₇	—	2	8	no	—
33	C₂²×D₆₇	—	2	16	no	—
34	C₂×C₄×C₆₇	—	2	4	no	—
35	C₂×C₄×C₆₇	—	2	4	no	—
36	C₂×C₄×C₆₇	—	2	8	no	—
37	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₆₇	#41	1	4	yes	—
40	C₂×Dic₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	2	8	no	—
42	D₂₆₈	—	4	16	no	—
43	D₂₆₈	—	4	8	no	—
44	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₆₇	C₂×#41	1	2	yes	—
46	C₂²×C₆₇	H(5,𝔽₂)	1	4	yes	—
47	C₂×D₆₇	—	4	4	no	—
48	C₂×D₆₇	—	4	8	no	—
49	C₂×D₆₇	—	4	16	no	—
50	C₄×C₆₇	—	2	4	no	—
51	C₄×C₆₇	—	4	4	no	—
52	C₄×C₆₇	#124	1	4	yes	—
53	Dic₆₇	—	4	4	no	—
54	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
55	C₂×C₆₇	#535	1	2	yes	—
56	C₂×C₆₇	#540	1	4	yes	—
57	D₆₇	—	8	4	no	—
58	#2641	—	67	1	no	maximal, Sylow
59	C₆₇	#2641	1	1	yes	characteristic, complemented, Sylow
60	C₂×#127	—	67	1	no	—
61	C₂×#125	D₆₇	1	1	yes	characteristic, complemented
62	#394	—	67	8	no	—
63	#390	—	67	4	no	—
64	#376	—	67	1	no	—
65	C₂²×D₄	—	67	4	no	—
66	C₂²×D₄	—	67	2	no	—
67	#127	—	134	4	no	—
68	#125	—	2	4	no	—
69	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
70	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
71	C₂×C₂²⋊C₄	—	67	16	no	—
72	C₂×C₄²	—	67	2	no	—
73	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
74	#95	—	67	4	no	—
75	C₂⁴	—	67	4	no	—
76	C₂×D₄	—	268	8	no	—
77	C₂×D₄	—	134	8	no	—
78	C₂×D₄	—	134	4	no	—
79	C₂×D₄	—	134	8	no	—
80	C₂²×C₄	—	67	4	no	—
81	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
82	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
84	C₄⋊C₄	—	2	8	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₂²⋊C₄	—	268	16	no	—
87	C₄²	—	134	4	no	—
88	C₄²	C₂²⋊D₆₇	1	4	yes	—
89	C₂³	—	134	4	no	—
90	C₂³	—	134	8	no	—
91	C₂³	—	134	16	no	—
92	C₂³	C₂³×D₆₇	1	1	yes	characteristic
93	D₄	—	268	16	no	—
94	D₄	—	268	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	134	4	no	—
99	C₂×C₄	—	134	8	no	—
100	C₂×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
102	C₂×C₄	#11761629	1	4	yes	—
103	C₂²	—	268	16	no	—
104	C₂²	—	268	4	no	—
105	C₂²	—	268	8	no	—
106	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
107	C₂²	C₂×#11761629	1	2	yes	—
108	C₂²	#14524089	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	268	4	no	—
112	C₄	#14524032	1	4	yes	—
113	C₂	—	536	4	no	—
114	C₂	C₂×#14524032	1	1	yes	characteristic
115	C₂	#16863960	1	2	yes	—
116	C₂	#16864002	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524040

C₂

yes

maximal, characteristic

#16863313

C₂

yes

maximal, complemented

#16863297

C₂

yes

maximal, complemented

#16863242

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

—

C₂²×D₂₆₈

C₂²

yes

—

#14524040

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

#125×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

C₂³

yes

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

C₂³

yes

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#124

yes

—

Dic₆₇

—

C₂×C₆₇

C₂×#124

yes

characteristic

C₂×C₆₇

#535

yes

—

C₂×C₆₇

#540

yes

—

D₆₇

—

#2641

—

maximal, Sylow

C₆₇

#2641

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#125

D₆₇

yes

characteristic, complemented

#394

—

#390

—

#376

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

134

—

#125

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#95

—

C₂⁴

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

268

—

C₄²

—

134

—

C₄²

C₂²⋊D₆₇

yes

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

C₂³×D₆₇

yes

characteristic

D₄

—

268

—

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

100

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

102

C₂×C₄

#11761629

yes

—

103

C₂²

—

268

—

104

C₂²

—

268

—

105

C₂²

—

268

—

106

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

107

C₂²

C₂×#11761629

yes

—

108

C₂²

#14524089

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

268

—

112

C₄

#14524032

yes

—

113

C₂

—

536

—

114

C₂

C₂×#14524032

yes

characteristic

115

C₂

#16863960

yes

—

116

C₂

#16864002

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×#125×C₆₇	C₂
hypercenter	C₂×#125	D₆₇
hypocenter	C₆₇	#2641
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×#125×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #16863313	C₂	semidirect
3. #16863297	C₂	semidirect
4. #16863242	C₂	semidirect
5. C₂×#125×C₆₇	C₂	semidirect
59. C₆₇	#2641	semidirect
61. C₂×#125	D₆₇	semidirect

H ≅ G/N

type

2. #16863313

C₂

semidirect

3. #16863297

C₂

semidirect

4. #16863242

C₂

semidirect

5. C₂×#125×C₆₇

C₂

semidirect

59. C₆₇

#2641

semidirect

61. C₂×#125

D₆₇

semidirect

Subgroups — #22566155

Conjugacy classes of subgroups

Complements