Group #30770212 — subgroups

Subgroups — #30770212

Order 17152.

188 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22562484	C₂	1	1	yes	maximal, characteristic
2	#22559613	C₂	1	2	yes	maximal
3	#22558283	C₂	1	1	yes	maximal, characteristic
4	#22558266	C₂	1	2	yes	maximal
5	C₆₇×#2183	C₂	1	1	yes	maximal, characteristic
6	#16863801	C₂²	1	2	yes	—
7	#16863691	—	2	2	no	—
8	#16863222	—	2	1	no	—
9	#16863222	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	#16863144	—	2	2	no	—
12	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₆₇	—	2	1	no	—
14	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#346×C₆₇	—	2	1	no	—
16	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	#14524043	—	2	2	no	—
18	#14523977	—	2	2	no	—
19	#14523977	—	2	2	no	—
20	#14523976	—	4	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	#14523967	—	4	2	no	—
23	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₇	—	2	1	no	—
25	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	1	no	—
28	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	—	2	1	no	—
30	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₃₄	—	4	2	no	—
32	C₂×C₆₇⋊C₈	—	4	2	no	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
41	C₄²×C₆₇	—	2	1	no	—
42	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₆₇	—	2	1	no	—
44	C₁₆×C₆₇	—	4	1	no	—
45	C₁₆×C₆₇	Q₁₆	1	2	yes	—
46	Q₈×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	1	no	—
48	C₂×C₄×C₆₇	—	2	1	no	—
49	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
52	C₂×Dic₆₇	—	8	2	no	—
53	Dic₁₃₄	—	16	2	no	—
54	C₈×C₆₇	—	2	1	no	—
55	C₈×C₆₇	—	4	1	no	—
56	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₆₇	#137	1	1	yes	characteristic
58	C₆₇⋊C₈	—	8	2	no	—
59	C₂²×C₆₇	#466	1	1	yes	characteristic
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	4	1	no	—
62	C₄×C₆₇	—	4	1	no	—
63	C₄×C₆₇	#462	1	1	yes	characteristic
64	C₄×C₆₇	#473	1	1	yes	characteristic
65	C₄×C₆₇	#510	1	2	yes	—
66	Dic₆₇	—	16	2	no	—
67	#12461	—	67	1	no	maximal, Sylow
68	C₂×C₆₇	#1674	1	1	yes	characteristic
69	C₂×C₆₇	#2219	1	1	yes	characteristic
70	C₂×C₆₇	#2221	1	1	yes	characteristic
71	#2252	—	67	1	no	—
72	#2183	D₆₇	1	1	yes	characteristic
73	#1676	—	67	2	no	—
74	#1372	—	67	1	no	—
75	#1365	—	67	2	no	—
76	C₆₇	#12461	1	1	yes	characteristic, complemented, Sylow
77	#500	—	67	2	no	—
78	#477	—	134	2	no	—
79	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
80	#366	—	134	1	no	—
81	#366	—	134	1	no	—
82	C₄⋊C₁₆	—	2	1	no	—
83	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
84	#346	—	2	1	no	—
85	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
86	#335	—	67	2	no	—
87	#333	—	134	2	no	—
88	C₄⋊Q₈	—	134	2	no	—
89	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
92	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
93	#107	—	268	2	no	—
94	#107	—	134	2	no	—
95	#107	—	134	2	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	134	2	no	—
98	C₄⋊C₈	D₂₆₈	1	1	yes	characteristic
99	#103	—	268	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	C₂×Q₈	—	268	2	no	—
103	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	268	2	no	—
106	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
107	C₂×C₈	#11761628	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	268	2	no	—
110	C₄⋊C₄	—	268	2	no	—
111	C₄⋊C₄	D₅₃₆	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	1	no	—
116	C₁₆	#11761618	1	2	yes	—
117	Q₈	—	2	2	no	—
118	Q₈	—	1072	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	536	2	no	—
122	C₂×C₄	#14524017	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
124	C₂×C₄	#14524065	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	536	2	no	—
128	C₈	C₂×#11761618	1	1	yes	characteristic
129	C₈	#14524073	1	1	yes	characteristic
130	C₂²	#16863632	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	1072	2	no	—
135	C₄	#16863606	1	1	yes	characteristic
136	C₄	#16863671	1	1	yes	characteristic
137	C₄	#16863839	1	2	yes	—
138	C₂	#22559603	1	1	yes	characteristic
139	C₂	#22562317	1	1	yes	characteristic
140	C₂	#22562329	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562484

C₂

yes

maximal, characteristic

#22559613

C₂

yes

maximal

#22558283

C₂

yes

maximal, characteristic

#22558266

C₂

yes

maximal

C₆₇×#2183

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

C₂²

yes

—

#16863144

—

C₈×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₁₆×C₆₇

—

C₄⋊C₁₆×C₆₇

C₂²

yes

characteristic

#346×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

—

—

—

—

C₄⋊(C₆₇⋊C₈)

—

#14523967

—

C₄×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₈

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

Q₁₆

yes

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#121

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₁₆

yes

characteristic

C₈×C₆₇

#137

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

#466

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

—

—

—

C₆₇

#12461

yes

characteristic, complemented, Sylow

#500

—

#477

—

134

—

C₈×Q₈

C₂×D₆₇

yes

characteristic

#366

—

134

—

#366

—

134

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₆₇

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

#335

—

#333

—

134

—

C₄⋊Q₈

—

134

—

C₄×Q₈

D₂₆₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

#107

—

268

—

#107

—

134

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

D₂₆₈

yes

characteristic

#103

—

268

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₆₇

yes

characteristic

102

C₂×Q₈

—

268

—

103

C₂×Q₈

D₅₃₆

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

268

—

106

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

107

C₂×C₈

#11761628

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

268

—

110

C₄⋊C₄

—

268

—

111

C₄⋊C₄

D₅₃₆

yes

characteristic

112

C₄²

—

113

C₄²

C₂×D₂₆₈

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

#11761618

yes

—

117

Q₈

—

118

Q₈

—

1072

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

536

—

122

C₂×C₄

#14524017

yes

characteristic

123

C₂×C₄

C₂×D₅₃₆

yes

characteristic

124

C₂×C₄

#14524065

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

536

—

128

C₈

C₂×#11761618

yes

characteristic

129

C₈

#14524073

yes

characteristic

130

C₂²

#16863632

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

1072

—

135

C₄

#16863606

yes

characteristic

136

C₄

#16863671

yes

characteristic

137

C₄

#16863839

yes

—

138

C₂

#22559603

yes

characteristic

139

C₂

#22562317

yes

characteristic

140

C₂

#22562329

yes

characteristic

	subgroup	G/H
center	C₂²	#16863632
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2183	C₂
socle, prime radical	C₂²×C₆₇	#466
hypercenter	#2183	D₆₇
hypocenter	C₆₇	#12461
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#16863632

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2183

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
76. C₆₇	#12461	semidirect

H ≅ G/N

type

76. C₆₇

#12461

semidirect

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

subgroup	index
#22562484	2
#22559613	2
#22558283	2
#22558266	2
C₆₇×#2183	2
#12461	67

subgroup

index

C₆₇×#2183

Subgroups — #30770212

Conjugacy classes of subgroups

Complements

Maximal subgroups