Group #30772228 — subgroups

Subgroups — #30772228

Order 17152.

234 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 111 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	#22560333	C₂	1	2	yes	maximal
2	#22558313	C₂	1	1	yes	maximal, characteristic
3	#22558311	C₂	1	1	yes	maximal, characteristic
4	#22558305	C₂	1	2	yes	maximal
5	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	C₂²	1	2	yes	—
7	#16863434	C₂²	1	2	yes	—
8	#16863167	—	2	2	no	—
9	#16863152	—	2	2	no	—
10	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#405×C₆₇	C₂²	1	1	yes	characteristic
13	#346×C₆₇	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₆₇	C₄	1	2	yes	complemented
15	C₂×#11761612	—	2	2	no	—
16	#14524002	—	2	2	no	—
17	#14523979	—	2	2	no	—
18	#14523979	—	2	2	no	—
19	#14523977	—	2	2	no	—
20	#14523975	—	2	2	no	—
21	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
23	#117×C₆₇	D₄	1	1	yes	characteristic
24	M₃₂×C₆₇	—	2	2	no	—
25	C₂×C₁₆×C₆₇	—	2	2	no	—
26	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	#97×C₆₇	Q₈	1	2	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
31	#11761612	—	4	2	no	—
32	#11761612	—	4	2	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₂²⋊Dic₆₇	—	4	2	no	—
36	C₄×Dic₆₇	—	4	2	no	—
37	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₆₇	—	2	2	no	—
39	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₆₇	#41	1	2	yes	—
43	C₂×C₈×C₆₇	#41	1	2	yes	—
44	C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂²⋊C₄×C₆₇	—	2	1	no	—
46	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₁₆×C₆₇	—	2	1	no	—
48	C₁₆×C₆₇	—	2	1	no	—
49	C₁₆×C₆₇	—	2	2	no	—
50	C₂³×C₆₇	#135	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	2	2	no	—
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#124	1	2	yes	—
54	C₂×C₄×C₆₇	#135	1	2	yes	—
55	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
56	C₂×Dic₆₇	—	8	2	no	—
57	C₈×C₆₇	—	2	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	—	8	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	—	4	2	no	—
66	C₄×C₆₇	#382	1	1	yes	characteristic
67	C₄×C₆₇	#427	1	1	yes	characteristic
68	C₄×C₆₇	#488	1	2	yes	—
69	Dic₆₇	—	16	2	no	—
70	#12893	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	—	2	1	no	—
73	C₂×C₆₇	#1835	1	1	yes	characteristic
74	#2108	D₆₇	1	1	yes	characteristic
75	#1833	—	67	2	no	—
76	#1385	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
82	C₂×#108	—	67	2	no	—
83	#427	—	67	2	no	—
84	#405	C₂×D₆₇	1	1	yes	characteristic
85	#346	Dic₆₇	1	2	yes	complemented
86	C₄×C₁₆	Dic₆₇	1	2	yes	complemented
87	#344	—	134	2	no	—
88	H(3,ℤ/4)	—	134	2	no	—
89	C₂×M₁₆	—	134	2	no	—
90	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	#117	—	134	2	no	—
93	#117	D₂₆₈	1	1	yes	characteristic
94	M₃₂	—	2	2	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
98	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
99	#108	—	134	2	no	—
100	#108	—	134	2	no	—
101	#107	—	134	2	no	—
102	#106	—	134	2	no	—
103	#97	Dic₁₃₄	1	2	yes	—
104	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
105	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	268	2	no	—
108	M₁₆	—	268	2	no	—
109	C₂×C₈	—	268	2	no	—
110	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
111	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
113	C₂×C₈	#11761628	1	2	yes	—
114	C₂×C₈	#11761628	1	2	yes	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	268	2	no	—
117	C₂²⋊C₄	—	2	1	no	—
118	C₂²⋊C₄	—	268	2	no	—
119	C₄²	—	268	2	no	—
120	C₄²	C₄⋊Dic₆₇	1	2	yes	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	2	1	no	—
123	C₁₆	—	2	2	no	—
124	C₂³	#14524065	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	536	2	no	—
127	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
128	C₂×C₄	#14524031	1	2	yes	—
129	C₂×C₄	#14524065	1	1	yes	characteristic
130	C₂×C₄	#14524065	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	536	2	no	—
133	C₈	#14524002	1	1	yes	characteristic
134	C₈	#14524002	1	1	yes	characteristic
135	C₈	#14524035	1	2	yes	—
136	C₂²	—	4	1	no	—
137	C₂²	#16863434	1	1	yes	characteristic
138	C₂²	#16863751	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	1072	2	no	—
141	C₄	#16863262	1	1	yes	characteristic
142	C₄	#16863434	1	1	yes	characteristic
143	C₄	#16863751	1	2	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	2	no	—
146	C₂	#22560340	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560333

