Group #31771180 — subgroups

Subgroups — #31771180

Order 17216.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#16864403	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575451	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575442	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575441	C₂	1	1	yes	maximal, characteristic, complemented
6	#22575438	C₂	1	4	yes	maximal, complemented
7	#22575431	C₂	1	4	yes	maximal, complemented
8	#22575410	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864403	—	2	4	no	—
11	C₂×Dic₅₃₈	C₂²	1	2	yes	—
12	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
13	Q₈×D₂₆₉	—	2	4	no	—
14	C₂×D₁₀₇₆	C₂²	1	2	yes	—
15	C₂×D₁₀₇₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
17	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
18	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
19	#16864381	C₂²	1	2	yes	—
20	#16864381	C₂²	1	4	yes	—
21	#16864379	C₂²	1	2	yes	—
22	#16864379	C₂²	1	4	yes	—
23	#16864379	C₂²	1	4	yes	—
24	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₂₆₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₉	C₂²	1	4	yes	—
28	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
29	C₂²×D₂₆₉	C₂³	1	2	yes	—
30	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
31	Dic₅₃₈	—	2	2	no	—
32	Dic₅₃₈	—	2	4	no	—
33	D₁₀₇₆	—	2	4	no	—
34	D₁₀₇₆	—	2	2	no	—
35	C₂×Dic₂₆₉	C₂³	1	2	yes	—
36	C₂×Dic₂₆₉	C₂³	1	2	yes	—
37	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
38	C₄×D₂₆₉	—	2	4	no	—
39	C₄×D₂₆₉	—	2	4	no	—
40	C₄×D₂₆₉	D₄	1	4	yes	—
41	Q₈×C₂₆₉	—	2	2	no	—
42	Q₈×C₂₆₉	—	2	2	no	—
43	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
44	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
45	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
46	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
47	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
48	C₂×D₂₆₉	—	2	2	no	—
49	C₂×D₂₆₉	—	2	2	no	—
50	C₂×D₂₆₉	—	2	4	no	—
51	C₂×D₂₆₉	C₂×D₄	1	2	yes	—
52	C₄×C₂₆₉	—	2	2	no	—
53	C₄×C₂₆₉	—	2	2	no	—
54	C₄×C₂₆₉	—	2	2	no	—
55	C₄×C₂₆₉	C₂×D₄	1	2	yes	—
56	Dic₂₆₉	—	2	2	no	—
57	Dic₂₆₉	—	2	2	no	—
58	Dic₂₆₉	C₂×D₄	1	2	yes	—
59	C₂×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
61	D₂₆₉	—	2	2	no	—
62	D₂₆₉	—	4	2	no	—
63	C₂₆₉	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	269	1	no	maximal, Sylow
65	C₂×#41	—	269	1	no	—
66	C₂²×Q₈	—	269	1	no	—
67	C₄⋊Q₈	—	269	1	no	—
68	C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic, complemented
69	#124	—	269	1	no	—
70	#124	—	269	1	no	—
71	#123	—	269	4	no	—
72	C₂²⋊Q₈	—	269	4	no	—
73	#117	—	269	1	no	—
74	#41	—	538	4	no	—
75	C₂×Q₈	—	269	2	no	—
76	C₂×Q₈	—	269	1	no	—
77	C₂×Q₈	—	538	4	no	—
78	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₂₆₉	1	1	yes	characteristic
80	C₂×D₄	—	269	2	no	—
81	C₂×D₄	—	269	1	no	—
82	C₂²×C₄	—	269	1	no	—
83	C₂²×C₄	—	269	2	no	—
84	C₂²×C₄	—	269	2	no	—
85	C₄⋊C₄	—	269	2	no	—
86	C₄⋊C₄	—	269	4	no	—
87	C₄⋊C₄	C₂×D₂₆₉	1	4	yes	—
88	C₂²⋊C₄	—	269	2	no	—
89	C₂²⋊C₄	—	269	4	no	—
90	C₂²⋊C₄	—	269	4	no	—
91	C₄²	—	269	1	no	—
92	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
93	C₂³	—	269	1	no	—
94	C₂³	—	269	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	538	4	no	—
98	Q₈	—	538	2	no	—
99	D₄	—	538	4	no	—
100	D₄	—	538	2	no	—
101	C₂×C₄	—	269	1	no	—
102	C₂×C₄	—	269	4	no	—
103	C₂×C₄	—	269	2	no	—
104	C₂×C₄	—	269	2	no	—
105	C₂×C₄	—	538	4	no	—
106	C₂×C₄	—	538	4	no	—
107	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
108	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
109	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
110	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
111	C₂²	—	269	2	no	—
112	C₂²	—	538	2	no	—
113	C₂²	—	538	2	no	—
114	C₂²	—	538	4	no	—
115	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	269	2	no	—
120	C₄	—	538	2	no	—
121	C₄	—	538	2	no	—
122	C₄	D₄×D₂₆₉	1	2	yes	—
123	C₂	—	538	2	no	—
124	C₂	—	1076	2	no	—
125	C₂	C₂×D₄×D₂₆₉	1	1	yes	characteristic
126	C₂	#22575501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16864403

