Group #22570952 — subgroups

Subgroups — #22570952

Order 8576.

358 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16863951	C₂	1	2	yes	maximal
2	#16863794	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863746	C₂	1	2	yes	maximal, complemented
4	#16863651	C₂	1	4	yes	maximal
5	#16863503	C₂	1	4	yes	maximal, complemented
6	C₄×#11761611	C₂	1	1	yes	maximal, characteristic
7	#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761611	C₂²	1	2	yes	—
9	#14524043	C₂²	1	2	yes	—
10	#14524034	—	2	2	no	—
11	#14524023	—	2	2	no	—
12	#14524019	C₂²	1	4	yes	—
13	Q₈×Dic₆₇	C₂²	1	4	yes	—
14	#14524008	—	2	2	no	—
15	#14524001	—	2	2	no	—
16	C₄²×D₆₇	C₂²	1	1	yes	characteristic
17	#14523975	C₂²	1	4	yes	—
18	#14523967	C₂²	1	8	yes	—
19	#14523953	C₂²	1	2	yes	—
20	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₆₇	C₂²	1	4	yes	—
22	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×C₄×D₆₇	D₄	1	2	yes	—
26	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
27	#11761611	—	2	4	no	—
28	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₆₇	—	2	2	no	—
30	C₄⋊Dic₆₇	—	2	4	no	—
31	C₄⋊Dic₆₇	C₂³	1	4	yes	—
32	Dic₆₇⋊C₄	—	2	4	no	—
33	(C₂×D₆₇)⋊C₄	—	2	4	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₄×Dic₆₇	D₄	1	2	yes	—
36	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
38	Q₁₆×C₆₇	—	2	8	no	—
39	C₂×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₆₇	—	2	8	no	—
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₆₇	—	2	4	no	—
49	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
50	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
51	C₄×D₆₇	—	2	4	no	—
52	C₄×D₆₇	—	2	2	no	—
53	Dic₁₃₄	—	4	4	no	—
54	C₈×C₆₇	#41	1	4	yes	—
55	C₆₇⋊C₈	—	2	2	no	—
56	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₆₇	—	2	1	no	—
58	C₂×D₆₇	—	2	2	no	—
59	C₄×C₆₇	—	4	4	no	—
60	C₄×C₆₇	#137	1	4	yes	—
61	C₄×C₆₇	C₂×#41	1	2	yes	—
62	Dic₆₇	—	2	1	no	—
63	Dic₆₇	—	2	2	no	—
64	Dic₆₇	—	4	4	no	—
65	C₂×C₆₇	#532	1	1	yes	characteristic
66	C₂×C₆₇	C₂×#137	1	2	yes	—
67	D₆₇	—	4	1	no	—
68	#3185	—	67	1	no	maximal, Sylow
69	C₆₇	#3185	1	1	yes	characteristic, complemented, Sylow
70	#533	—	67	2	no	—
71	#498	—	67	1	no	—
72	#494	D₆₇	1	1	yes	characteristic, complemented
73	#487	—	67	2	no	—
74	#470	—	67	4	no	—
75	#441	—	67	4	no	—
76	C₄×M₁₆	—	67	1	no	—
77	C₂×Q₁₆	C₂×D₆₇	1	4	yes	—
78	C₂×M₁₆	—	67	2	no	—
79	C₄⋊Q₈	—	67	2	no	—
80	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
81	#125	—	134	2	no	—
82	C₂²⋊Q₈	—	67	4	no	—
83	C₂²⋊Q₈	—	134	2	no	—
84	C₄×Q₈	—	67	4	no	—
85	C₄×Q₈	—	134	2	no	—
86	#117	—	134	2	no	—
87	C₂×C₄²	—	67	1	no	—
88	#106	—	67	4	no	—
89	#103	—	67	8	no	—
90	#97	—	67	2	no	—
91	C₄×C₈	—	67	1	no	—
92	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	134	2	no	—
94	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
95	C₂²×C₄	—	67	2	no	—
96	C₂²×C₄	—	67	1	no	—
97	Q₁₆	—	2	8	no	—
98	M₁₆	—	134	4	no	—
99	C₂×C₈	—	67	2	no	—
100	C₂×C₈	C₂²×D₆₇	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	67	4	no	—
104	C₄⋊C₄	—	134	4	no	—
105	C₄⋊C₄	—	134	4	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₂²⋊C₄	—	134	4	no	—
108	C₄²	—	67	2	no	—
109	C₄²	—	67	1	no	—
110	C₄²	—	134	4	no	—
111	C₄²	C₂²×D₆₇	1	1	yes	characteristic
112	C₂³	—	67	1	no	—
113	Q₈	—	2	8	no	—
114	Q₈	—	268	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	67	2	no	—
117	C₂×C₄	—	67	1	no	—
118	C₂×C₄	—	134	4	no	—
119	C₂×C₄	—	134	4	no	—
120	C₂×C₄	—	134	2	no	—
121	C₂×C₄	D₄×D₆₇	1	2	yes	—
122	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
123	C₈	—	134	2	no	—
124	C₈	#11761627	1	4	yes	—
125	C₂²	—	134	2	no	—
126	C₂²	—	134	1	no	—
127	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	268	4	no	—
130	C₄	—	134	2	no	—
131	C₄	—	134	1	no	—
132	C₄	#14524075	1	4	yes	—
133	C₄	C₂×#11761627	1	2	yes	—
134	C₂	—	268	1	no	—
135	C₂	#16863944	1	1	yes	characteristic
136	C₂	C₂×#14524075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863951