C₂

yes

maximal

#22558313

C₂

yes

maximal, characteristic

#22558311

C₂

yes

maximal, characteristic

#22558305

C₂

yes

maximal

C₆₇×#2108

C₂

yes

maximal, characteristic

C₂×#14523979

C₂²

yes

—

#16863434

C₂²

yes

—

#16863167

—

#16863152

—

C₂×M₃₂×C₆₇

C₂²

yes

characteristic

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#405×C₆₇

C₂²

yes

characteristic

#346×C₆₇

C₄

yes

complemented

C₄×C₁₆×C₆₇

C₄

yes

—

—

—

—

—

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#117×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#97×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

#135

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#135

yes

characteristic

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

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12893

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1835

yes

characteristic

#2108

D₆₇

yes

—

—

—

—

C₆₇

#12893

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₆₇

yes

characteristic

C₂²×C₁₆

C₂×D₆₇

yes

—

—

C₂×D₆₇

yes

characteristic

#346

Dic₆₇

yes

complemented

C₄×C₁₆

Dic₆₇

yes

complemented

#344

—

134

—

H(3,ℤ/4)

—

134

—

C₂×M₁₆

—

134

—

C₂×M₁₆

D₂₆₈

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

#117

—

134

—

#117

D₂₆₈

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

C₂×Dic₆₇

yes

—

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

#108

—

134

—

100

#108

—

134

—

101

#107

—

134

—

102

#106

—

134

—

103

#97

Dic₁₃₄

yes

—

104

C₄×C₈

C₂×Dic₆₇

yes

—

105

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

106

M₁₆

—

107

M₁₆

—

268

—

108

M₁₆

—

268

—

109

C₂×C₈

—

268

—

110

C₂×C₈

C₄⋊Dic₆₇

yes

—

111

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

112

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

113

C₂×C₈

#11761628

yes

—

114

C₂×C₈

#11761628

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

268

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

268

—

119

C₄²

—

268

—

120

C₄²

C₄⋊Dic₆₇

yes

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#14524065

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

536

—

127

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

128

C₂×C₄

#14524031

yes

—

129

C₂×C₄

#14524065

yes

characteristic

130

C₂×C₄

#14524065

yes

—

131

C₈

—

132

C₈

—

536

—

133

C₈

#14524002

yes

characteristic

134

C₈

#14524002

yes

characteristic

135

C₈

#14524035

yes

—

136

C₂²

—

137

C₂²

#16863434

yes

characteristic

138

C₂²

#16863751

yes

—

139

C₄

—

140

C₄

—

1072

—

141

C₄

#16863262

yes

characteristic

142

C₄

#16863434

yes

characteristic

143

C₄

#16863751

yes

—

144

C₂

—

145

C₂

—

146

C₂

#22560340

yes

characteristic

	subgroup	G/H
center	C₄	#16863434
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2108	C₂
socle	C₂×C₆₇	#1835
prime radical	C₂³×C₆₇	#135
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#12893
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

subgroup

G/H

center

C₄

#16863434

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#2108

C₂

C₂×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

N	H ≅ G/N	type
13. #346×C₆₇	C₄	semidirect
14. C₄×C₁₆×C₆₇	C₄	semidirect
79. C₆₇	#12893	semidirect
85. #346	Dic₆₇	semidirect
86. C₄×C₁₆	Dic₆₇	semidirect

H ≅ G/N

type

13. #346×C₆₇

C₄

semidirect

14. C₄×C₁₆×C₆₇

C₄

semidirect

79. C₆₇

#12893

semidirect

85. #346

Dic₆₇

semidirect

86. C₄×C₁₆

Dic₆₇

semidirect

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

subgroup	index
#22560333	2
#22558313	2
#22558311	2
#22558305	2
C₆₇×#2108	2
#12893	67

subgroup

index

C₆₇×#2108

Subgroups — #30772228

Conjugacy classes of subgroups

Complements

Maximal subgroups