Group #30772229 — subgroups

Subgroups — #30772229

Order 17152.

234 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22561808	C₂	1	1	yes	maximal, characteristic
2	#22560334	C₂	1	2	yes	maximal
3	#22558311	C₂	1	1	yes	maximal, characteristic
4	#22558306	C₂	1	2	yes	maximal
5	C₆₇×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	#16863434	C₂²	1	2	yes	—
8	#16863351	C₂²	1	1	yes	characteristic
9	#16863171	C₄	1	2	yes	complemented
10	C₁₆×Dic₆₇	C₄	1	2	yes	complemented
11	#16863168	—	2	2	no	—
12	#16863153	—	2	2	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₇	C₂²	1	2	yes	—
15	C₂×#11761612	D₄	1	1	yes	characteristic
16	#14524002	—	2	2	no	—
17	#14524002	D₄	1	1	yes	characteristic
18	C₂×C₆₇⋊C₁₆	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
21	#14523977	—	2	2	no	—
22	#14523975	—	2	2	no	—
23	#14523953	Q₈	1	2	yes	—
24	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	M₃₂×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	#108×C₆₇	—	2	2	no	—
30	#108×C₆₇	—	2	2	no	—
31	#11761612	—	2	2	no	—
32	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
33	C₄⋊Dic₆₇	—	2	1	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₂²⋊Dic₆₇	—	2	1	no	—
36	C₂²⋊Dic₆₇	—	4	2	no	—
37	C₄×Dic₆₇	—	4	2	no	—
38	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
39	C₆₇⋊C₁₆	—	2	1	no	—
40	C₆₇⋊C₁₆	—	2	1	no	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	M₁₆×C₆₇	—	4	2	no	—
43	M₁₆×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	—	4	2	no	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₂×C₈×C₆₇	#41	1	2	yes	—
49	C₁₆×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	#135	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₇	#124	1	2	yes	—
53	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#135	1	2	yes	—
55	C₂×Dic₆₇	—	2	2	no	—
56	C₂×Dic₆₇	—	8	2	no	—
57	C₈×C₆₇	—	8	2	no	—
58	C₈×C₆₇	#117	1	1	yes	characteristic
59	C₈×C₆₇	#117	1	1	yes	characteristic
60	C₈×C₆₇	#125	1	2	yes	—
61	C₆₇⋊C₈	—	2	2	no	—
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	#427	1	1	yes	characteristic
64	C₂²×C₆₇	#488	1	2	yes	—
65	C₄×C₆₇	#382	1	1	yes	characteristic
66	C₄×C₆₇	#427	1	1	yes	characteristic
67	C₄×C₆₇	#488	1	2	yes	—
68	Dic₆₇	—	4	2	no	—
69	Dic₆₇	—	16	2	no	—
70	#12893	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	—	2	1	no	—
72	C₂×C₆₇	—	2	2	no	—
73	C₂×C₆₇	#1835	1	1	yes	characteristic
74	#2108	—	67	1	no	—
75	#1833	—	67	2	no	—
76	#1385	D₆₇	1	1	yes	characteristic
77	#1384	—	67	1	no	—
78	#1382	—	67	2	no	—
79	C₆₇	#12893	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
81	C₂²×C₁₆	—	67	1	no	—
82	C₂×#108	C₂×D₆₇	1	2	yes	—
83	#427	—	67	2	no	—
84	#405	—	67	1	no	—
85	#346	—	67	2	no	—
86	C₄×C₁₆	—	67	2	no	—
87	#344	—	134	2	no	—
88	H(3,ℤ/4)	—	134	2	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	67	1	no	—
91	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	#117	—	67	1	no	—
93	#117	—	134	2	no	—
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	67	1	no	—
96	C₂×C₁₆	—	67	1	no	—
97	C₂×C₁₆	—	134	2	no	—
98	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
99	#108	—	2	2	no	—
100	#108	—	2	2	no	—
101	#107	—	134	2	no	—
102	#106	—	134	2	no	—
103	#97	—	67	2	no	—
104	C₄×C₈	—	67	2	no	—
105	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	134	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	67	2	no	—
111	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
113	C₂×C₈	#11761627	1	2	yes	—
114	C₂×C₈	#11761628	1	2	yes	—
115	C₄⋊C₄	—	268	2	no	—
116	C₄⋊C₄	—	134	1	no	—
117	C₂²⋊C₄	—	268	2	no	—
118	C₂²⋊C₄	—	134	1	no	—
119	C₄²	—	268	2	no	—
120	C₄²	—	67	2	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	134	1	no	—
123	C₁₆	—	134	1	no	—
124	C₂³	#14524066	1	1	yes	characteristic
125	C₂×C₄	—	536	2	no	—
126	C₂×C₄	—	134	2	no	—
127	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
128	C₂×C₄	#14524029	1	2	yes	—
129	C₂×C₄	#14524062	1	1	yes	characteristic
130	C₂×C₄	#14524064	1	2	yes	—
131	C₈	—	8	2	no	—
132	C₈	—	134	2	no	—
133	C₈	#14523999	1	1	yes	characteristic
134	C₈	#14524000	1	1	yes	characteristic
135	C₈	#14524033	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#16863433	1	1	yes	characteristic
138	C₂²	#16863752	1	2	yes	—
139	C₄	—	268	2	no	—
140	C₄	—	1072	2	no	—
141	C₄	#16863260	1	1	yes	characteristic
142	C₄	#16863431	1	1	yes	characteristic
143	C₄	#16863749	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#22560338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561808

