Group #30749607 — subgroups

Subgroups — #30749607

Order 17152.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#22560369	C₂	1	1	yes	maximal, characteristic
2	#22560336	C₂	1	4	yes	maximal
3	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523977	—	2	2	no	—
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
7	#16863266	C₂²	1	2	yes	—
8	#16863158	—	2	8	no	—
9	C₈×C₆₇⋊C₈	C₄	1	4	yes	complemented
10	C₂×#107×C₆₇	—	2	2	no	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#384×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₆₇	—	2	4	no	—
15	C₂×C₄⋊Dic₆₇	—	2	2	no	—
16	#14523977	—	4	4	no	—
17	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₆₇	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
22	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
23	#107×C₆₇	—	4	4	no	—
24	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₇	Q₈	1	2	yes	—
26	C₂²×Dic₆₇	—	4	2	no	—
27	C₂×C₆₇⋊C₈	—	2	4	no	—
28	C₂×C₆₇⋊C₈	—	2	4	no	—
29	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	—
30	C₄⋊Dic₆₇	—	4	4	no	—
31	C₄⋊Dic₆₇	—	4	8	no	—
32	C₂²×C₄×C₆₇	—	4	2	no	—
33	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
34	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₆₇	D₈	1	4	yes	—
37	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
38	C₄⋊C₄×C₆₇	—	4	4	no	—
39	C₄⋊C₄×C₆₇	—	4	8	no	—
40	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
42	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₆₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	8	2	no	—
45	C₂×C₄×C₆₇	—	8	4	no	—
46	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
52	C₂×C₄×C₆₇	#135	1	4	yes	—
53	C₂×C₄×C₆₇	#135	1	4	yes	—
54	C₂×Dic₆₇	—	8	2	no	—
55	C₂×Dic₆₇	—	8	4	no	—
56	C₈×C₆₇	#107	1	8	yes	—
57	C₆₇⋊C₈	—	2	8	no	—
58	C₂²×C₆₇	#384	1	1	yes	characteristic
59	C₂²×C₆₇	#493	1	1	yes	characteristic
60	C₂²×C₆₇	#494	1	1	yes	characteristic
61	C₂²×C₆₇	#495	1	2	yes	—
62	C₂²×C₆₇	#495	1	2	yes	—
63	C₄×C₆₇	—	16	2	no	—
64	C₄×C₆₇	C₂×#107	1	2	yes	—
65	C₄×C₆₇	#427	1	4	yes	—
66	C₄×C₆₇	#499	1	4	yes	—
67	C₄×C₆₇	#500	1	2	yes	—
68	Dic₆₇	—	16	2	no	—
69	#8816	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	#1709	1	1	yes	characteristic
71	C₂×C₆₇	#1709	1	1	yes	characteristic
72	C₂×C₆₇	#1710	1	2	yes	—
73	C₂×C₆₇	#1845	1	1	yes	characteristic
74	C₂×C₆₇	#1846	1	2	yes	—
75	#1845	—	67	1	no	—
76	#1845	D₆₇	1	1	yes	characteristic
77	#1834	—	67	4	no	—
78	C₂×C₈²	—	67	1	no	—
79	C₆₇	#8816	1	1	yes	characteristic, complemented, Sylow
80	C₂×#107	—	2	2	no	—
81	C₂×#107	—	134	2	no	—
82	C₂×C₄×C₈	—	67	2	no	—
83	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
84	#384	—	67	2	no	—
85	#384	C₂×D₆₇	1	2	yes	—
86	#340	—	134	8	no	—
87	C₈²	—	67	4	no	—
88	C₂²×C₈	—	67	4	no	—
89	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	134	4	no	—
93	C₂×C₄⋊C₄	—	134	2	no	—
94	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
95	#107	—	4	4	no	—
96	#107	—	268	4	no	—
97	C₄×C₈	—	67	4	no	—
98	C₄×C₈	—	67	4	no	—
99	C₄×C₈	Dic₁₃₄	1	2	yes	—
100	C₄×C₈	C₄×D₆₇	1	2	yes	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	—	268	2	no	—
103	C₂²×C₄	D₄×D₆₇	1	2	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂×C₈	—	67	8	no	—
106	C₂×C₈	—	134	4	no	—
107	C₂×C₈	—	134	4	no	—
108	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
109	C₂×C₈	#11761613	1	4	yes	—
110	C₂×C₈	#11761618	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	268	4	no	—
114	C₄⋊C₄	—	268	8	no	—
115	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
116	C₄²	Q₈×D₆₇	1	2	yes	—
117	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
118	C₂³	#14524039	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	536	2	no	—
122	C₂×C₄	—	536	4	no	—
123	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
124	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#14524041	1	1	yes	characteristic
126	C₂×C₄	#14524042	1	2	yes	—
127	C₂×C₄	C₂×#11761613	1	2	yes	—
128	C₂×C₄	C₂×#11761618	1	2	yes	—
129	C₂×C₄	#14524062	1	4	yes	—
130	C₂×C₄	#14524066	1	4	yes	—
131	C₈	—	134	8	no	—
132	C₈	#14523976	1	8	yes	—
133	C₂²	#16863265	1	1	yes	characteristic
134	C₂²	#16863772	1	1	yes	characteristic
135	C₂²	#16863775	1	1	yes	characteristic
136	C₂²	#16863779	1	2	yes	—
137	C₂²	#16863781	1	2	yes	—
138	C₄	—	16	2	no	—
139	C₄	—	1072	2	no	—
140	C₄	C₂×#14523976	1	2	yes	—
141	C₄	#16863433	1	4	yes	—
142	C₄	#16863797	1	4	yes	—
143	C₄	#16863799	1	2	yes	—
144	C₂	#22559814	1	1	yes	characteristic
145	C₂	#22559816	1	1	yes	characteristic
146	C₂	#22559818	1	2	yes	—
147	C₂	#22560368	1	1	yes	characteristic
148	C₂	#22560372	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560369