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₆₉

C₂

yes

maximal, characteristic, complemented

#22575451

C₂

yes

maximal, characteristic, complemented

#22575442

C₂

yes

maximal, characteristic, complemented

#22575441

C₂

yes

maximal, characteristic, complemented

#22575438

C₂

yes

maximal, complemented

#22575431

C₂

yes

maximal, complemented

#22575410

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₂₆₉

C₂

yes

maximal, characteristic, complemented

#16864403

—

C₂×Dic₅₃₈

C₂²

yes

—

C₂×Dic₅₃₈

C₂²

yes

characteristic

Q₈×D₂₆₉

—

C₂×D₁₀₇₆

C₂²

yes

—

C₂×D₁₀₇₆

C₂²

yes

characteristic

C₂×C₄×D₂₆₉

C₂²

yes

—

C₂×C₄×D₂₆₉

C₂²

yes

—

C₂×C₄×D₂₆₉

C₂²

yes

characteristic

#16864381

C₂²

yes

—

#16864381

C₂²

yes

—

#16864379

C₂²

yes

—

#16864379

C₂²

yes

—

#16864379

C₂²

yes

—

C₄×Dic₂₆₉

C₂²

yes

characteristic

C₂×Q₈×C₂₆₉

C₂²

yes

characteristic

C₂×Q₈×C₂₆₉

C₂²

yes

characteristic

C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₄²×C₂₆₉

C₂²

yes

characteristic

C₂²×D₂₆₉

C₂³

yes

—

C₂²×D₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

Dic₅₃₈

—

D₁₀₇₆

—

D₁₀₇₆

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

—

C₂×Dic₂₆₉

C₂³

yes

characteristic

C₄×D₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

D₄

yes

—

Q₈×C₂₆₉

—

Q₈×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

—

C₂×C₄×C₂₆₉

C₂³

yes

characteristic

C₂²×C₂₆₉

C₂⁴

yes

characteristic

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₂×D₂₆₉

C₂×D₄

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×D₄

yes

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

C₂×D₄

yes

—

C₂×C₂₆₉

C₂²×D₄

yes

characteristic

C₂×C₂₆₉

2₋¹⁺⁴

yes

—

D₂₆₉

—

D₂₆₉

—

C₂₆₉

#536

yes

characteristic, complemented, Sylow

#536

—

269

maximal, Sylow

C₂×#41

—

269

—

C₂²×Q₈

—

269

—

C₄⋊Q₈

—

269

—

C₄⋊Q₈

D₂₆₉

yes

characteristic, complemented

#124

—

269

—

#124

—

269

—

#123

—

269

—

C₂²⋊Q₈

—

269

—

#117

—

269

—

#41

—

538

—

C₂×Q₈

—

269

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

C₂×Q₈

C₂×D₂₆₉

yes

characteristic

C₂×Q₈

C₂×D₂₆₉

yes

characteristic

C₂×D₄

—

269

—

C₂×D₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

—

269

—

C₂²×C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

269

—

C₄⋊C₄

C₂×D₂₆₉

yes

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

—

269

—

C₂²⋊C₄

—

269

—

C₄²

—

269

—

C₄²

C₂×D₂₆₉

yes

characteristic

C₂³

—

269

—

C₂³

—

269

—

Q₈

—

Q₈

—

Q₈

—

538

—

Q₈

—

538

—

D₄

—

538

—

100

D₄

—

538

—

101

C₂×C₄

—

269

—

102

C₂×C₄

—

269

—

103

C₂×C₄

—

269

—

104

C₂×C₄

—

269

—

105

C₂×C₄

—

538

—

106

C₂×C₄

—

538

—

107

C₂×C₄

C₂²×D₂₆₉

yes

—

108

C₂×C₄

C₂²×D₂₆₉

yes

—

109

C₂×C₄

C₂²×D₂₆₉

yes

—

110

C₂×C₄

C₂²×D₂₆₉

yes

characteristic

111

C₂²

—

269

—

112

C₂²

—

538

—

113

C₂²

—

538

—

114

C₂²

—

538

—

115

C₂²

C₂³×D₂₆₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

269

—

120

C₄

—

538

—

121

C₄

—

538

—

122

C₄

D₄×D₂₆₉

yes

—

123

C₂

—

538

—

124

C₂

—

1076

—

125

C₂

C₂×D₄×D₂₆₉

yes

characteristic

126

C₂

#22575501

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	C₄⋊Q₈×C₂₆₉	C₂
prime radical	C₂×#16864403	C₂
hypercenter	C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	#536
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₆₉

commutator, socle, cosocle

C₂²×C₂₆₉

C₂⁴

Fitting

C₄⋊Q₈×C₂₆₉

C₂

prime radical

C₂×#16864403

C₂

hypercenter

C₄⋊Q₈

D₂₆₉

C₂₆₉

G ▷ C₂²×C₂₆₉ ▷ C₂₆₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₂₆₉	C₂	semidirect
3. #22575451	C₂	semidirect
4. #22575442	C₂	semidirect
5. #22575441	C₂	semidirect
6. #22575438	C₂	semidirect
7. #22575431	C₂	semidirect
8. #22575410	C₂	semidirect
9. C₄⋊Q₈×C₂₆₉	C₂	semidirect
63. C₂₆₉	#536	semidirect
68. C₄⋊Q₈	D₂₆₉	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₂₆₉

C₂

semidirect

3. #22575451

C₂

semidirect

4. #22575442

C₂

semidirect

5. #22575441

C₂

semidirect

6. #22575438

C₂

semidirect

7. #22575431

C₂

semidirect

8. #22575410

C₂

semidirect

9. C₄⋊Q₈×C₂₆₉

C₂

semidirect

63. C₂₆₉

#536

semidirect

68. C₄⋊Q₈

D₂₆₉

semidirect

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

subgroup	index
C₂×#16864403	2
C₂×Q₈×D₂₆₉	2
#22575451	2
#22575442	2
#22575441	2
#22575438	2
#22575431	2
#22575410	2
C₄⋊Q₈×C₂₆₉	2
#536	269

subgroup

index

C₂×Q₈×D₂₆₉

C₄⋊Q₈×C₂₆₉

269

Subgroups — #31771180

Conjugacy classes of subgroups

Complements

Maximal subgroups