C₂

yes

maximal, characteristic

#22560334

C₂

yes

maximal

#22558311

C₂

yes

maximal, characteristic

#22558306

C₂

yes

maximal

C₆₇×#1385

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

#16863434

C₂²

yes

—

#16863351

C₂²

yes

characteristic

#16863171

C₄

yes

complemented

C₁₆×Dic₆₇

C₄

yes

complemented

#16863168

—

#16863153

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂×#11761612

D₄

yes

characteristic

#14524002

—

#14524002

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

—

Q₈

yes

—

C₈×Dic₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₄⋊C₄

yes

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#427

yes

characteristic

C₂²×C₆₇

#488

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

Dic₆₇

—

#12893

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1384

—

#1382

—

C₆₇

#12893

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₆₇

yes

—

#427

—

#405

—

#346

—

C₄×C₁₆

—

#344

—

134

—

H(3,ℤ/4)

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

—

#117

—

134

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₄×D₆₇

yes

—

#108

—

100

#108

—

101

#107

—

134

—

102

#106

—

134

—

103

#97

—

104

C₄×C₈

—

105

C₂²×C₄

D₄×D₆₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

134

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

112

C₂×C₈

Q₈×D₆₇

yes

characteristic

113

C₂×C₈

#11761627

yes

—

114

C₂×C₈

#11761628

yes

—

115

C₄⋊C₄

—

268

—

116

C₄⋊C₄

—

134

—

117

C₂²⋊C₄

—

268

—

118

C₂²⋊C₄

—

134

—

119

C₄²

—

268

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

134

—

123

C₁₆

—

134

—

124

C₂³

#14524066

yes

characteristic

125

C₂×C₄

—

536

—

126

C₂×C₄

—

134

—

127

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

128

C₂×C₄

#14524029

yes

—

129

C₂×C₄

#14524062

yes

characteristic

130

C₂×C₄

#14524064

yes

—

131

C₈

—

132

C₈

—

134

—

133

C₈

#14523999

yes

characteristic

134

C₈

#14524000

yes

characteristic

135

C₈

#14524033

yes

—

136

C₂²

—

137

C₂²

#16863433

yes

characteristic

138

C₂²

#16863752

yes

—

139

C₄

—

268

—

140

C₄

—

1072

—

141

C₄

#16863260

yes

characteristic

142

C₄

#16863431

yes

characteristic

143

C₄

#16863749

yes

—

144

C₂

—

145

C₂

—

146

C₂

#22560338

yes

characteristic

	subgroup	G/H
center	C₄	#16863431
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1385	C₂
socle	C₂×C₆₇	#1835
prime radical	C₂³×C₆₇	#135
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1385	D₆₇
hypocenter	C₆₇	#12893
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#16863431

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1385

C₂

C₂×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
9. #16863171	C₄	semidirect
10. C₁₆×Dic₆₇	C₄	semidirect
79. C₆₇	#12893	semidirect

H ≅ G/N

type

9. #16863171

C₄

semidirect

10. C₁₆×Dic₆₇

C₄

semidirect

79. C₆₇

#12893

semidirect

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

subgroup	index
#22561808	2
#22560334	2
#22558311	2
#22558306	2
C₆₇×#1385	2
#12893	67

subgroup

index

C₆₇×#1385

Subgroups — #30772229

Conjugacy classes of subgroups

Complements

Maximal subgroups