Group #30770565 — subgroups

Subgroups — #30770565

Order 17152.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22562464	C₂	1	2	yes	maximal
2	#22562461	C₂	1	2	yes	maximal, complemented
3	#22559827	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558126	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1709	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523989	—	2	4	no	—
7	C₂×#14523987	—	2	4	no	—
8	#16863801	—	2	2	no	—
9	#16863776	C₂²	1	2	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#494×C₆₇	C₂²	1	1	yes	characteristic
12	#493×C₆₇	C₂²	1	1	yes	characteristic
13	#486×C₆₇	—	2	2	no	—
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₆₈	—	2	2	no	—
16	C₂×Dic₂₆₈	—	2	2	no	—
17	#14524043	—	2	2	no	—
18	#14523989	—	4	8	no	—
19	#14523987	—	4	8	no	—
20	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
21	#14523977	—	4	4	no	—
22	C₂×Q₁₆×C₆₇	—	2	2	no	—
23	C₂×D₈×C₆₇	—	2	2	no	—
24	C₄⋊Q₈×C₆₇	—	2	1	no	—
25	#127×C₆₇	—	2	1	no	—
26	#102×C₆₇	—	4	4	no	—
27	C₄×C₈×C₆₇	D₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	4	2	no	—
30	Dic₂₆₈	—	4	4	no	—
31	Dic₂₆₈	—	4	4	no	—
32	C₄⋊Dic₆₇	—	4	4	no	—
33	C₆₇⋊C₁₆	D₈	1	8	yes	—
34	C₂×Q₈×C₆₇	—	4	1	no	—
35	C₂×D₄×C₆₇	—	4	1	no	—
36	C₂×D₄×C₆₇	—	4	2	no	—
37	Q₁₆×C₆₇	—	4	4	no	—
38	D₈×C₆₇	—	4	4	no	—
39	C₂×C₈×C₆₇	D₈	1	4	yes	—
40	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₆₇	—	4	2	no	—
42	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂³×C₆₇	—	8	1	no	—
44	Q₈×C₆₇	—	8	2	no	—
45	D₄×C₆₇	—	8	2	no	—
46	D₄×C₆₇	—	8	4	no	—
47	C₂×C₄×C₆₇	—	8	1	no	—
48	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
50	C₂×Dic₆₇	—	8	2	no	—
51	Dic₁₃₄	—	8	2	no	—
52	Dic₁₃₄	—	8	2	no	—
53	C₈×C₆₇	—	2	4	no	—
54	C₈×C₆₇	C₂×D₈	1	4	yes	—
55	C₂²×C₆₇	—	16	1	no	—
56	C₂²×C₆₇	—	16	2	no	—
57	C₂²×C₆₇	#493	1	1	yes	characteristic
58	C₄×C₆₇	—	16	1	no	—
59	C₄×C₆₇	#493	1	2	yes	—
60	C₄×C₆₇	#510	1	4	yes	—
61	Dic₆₇	—	16	2	no	—
62	#12520	—	67	1	no	maximal, Sylow
63	C₂×C₆₇	—	32	1	no	—
64	C₂×C₆₇	#1712	1	1	yes	characteristic
65	C₂×C₆₇	#2251	1	2	yes	—
66	#2248	—	67	2	no	—
67	#2247	—	67	2	no	—
68	#1713	—	67	1	no	—
69	#1709	D₆₇	1	1	yes	characteristic
70	#1311	—	67	1	no	—
71	C₆₇	#12520	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	134	4	no	—
73	C₂×SD₃₂	—	134	4	no	—
74	#500	—	134	2	no	—
75	#494	—	67	2	no	—
76	#494	C₂×D₆₇	1	1	yes	characteristic
77	#493	C₂×D₆₇	1	1	yes	characteristic
78	#486	—	2	2	no	—
79	C₄×C₁₆	—	67	2	no	—
80	C₈²	C₂×D₆₇	1	1	yes	characteristic
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	134	2	no	—
83	C₂×Q₁₆	—	134	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	—	134	2	no	—
87	#127	—	2	1	no	—
88	Q₃₂	—	268	8	no	—
89	SD₃₂	—	268	8	no	—
90	C₂×C₁₆	—	67	4	no	—
91	#107	—	268	4	no	—
92	#102	—	4	4	no	—
93	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
94	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	268	2	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	4	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	268	4	no	—
101	Q₁₆	—	268	4	no	—
102	D₈	—	4	4	no	—
103	C₂×C₈	#11761613	1	4	yes	—
104	C₂×C₈	D₄×D₆₇	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	268	4	no	—
107	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
108	C₁₆	—	67	8	no	—
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	536	2	no	—
112	Q₈	—	536	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	4	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	536	2	no	—
117	C₂×C₄	#14524039	1	1	yes	characteristic
118	C₂×C₄	C₂×#11761613	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	D₈×D₆₇	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#16863772	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	1072	2	no	—
126	C₄	#16863771	1	2	yes	—
127	C₄	#16863836	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#22559823	1	1	yes	characteristic
130	C₂	#22562481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562464

