Group #30745174 — subgroups

Subgroups — #30745174

Order 17152.

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

normal subgroups only

This group has 414 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	#22561747	C₂	1	3	yes	maximal
2	#22558035	C₂	1	3	yes	maximal
3	C₆₇×#2811	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523976	—	2	3	no	—
5	C₂×#14523967	—	2	3	no	—
6	#16863266	C₂²	1	1	yes	characteristic
7	#16863151	C₂²	1	3	yes	—
8	#395×C₆₇	—	2	3	no	—
9	#391×C₆₇	—	2	3	no	—
10	#391×C₆₇	C₂²	1	3	yes	—
11	C₂²×C₆₇⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₆₇	—	2	3	no	—
13	#14523976	—	4	6	no	—
14	#14523967	—	4	6	no	—
15	C₂²×Q₈×C₆₇	—	2	1	no	—
16	C₂²×Q₈×C₆₇	D₄	1	3	yes	—
17	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
18	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
19	C₂×C₄²×C₆₇	—	2	1	no	—
20	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
21	#95×C₆₇	—	2	3	no	—
22	#95×C₆₇	—	2	6	no	—
23	C₂²×Dic₆₇	—	4	1	no	—
24	C₂×C₆₇⋊C₈	—	2	6	no	—
25	C₂×C₆₇⋊C₈	—	4	3	no	—
26	C₂×C₆₇⋊C₈	—	4	3	no	—
27	C₄⋊Dic₆₇	—	4	3	no	—
28	C₄⋊Dic₆₇	—	4	3	no	—
29	C₂×Q₈×C₆₇	—	2	3	no	—
30	C₂×Q₈×C₆₇	—	4	3	no	—
31	C₂×Q₈×C₆₇	—	4	3	no	—
32	C₂×Q₈×C₆₇	—	8	3	no	—
33	C₂×Q₈×C₆₇	Q₁₆	1	6	yes	—
34	C₂²×C₄×C₆₇	—	2	3	no	—
35	C₂²×C₄×C₆₇	—	2	3	no	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
37	C₄⋊C₄×C₆₇	—	4	3	no	—
38	C₄⋊C₄×C₆₇	—	8	3	no	—
39	C₄²×C₆₇	—	4	1	no	—
40	C₄²×C₆₇	—	8	1	no	—
41	C₂³×C₆₇	#120	1	1	yes	characteristic
42	Q₈×C₆₇	—	4	6	no	—
43	Q₈×C₆₇	—	8	3	no	—
44	Q₈×C₆₇	—	8	4	no	—
45	C₂×C₄×C₆₇	—	2	3	no	—
46	C₂×C₄×C₆₇	—	2	3	no	—
47	C₂×C₄×C₆₇	—	2	6	no	—
48	C₂×C₄×C₆₇	—	4	3	no	—
49	C₂×C₄×C₆₇	—	4	3	no	—
50	C₂×C₄×C₆₇	—	4	3	no	—
51	C₂×C₄×C₆₇	—	4	6	no	—
52	C₂×C₄×C₆₇	C₂×Q₁₆	1	3	yes	—
53	C₂×C₄×C₆₇	#137	1	3	yes	—
54	C₂×Dic₆₇	—	8	3	no	—
55	C₆₇⋊C₈	—	4	6	no	—
56	C₂²×C₆₇	#451	1	3	yes	—
57	C₂²×C₆₇	#453	1	3	yes	—
58	C₂²×C₆₇	#458	1	1	yes	characteristic
59	C₄×C₆₇	—	2	3	no	—
60	C₄×C₆₇	—	2	3	no	—
61	C₄×C₆₇	—	4	6	no	—
62	C₄×C₆₇	—	8	3	no	—
63	Dic₆₇	—	16	1	no	—
64	#7837	—	67	1	no	maximal, Sylow
65	C₂×C₆₇	#1614	1	3	yes	—
66	C₂×C₆₇	#1633	1	3	yes	—
67	C₂×C₆₇	#1655	1	1	yes	characteristic
68	#2811	D₆₇	1	1	yes	characteristic
69	#2095	—	67	3	no	—
70	#1272	—	67	3	no	—
71	C₆₇	#7837	1	1	yes	characteristic, complemented, Sylow
72	C₂×#107	—	134	3	no	—
73	C₂×#103	—	134	3	no	—
74	#395	—	2	3	no	—
75	#391	—	2	3	no	—
76	#391	C₂×D₆₇	1	3	yes	—
77	#384	—	67	1	no	—
78	#336	—	67	3	no	—
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×Q₈	C₂²⋊D₆₇	1	3	yes	—
81	C₂²×C₈	—	134	3	no	—
82	C₂×C₄⋊C₄	—	2	3	no	—
83	C₂×C₄⋊C₄	—	134	3	no	—
84	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
87	#107	—	268	6	no	—
88	#103	—	268	6	no	—
89	#95	—	2	3	no	—
90	#95	—	2	6	no	—
91	C₂×Q₈	—	2	3	no	—
92	C₂×Q₈	—	4	3	no	—
93	C₂×Q₈	—	4	3	no	—
94	C₂×Q₈	—	8	3	no	—
95	C₂×Q₈	#11761618	1	6	yes	—
96	C₂²×C₄	—	2	3	no	—
97	C₂²×C₄	—	2	3	no	—
98	C₂²×C₄	—	268	1	no	—
99	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
100	C₂×C₈	—	268	3	no	—
101	C₂×C₈	—	268	3	no	—
102	C₂×C₈	—	134	6	no	—
103	C₄⋊C₄	—	4	3	no	—
104	C₄⋊C₄	—	8	3	no	—
105	C₄⋊C₄	—	268	3	no	—
106	C₄⋊C₄	—	268	3	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	8	1	no	—
109	C₂³	#14524012	1	1	yes	characteristic
110	Q₈	—	4	6	no	—
111	Q₈	—	8	3	no	—
112	Q₈	—	8	4	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	4	3	no	—
117	C₂×C₄	—	4	3	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	6	no	—
120	C₂×C₄	—	536	3	no	—
121	C₂×C₄	C₂×#11761618	1	3	yes	—
122	C₂×C₄	#14524074	1	3	yes	—
123	C₈	—	268	6	no	—
124	C₂²	#16863552	1	3	yes	—
125	C₂²	#16863564	1	3	yes	—
126	C₂²	#16863586	1	1	yes	characteristic
127	C₄	—	2	3	no	—
128	C₄	—	2	3	no	—
129	C₄	—	4	6	no	—
130	C₄	—	8	3	no	—
131	C₄	—	1072	1	no	—
132	C₂	#22559234	1	3	yes	—
133	C₂	#22559349	1	3	yes	—
134	C₂	#22559498	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561747