C₂

yes

maximal

#16863794

C₂

yes

maximal, characteristic, complemented

#16863746

C₂

yes

maximal, complemented

#16863651

C₂

yes

maximal

#16863503

C₂

yes

maximal, complemented

C₄×#11761611

C₂

yes

maximal, characteristic

#494×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761611

C₂²

yes

—

#14524043

C₂²

yes

—

#14524034

—

#14524023

—

#14524019

C₂²

yes

—

Q₈×Dic₆₇

C₂²

yes

—

#14524008

—

#14524001

—

C₄²×D₆₇

C₂²

yes

characteristic

#14523975

C₂²

yes

—

#14523967

C₂²

yes

—

#14523953

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

characteristic

#11761611

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

C₂³

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×D₄

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#137

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#532

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

D₆₇

—

#3185

—

maximal, Sylow

C₆₇

#3185

yes

characteristic, complemented, Sylow

#533

—

#498

—

#494

D₆₇

yes

characteristic, complemented

#487

—

#470

—

#441

—

C₄×M₁₆

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

—

#125

—

134

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

C₄×Q₈

—

134

—

#117

—

134

—

C₂×C₄²

—

#106

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

C₂×Q₈

C₂²×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

M₁₆

—

134

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₆₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₂²⋊C₄

—

134

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

134

—

111

C₄²

C₂²×D₆₇

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

268

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

134

—

119

C₂×C₄

—

134

—

120

C₂×C₄

—

134

—

121

C₂×C₄

D₄×D₆₇

yes

—

122

C₂×C₄

C₂³×D₆₇

yes

characteristic

123

C₈

—

134

—

124

C₈

#11761627

yes

—

125

C₂²

—

134

—

126

C₂²

—

134

—

127

C₂²

C₂×D₄×D₆₇

yes

characteristic

128

C₄

—

129

C₄

—

268

—

130

C₄

—

134

—

131

C₄

—

134

—

132

C₄

#14524075

yes

—

133

C₄

C₂×#11761627

yes

—

134

C₂

—

268

—

135

C₂

#16863944

yes

characteristic

136

C₂

C₂×#14524075

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#494	D₆₇
hypocenter	C₆₇	#3185
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#494×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #16863794	C₂	semidirect
3. #16863746	C₂	semidirect
5. #16863503	C₂	semidirect
7. #494×C₆₇	C₂	semidirect
69. C₆₇	#3185	semidirect
72. #494	D₆₇	semidirect

H ≅ G/N

type

2. #16863794

C₂

semidirect

3. #16863746

C₂

semidirect

5. #16863503

C₂

semidirect

7. #494×C₆₇

C₂

semidirect

69. C₆₇

#3185

semidirect

72. #494

D₆₇

semidirect

Subgroups — #22570952

Conjugacy classes of subgroups

Complements