C₂

yes

maximal

#22562461

C₂

yes

maximal, complemented

#22559827

C₂

yes

maximal, characteristic, complemented

#22558126

C₂

yes

maximal, characteristic, complemented

C₆₇×#1709

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#494×C₆₇

C₂²

yes

characteristic

#493×C₆₇

C₂²

yes

characteristic

#486×C₆₇

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

#14523989

—

#14523987

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

C₂×Q₁₆×C₆₇

—

C₂×D₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

#127×C₆₇

—

#102×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

D₈

yes

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

Q₁₆×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#493

yes

—

C₄×C₆₇

#510

yes

—

Dic₆₇

—

#12520

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1712

yes

characteristic

C₂×C₆₇

#2251

yes

—

—

—

—

D₆₇

yes

characteristic

#1311

—

C₆₇

#12520

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

134

—

C₂×SD₃₂

—

134

—

#500

—

134

—

#494

—

#494

C₂×D₆₇

yes

characteristic

#493

C₂×D₆₇

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

#127

—

Q₃₂

—

268

—

SD₃₂

—

268

—

C₂×C₁₆

—

#107

—

268

—

#102

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

268

—

101

Q₁₆

—

268

—

102

D₈

—

103

C₂×C₈

#11761613

yes

—

104

C₂×C₈

D₄×D₆₇

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

268

—

107

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

108

C₁₆

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

536

—

112

Q₈

—

536

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

536

—

117

C₂×C₄

#14524039

yes

characteristic

118

C₂×C₄

C₂×#11761613

yes

—

119

C₈

—

120

C₈

D₈×D₆₇

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#16863772

yes

characteristic

124

C₄

—

125

C₄

—

1072

—

126

C₄

#16863771

yes

—

127

C₄

#16863836

yes

—

128

C₂

—

129

C₂

#22559823

yes

characteristic

130

C₂

#22562481

yes

—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1709	C₂
socle	C₂²×C₆₇	#493
prime radical	#493×C₆₇	C₂²
hypercenter	#1709	D₆₇
hypocenter	C₆₇	#12520
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1709

C₂

socle

C₂²×C₆₇

#493

prime radical

#493×C₆₇

C₂²

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #22562461	C₂	semidirect
3. #22559827	C₂	semidirect
4. #22558126	C₂	semidirect
71. C₆₇	#12520	semidirect

H ≅ G/N

type

2. #22562461

C₂

semidirect

3. #22559827

C₂

semidirect

4. #22558126

C₂

semidirect

71. C₆₇

#12520

semidirect

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

subgroup	index
#22562464	2
#22562461	2
#22559827	2
#22558126	2
C₆₇×#1709	2
#12520	67

subgroup

index

C₆₇×#1709

Subgroups — #30770565

Conjugacy classes of subgroups

Complements

Maximal subgroups