C₂

yes

maximal, characteristic

#22560336

C₂

yes

maximal

C₂×C₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#1845

C₂

yes

maximal, characteristic

C₂×#14523977

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863266

C₂²

yes

—

#16863158

—

C₈×C₆₇⋊C₈

C₄

yes

complemented

C₂×#107×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#107

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

#495

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#107

yes

—

C₄×C₆₇

#427

yes

—

C₄×C₆₇

#499

yes

—

C₄×C₆₇

#500

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#1845

yes

characteristic

C₂×C₆₇

#1846

yes

—

#1845

—

#1845

D₆₇

yes

characteristic

#1834

—

C₂×C₈²

—

C₆₇

#8816

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#107

—

134

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#384

—

#384

C₂×D₆₇

yes

—

#340

—

134

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#107

—

#107

—

268

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₃₄

yes

—

100

C₄×C₈

C₄×D₆₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

268

—

103

C₂²×C₄

D₄×D₆₇

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

134

—

107

C₂×C₈

—

134

—

108

C₂×C₈

Dic₆₇⋊C₄

yes

—

109

C₂×C₈

#11761613

yes

—

110

C₂×C₈

#11761618

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

268

—

114

C₄⋊C₄

—

268

—

115

C₄²

C₂×C₄×D₆₇

yes

characteristic

116

C₄²

Q₈×D₆₇

yes

—

117

C₄²

C₂×Dic₁₃₄

yes

characteristic

118

C₂³

#14524039

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

536

—

122

C₂×C₄

—

536

—

123

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

124

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

125

C₂×C₄

#14524041

yes

characteristic

126

C₂×C₄

#14524042

yes

—

127

C₂×C₄

C₂×#11761613

yes

—

128

C₂×C₄

C₂×#11761618

yes

—

129

C₂×C₄

#14524062

yes

—

130

C₂×C₄

#14524066

yes

—

131

C₈

—

134

—

132

C₈

#14523976

yes

—

133

C₂²

#16863265

yes

characteristic

134

C₂²

#16863772

yes

characteristic

135

C₂²

#16863775

yes

characteristic

136

C₂²

#16863779

yes

—

137

C₂²

#16863781

yes

—

138

C₄

—

139

C₄

—

1072

—

140

C₄

C₂×#14523976

yes

—

141

C₄

#16863433

yes

—

142

C₄

#16863797

yes

—

143

C₄

#16863799

yes

—

144

C₂

#22559814

yes

characteristic

145

C₂

#22559816

yes

characteristic

146

C₂

#22559818

yes

—

147

C₂

#22560368

yes

characteristic

148

C₂

#22560372

yes

—

	subgroup	G/H
center	C₂³	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1845	C₂
socle, prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#8816
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1845

C₂

socle, prime radical

C₂³×C₆₇

#127

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
9. C₈×C₆₇⋊C₈	C₄	semidirect
79. C₆₇	#8816	semidirect

H ≅ G/N

type

9. C₈×C₆₇⋊C₈

C₄

semidirect

79. C₆₇

#8816

semidirect

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

subgroup	index
#22560369	2
#22560336	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1845	2
#8816	67

subgroup

index

#22560369

#22560336

C₂×C₈×C₆₇⋊C₈

C₆₇×#1845

#8816

Subgroups — #30749607

Conjugacy classes of subgroups

Complements

Maximal subgroups