Group #22570705 — subgroups

Subgroups — #498×D₆₇ #22570705

Order 8576.

424 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 165 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⋊Q₈×D₆₇	C₂	1	2	yes	maximal
2	#16863794	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863793	C₂	1	4	yes	maximal, complemented
4	#16863792	C₂	1	2	yes	maximal, complemented
5	#106×D₆₇	C₂	1	4	yes	maximal
6	C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
7	#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₇	—	2	2	no	—
9	C₂×C₈×D₆₇	C₂²	1	2	yes	—
10	#14524043	C₂²	1	2	yes	—
11	#14524042	—	2	2	no	—
12	#14524042	C₂²	1	4	yes	—
13	#14524041	—	2	2	no	—
14	C₄⋊C₄×D₆₇	—	2	2	no	—
15	C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
16	C₄²×D₆₇	C₂²	1	1	yes	characteristic
17	#14523975	C₂²	1	4	yes	—
18	#14523974	C₂²	1	8	yes	—
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₆₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	#106×C₆₇	C₂²	1	4	yes	—
23	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	—	2	4	no	—
26	Q₈×D₆₇	—	4	8	no	—
27	C₂×C₄×D₆₇	—	2	4	no	—
28	C₂×C₄×D₆₇	D₄	1	2	yes	—
29	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
30	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₈×D₆₇	Q₈	1	8	yes	complemented
32	C₄⋊Dic₆₇	—	2	2	no	—
33	C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	Dic₆₇⋊C₄	—	2	4	no	—
35	Dic₆₇⋊C₄	—	2	4	no	—
36	C₄×Dic₆₇	D₄	1	2	yes	—
37	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
42	C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₆₇	—	4	4	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	1	yes	characteristic
50	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
51	C₄×D₆₇	—	4	8	no	—
52	C₄×D₆₇	SD₁₆	1	8	yes	—
53	C₄×D₆₇	C₂×Q₈	1	4	yes	—
54	Dic₁₃₄	—	4	4	no	—
55	Dic₁₃₄	—	4	8	no	—
56	C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
57	C₆₇⋊C₈	C₂×Q₈	1	4	yes	complemented
58	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₆₇	C₄⋊Q₈	1	2	yes	—
60	C₂×D₆₇	C₂×SD₁₆	1	4	yes	—
61	C₄×C₆₇	—	4	4	no	—
62	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
63	C₄×C₆₇	C₂²×Q₈	1	2	yes	—
64	Dic₆₇	—	4	4	no	—
65	Dic₆₇	C₄⋊Q₈	1	2	yes	—
66	Dic₆₇	C₂×SD₁₆	1	4	yes	—
67	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₆₇	C₂²×SD₁₆	1	2	yes	—
69	D₆₇	#498	1	4	yes	direct product factor
70	C₂×#498	—	67	1	no	maximal, Sylow
71	C₆₇	C₂×#498	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊Q₈	—	67	2	no	—
73	#498	—	67	2	no	—
74	#498	—	67	4	no	—
75	#498	—	67	1	no	—
76	#498	D₆₇	1	1	yes	characteristic, direct product factor
77	C₂×#106	—	67	4	no	—
78	C₂×C₄×C₈	—	67	1	no	—
79	C₂²×Q₈	—	134	2	no	—
80	C₂²×C₈	—	67	2	no	—
81	C₄⋊Q₈	—	67	4	no	—
82	C₄⋊Q₈	—	67	2	no	—
83	C₄⋊Q₈	—	134	2	no	—
84	C₄⋊Q₈	—	134	2	no	—
85	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
86	C₂×C₄⋊C₄	—	67	4	no	—
87	C₂×C₄⋊C₄	—	134	2	no	—
88	C₂×C₄²	—	67	1	no	—
89	#106	—	67	4	no	—
90	#106	—	67	8	no	—
91	#106	C₂×D₆₇	1	4	yes	—
92	C₄×C₈	—	67	2	no	—
93	C₄×C₈	—	67	1	no	—
94	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	268	8	no	—
97	C₂×Q₈	—	134	4	no	—
98	C₂×Q₈	—	134	2	no	—
99	C₂²×C₄	—	67	2	no	—
100	C₂²×C₄	—	67	1	no	—
101	C₂²×C₄	—	134	4	no	—
102	C₂×C₈	—	67	2	no	—
103	C₂×C₈	—	67	8	no	—
104	C₂×C₈	C₂²×D₆₇	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	67	4	no	—
107	C₄⋊C₄	—	134	4	no	—
108	C₄⋊C₄	—	134	4	no	—
109	C₄⋊C₄	—	134	2	no	—
110	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
111	C₄²	—	67	2	no	—
112	C₄²	—	67	1	no	—
113	C₄²	C₂²×D₆₇	1	1	yes	characteristic
114	C₂³	—	67	1	no	—
115	Q₈	—	4	4	no	—
116	Q₈	—	268	8	no	—
117	Q₈	—	268	4	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	268	8	no	—
120	C₂×C₄	—	67	1	no	—
121	C₂×C₄	—	67	2	no	—
122	C₂×C₄	—	67	8	no	—
123	C₂×C₄	—	67	4	no	—
124	C₂×C₄	—	134	4	no	—
125	C₂×C₄	D₄×D₆₇	1	2	yes	—
126	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
127	C₈	—	67	4	no	—
128	C₈	Q₈×D₆₇	1	4	yes	complemented
129	C₂²	—	67	4	no	—
130	C₂²	—	67	2	no	—
131	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
132	C₄	—	4	4	no	—
133	C₄	—	268	4	no	—
134	C₄	—	67	4	no	—
135	C₄	—	67	2	no	—
136	C₄	SD₁₆×D₆₇	1	4	yes	—
137	C₄	C₂×Q₈×D₆₇	1	2	yes	—
138	C₂	—	67	4	no	—
139	C₂	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
140	C₂	C₂×SD₁₆×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₆₇

