Group #30746276 — subgroups

Subgroups — #30746276

Order 17152.

354 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 195 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	#22558492	C₂	1	2	yes	maximal
2	#22558040	C₂	1	2	yes	maximal
3	#22558038	C₂	1	2	yes	maximal
4	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
5	#16863449	C₂²	1	4	yes	—
6	#16863117	C₄	1	4	yes	complemented
7	C₈×C₆₇⋊C₈	C₄	1	4	yes	complemented
8	C₂×#97×C₆₇	C₄	1	2	yes	—
9	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#376×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
14	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
15	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
21	#97×C₆₇	—	2	2	no	—
22	#97×C₆₇	C₈	1	8	yes	complemented
23	C₄×C₈×C₆₇	—	2	4	no	—
24	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	#95×C₆₇	—	2	2	no	—
27	C₂×C₆₇⋊C₈	—	2	8	no	—
28	C₂²×C₄×C₆₇	—	2	2	no	—
29	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	#41	1	2	yes	—
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	C₄²	1	2	yes	—
34	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
38	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
39	C₄²×C₆₇	—	2	2	no	—
40	C₄²×C₆₇	C₂×C₈	1	4	yes	—
41	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
42	C₄²×C₆₇	#41	1	2	yes	—
43	C₂³×C₆₇	#117	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	—	4	4	no	—
47	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₆₇	#117	1	4	yes	—
49	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
52	C₈×C₆₇	—	2	4	no	—
53	C₈×C₆₇	C₄×C₈	1	4	yes	—
54	C₈×C₆₇	#97	1	2	yes	—
55	C₈×C₆₇	#97	1	2	yes	—
56	C₆₇⋊C₈	—	4	8	no	—
57	C₂²×C₆₇	—	2	2	no	—
58	C₂²×C₆₇	#376	1	1	yes	characteristic
59	C₂²×C₆₇	#431	1	2	yes	—
60	C₄×C₆₇	—	2	4	no	—
61	C₄×C₆₇	—	4	4	no	—
62	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
64	C₄×C₆₇	#431	1	2	yes	—
65	#8138	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	#1744	1	1	yes	characteristic
68	C₂×C₆₇	#1747	1	2	yes	—
69	#1726	D₆₇	1	1	yes	characteristic
70	#1450	—	67	2	no	—
71	#1275	—	67	2	no	—
72	#1274	—	67	2	no	—
73	C₆₇	#8138	1	1	yes	characteristic, complemented, Sylow
74	#431	—	67	4	no	—
75	C₂×#97	Dic₆₇	1	2	yes	—
76	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
77	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
79	#376	C₂×D₆₇	1	1	yes	characteristic
80	#322	—	67	4	no	—
81	C₈²	—	67	4	no	—
82	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
83	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
84	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
85	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
86	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
87	C₄⋊C₈	—	134	4	no	—
88	C₂²⋊C₈	—	134	4	no	—
89	#97	—	2	2	no	—
90	#97	C₆₇⋊C₈	1	8	yes	complemented
91	C₄×C₈	—	2	4	no	—
92	C₄×C₈	—	67	8	no	—
93	C₄×C₈	C₄×D₆₇	1	2	yes	—
94	C₄×C₈	C₄×D₆₇	1	2	yes	—
95	#95	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
98	C₂²×C₄	#11761628	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	134	8	no	—
102	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
103	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
104	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
105	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
106	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
107	C₂×C₈	#11761612	1	4	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
110	C₄²	C₂×C₄×D₆₇	1	2	yes	—
111	C₄²	#11761628	1	2	yes	—
112	C₂³	#14524002	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
117	C₂×C₄	#14524000	1	4	yes	—
118	C₂×C₄	#14524002	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
120	C₂×C₄	C₂×#11761612	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	268	8	no	—
123	C₈	C₄×C₆₇⋊C₈	1	4	yes	—
124	C₈	#14523954	1	2	yes	—
125	C₈	#14523954	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#16863242	1	1	yes	characteristic
128	C₂²	#16863449	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
132	C₄	C₂×#14523954	1	1	yes	characteristic
133	C₄	#16863449	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	#22559955	1	1	yes	characteristic
136	C₂	#22559964	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558492

C₂

yes

maximal

#22558040

C₂

yes

maximal

#22558038

C₂

yes

maximal

C₆₇×#1726

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

#16863117

C₄

yes

complemented

C₈×C₆₇⋊C₈

C₄

yes

complemented

C₂×#97×C₆₇

C₄

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#97×C₆₇

—

#97×C₆₇

C₈

yes

complemented

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

C₄²

yes

characteristic

C₂×C₈×C₆₇

C₄²

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×C₈

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₄×C₈

yes

—

C₈×C₆₇

#97

yes

—

C₈×C₆₇

#97

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#431

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄×C₈

yes

characteristic

C₄×C₆₇

C₂×#97

yes

characteristic

C₄×C₆₇

#431

yes

—

#8138

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1744

yes

characteristic

C₂×C₆₇

#1747

yes

—

#1726

D₆₇

yes

—

—

—

C₆₇

#8138

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

Dic₆₇

yes

—

C₂×#97

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

Dic₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

C₂×D₆₇

yes

characteristic

#322

—

C₈²

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

134

—

#97

—

#97

C₆₇⋊C₈

yes

complemented

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

C₄×D₆₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

#11761628

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

134

—

102

C₂×C₈

C₄×Dic₆₇

yes

—

103

C₂×C₈

C₄×Dic₆₇

yes

characteristic

104

C₂×C₈

C₄×Dic₆₇

yes

characteristic

105

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

106

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

107

C₂×C₈

#11761612

yes

—

108

C₄²

—

109

C₄²

C₂×C₆₇⋊C₈

yes

—

110

C₄²

C₂×C₄×D₆₇

yes

—

111

C₄²

#11761628

yes

—

112

C₂³

#14524002

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

117

C₂×C₄

#14524000

yes

—

118

C₂×C₄

#14524002

yes

characteristic

119

C₂×C₄

C₂²×C₆₇⋊C₈

yes

—

120

C₂×C₄

C₂×#11761612

yes

—

121

C₈

—

122

C₈

—

268

—

123

C₈

C₄×C₆₇⋊C₈

yes

—

124

C₈

#14523954

yes

—

125

C₈

#14523954

yes

—

126

C₂²

—

127

C₂²

#16863242

yes

characteristic

128

C₂²

#16863449

yes

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×C₆₇⋊C₈

yes

characteristic

132

C₄

C₂×#14523954

yes

characteristic

133

C₄

#16863449

yes

—

134

C₂

—

135

C₂

#22559955

yes

characteristic

136

C₂

#22559964

yes

—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1726	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#8138
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₄×Dic₆₇

commutator

C₄×C₆₇

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1726

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₈ ◁ #1726

N	H ≅ G/N	type
6. #16863117	C₄	semidirect
7. C₈×C₆₇⋊C₈	C₄	semidirect
22. #97×C₆₇	C₈	semidirect
73. C₆₇	#8138	semidirect
90. #97	C₆₇⋊C₈	semidirect

H ≅ G/N

type

6. #16863117

C₄

semidirect

7. C₈×C₆₇⋊C₈

C₄

semidirect

22. #97×C₆₇

C₈

semidirect

73. C₆₇

#8138

semidirect

90. #97

C₆₇⋊C₈

semidirect

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

subgroup	index
#22558492	2
#22558040	2
#22558038	2
C₆₇×#1726	2
#8138	67

subgroup

index

C₆₇×#1726

Subgroups — #30746276

Conjugacy classes of subgroups

Complements

Maximal subgroups