C₂

yes

maximal

#22558035

C₂

yes

maximal

C₆₇×#2811

C₂

yes

maximal, characteristic

C₂×#14523976

—

C₂×#14523967

—

#16863266

C₂²

yes

characteristic

#16863151

C₂²

yes

—

#395×C₆₇

—

#391×C₆₇

—

#391×C₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

—

#14523976

—

#14523967

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#120

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#451

yes

—

C₂²×C₆₇

#453

yes

—

C₂²×C₆₇

#458

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

#7837

—

maximal, Sylow

C₂×C₆₇

#1614

yes

—

C₂×C₆₇

#1633

yes

—

C₂×C₆₇

#1655

yes

characteristic

#2811

D₆₇

yes

characteristic

#2095

—

#1272

—

C₆₇

#7837

yes

characteristic, complemented, Sylow

C₂×#107

—

134

—

C₂×#103

—

134

—

#395

—

#391

—

#391

C₂×D₆₇

yes

—

#384

—

#336

—

C₂²×Q₈

—

C₂²×Q₈

C₂²⋊D₆₇

yes

—

C₂²×C₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#107

—

268

—

#103

—

268

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#11761618

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

100

C₂×C₈

—

268

—

101

C₂×C₈

—

268

—

102

C₂×C₈

—

134

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

268

—

106

C₄⋊C₄

—

268

—

107

C₄²

—

108

C₄²

—

109

C₂³

#14524012

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

536

—

121

C₂×C₄

C₂×#11761618

yes

—

122

C₂×C₄

#14524074

yes

—

123

C₈

—

268

—

124

C₂²

#16863552

yes

—

125

C₂²

#16863564

yes

—

126

C₂²

#16863586

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

1072

—

132

C₂

#22559234

yes

—

133

C₂

#22559349

yes

—

134

C₂

#22559498

yes

characteristic

	subgroup	G/H
center	C₂³	#14524012
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2811	C₂
socle, prime radical	C₂³×C₆₇	#120
hypercenter	#2811	D₆₇
hypocenter	C₆₇	#7837
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

subgroup

G/H

center

C₂³

#14524012

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2811

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂³×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

N	H ≅ G/N	type
71. C₆₇	#7837	semidirect

H ≅ G/N

type

71. C₆₇

#7837

semidirect

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

subgroup	index
#22561747	2
#22558035	2
C₆₇×#2811	2
#7837	67

subgroup

index

#22561747

#22558035

C₆₇×#2811

#7837

Subgroups — #30745174

Conjugacy classes of subgroups

Complements

Maximal subgroups