C₂

yes

maximal

#16863794

C₂

yes

maximal, characteristic, complemented

#16863793

C₂

yes

maximal, complemented

#16863792

C₂

yes

maximal, complemented

#106×D₆₇

C₂

yes

maximal

C₄×C₈×D₆₇

C₂

yes

maximal, characteristic

#498×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₇

—

C₂×C₈×D₆₇

C₂²

yes

—

#14524043

C₂²

yes

—

#14524042

—

#14524042

C₂²

yes

—

#14524041

—

C₄⋊C₄×D₆₇

—

C₄⋊C₄×D₆₇

C₂²

yes

—

C₄²×D₆₇

C₂²

yes

characteristic

#14523975

C₂²

yes

—

#14523974

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

#106×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Q₈×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

—

C₈×D₆₇

Q₈

yes

complemented

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

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

characteristic

C₂×Dic₆₇

C₂×D₄

yes

—

C₄×D₆₇

—

C₄×D₆₇

SD₁₆

yes

—

C₄×D₆₇

C₂×Q₈

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₈

yes

complemented

C₆₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

C₄⋊Q₈

yes

—

C₂×D₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂²×Q₈

yes

—

Dic₆₇

—

Dic₆₇

C₄⋊Q₈

yes

—

Dic₆₇

C₂×SD₁₆

yes

—

C₂×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

—

D₆₇

#498

yes

direct product factor

C₂×#498

—

maximal, Sylow

C₆₇

C₂×#498

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₆₇

yes

characteristic, direct product factor

C₂×#106

—

C₂×C₄×C₈

—

C₂²×Q₈

—

134

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

—

#106

—

#106

—

#106

C₂×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

134

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₆₇

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

—

134

—

109

C₄⋊C₄

—

134

—

110

C₄⋊C₄

C₂²×D₆₇

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂²×D₆₇

yes

characteristic

114

C₂³

—

115

Q₈

—

116

Q₈

—

268

—

117

Q₈

—

268

—

118

C₂×C₄

—

119

C₂×C₄

—

268

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

134

—

125

C₂×C₄

D₄×D₆₇

yes

—

126

C₂×C₄

C₂³×D₆₇

yes

characteristic

127

C₈

—

128

C₈

Q₈×D₆₇

yes

complemented

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×D₄×D₆₇

yes

characteristic

132

C₄

—

133

C₄

—

268

—

134

C₄

—

135

C₄

—

136

C₄

SD₁₆×D₆₇

yes

—

137

C₄

C₂×Q₈×D₆₇

yes

—

138

C₂

—

139

C₂

C₄⋊Q₈×D₆₇

yes

characteristic

140

C₂

C₂×SD₁₆×D₆₇

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#498×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#498	D₆₇
hypocenter	C₆₇	C₂×#498
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#498×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #16863794	C₂	semidirect
3. #16863793	C₂	semidirect
4. #16863792	C₂	semidirect
7. #498×C₆₇	C₂	semidirect
31. C₈×D₆₇	Q₈	semidirect
56. C₈×C₆₇	C₂×Q₈	semidirect
57. C₆₇⋊C₈	C₂×Q₈	semidirect
69. D₆₇	#498	direct (N × H)
71. C₆₇	C₂×#498	semidirect
76. #498	D₆₇	direct (N × H)
128. C₈	Q₈×D₆₇	semidirect

H ≅ G/N

type

2. #16863794

C₂

semidirect

3. #16863793

C₂

semidirect

4. #16863792

C₂

semidirect

7. #498×C₆₇

C₂

semidirect

31. C₈×D₆₇

Q₈

semidirect

56. C₈×C₆₇

C₂×Q₈

semidirect

57. C₆₇⋊C₈

C₂×Q₈

semidirect

69. D₆₇

#498

direct (N × H)

71. C₆₇

C₂×#498

semidirect

76. #498

D₆₇

direct (N × H)

128. C₈

Q₈×D₆₇

semidirect

Subgroups — #498×D₆₇ #22570705

Conjugacy classes of subgroups

Complements

Subgroups — #498×D67 #22570705

Conjugacy classes of subgroups

Complements

Subgroups — #498×D₆₇ #22570705