Group #16827849 — subgroups

Subgroups — #16827849

Order 3712.

752 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 213 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×#11745231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11745236	C₂	1	1	yes	maximal, characteristic
3	C₂×#11745237	C₂	1	4	yes	maximal, complemented
4	#14271065	C₂	1	8	yes	maximal, complemented
5	C₄³×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₁₆	C₂²	1	4	yes	—
7	C₂²×D₁₁₆	C₂²	1	2	yes	—
8	C₂²×Dic₅₈	C₂²	1	2	yes	—
9	C₂×C₄²×C₂₉	C₂²	1	4	yes	—
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
12	C₂×#90781	C₂²	1	16	yes	—
13	C₂×C₄⋊Dic₂₉	C₂²	1	4	yes	—
14	#11745237	—	2	16	no	—
15	#11745236	—	2	4	no	—
16	#11745231	—	2	4	no	—
17	C₂³×D₂₉	C₂³	1	4	yes	—
18	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
19	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
21	C₂²×Dic₂₉	C₂³	1	4	yes	—
22	C₂×D₁₁₆	—	2	8	no	—
23	C₂×D₁₁₆	—	2	4	no	—
24	C₂×D₁₁₆	—	2	8	no	—
25	C₂×D₁₁₆	—	2	16	no	—
26	C₂×Dic₅₈	—	2	4	no	—
27	C₂×Dic₅₈	—	2	8	no	—
28	C₄²×C₂₉	—	2	8	no	—
29	C₄²×C₂₉	D₄	1	4	yes	—
30	C₄²×C₂₉	D₄	1	8	yes	complemented
31	#90781	—	2	32	no	—
32	C₄⋊Dic₂₉	—	2	8	no	—
33	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
34	C₂²×D₂₉	—	2	8	no	—
35	C₂²×D₂₉	—	2	4	no	—
36	C₂²×D₂₉	—	2	16	no	—
37	C₂×C₄×C₂₉	—	2	8	no	—
38	C₂×C₄×C₂₉	C₂×D₄	1	8	yes	—
39	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₂₉	#41	1	8	yes	—
43	C₂×Dic₂₉	—	2	4	no	—
44	C₂×Dic₂₉	—	2	8	no	—
45	D₁₁₆	—	4	16	no	—
46	D₁₁₆	—	4	8	no	—
47	Dic₅₈	—	4	8	no	—
48	#2858	—	29	1	no	maximal, Sylow
49	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₂₉	C₂²×D₄	1	2	yes	—
51	C₂²×C₂₉	C₂×#41	1	4	yes	—
52	C₂×D₂₉	—	4	8	no	—
53	C₂×D₂₉	—	4	4	no	—
54	C₂×D₂₉	—	4	16	no	—
55	C₄×C₂₉	—	2	8	no	—
56	C₄×C₂₉	#124	1	8	yes	complemented
57	C₄×C₂₉	#127	1	4	yes	—
58	Dic₂₉	—	4	4	no	—
59	C₂×C₄⋊Q₈	—	29	1	no	—
60	C₂×#127	—	29	1	no	—
61	C₂×#124	—	29	4	no	—
62	#390	—	29	8	no	—
63	C₄³	D₂₉	1	1	yes	characteristic, complemented
64	C₂×C₂₉	C₂×#124	1	2	yes	—
65	C₂×C₂₉	C₂×#127	1	1	yes	characteristic
66	C₂×C₂₉	#532	1	4	yes	—
67	D₂₉	—	8	4	no	—
68	C₂²×Q₈	—	29	2	no	—
69	C₂²×D₄	—	29	2	no	—
70	C₂²×D₄	—	29	4	no	—
71	C₄⋊Q₈	—	58	4	no	—
72	#127	—	58	4	no	—
73	#124	—	58	16	no	—
74	C₂×C₄⋊C₄	—	29	4	no	—
75	C₂×C₂²⋊C₄	—	29	16	no	—
76	C₂×C₄²	C₂×D₂₉	1	4	yes	—
77	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₂₉	1	2	yes	—
79	C₂₉	#2858	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	—	29	4	no	—
81	C₂×Q₈	—	58	8	no	—
82	C₂×Q₈	—	58	4	no	—
83	C₂×D₄	—	58	8	no	—
84	C₂×D₄	—	58	4	no	—
85	C₂×D₄	—	58	8	no	—
86	C₂×D₄	—	58	16	no	—
87	C₂²×C₄	—	29	4	no	—
88	C₂²×C₄	C₂²×D₂₉	1	4	yes	—
89	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
91	C₄⋊C₄	—	58	8	no	—
92	C₂²⋊C₄	—	58	32	no	—
93	C₄²	—	2	8	no	—
94	C₄²	D₁₁₆	1	4	yes	—
95	C₄²	D₁₁₆	1	8	yes	complemented
96	C₂³	—	58	8	no	—
97	C₂³	—	58	4	no	—
98	C₂³	—	58	16	no	—
99	C₂³	C₂³×D₂₉	1	1	yes	characteristic
100	Q₈	—	116	8	no	—
101	D₄	—	116	8	no	—
102	D₄	—	116	16	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	58	8	no	—
105	C₂×C₄	—	58	4	no	—
106	C₂×C₄	C₂×D₁₁₆	1	8	yes	—
107	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
108	C₂×C₄	C₂×D₁₁₆	1	4	yes	—
109	C₂×C₄	C₂×D₁₁₆	1	4	yes	—
110	C₂×C₄	#90805	1	8	yes	—
111	C₂²	—	116	8	no	—
112	C₂²	—	116	4	no	—
113	C₂²	—	116	16	no	—
114	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
115	C₂²	C₂²×D₁₁₆	1	2	yes	—
116	C₂²	C₂×#90805	1	4	yes	—
117	C₄	—	2	8	no	—
118	C₄	—	116	4	no	—
119	C₄	#11745236	1	4	yes	—
120	C₄	#11745237	1	8	yes	complemented
121	C₂	—	232	4	no	—
122	C₂	C₂×#11745237	1	2	yes	—
123	C₂	C₂×#11745236	1	1	yes	characteristic
124	C₂	#14271713	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic, complemented

C₂×#11745236

C₂

yes

maximal, characteristic

C₂×#11745237

C₂

yes

maximal, complemented

#14271065

C₂

yes

maximal, complemented

C₄³×C₂₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₁₆

C₂²

yes

—

C₂²×D₁₁₆

C₂²

yes

—

C₂²×Dic₅₈

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄²×C₂₉

C₂²

yes

—

C₂×#90781

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

#11745237

—

#11745236

—

#11745231

—

C₂³×D₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

—

C₂²×Dic₂₉

C₂³

yes

—

C₂×D₁₁₆

—

C₂×D₁₁₆

—

C₂×D₁₁₆

—

C₂×D₁₁₆

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₄²×C₂₉

—

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

D₄

yes

complemented

#90781

—

C₄⋊Dic₂₉

—

C₂³×C₂₉

C₂⁴

yes

characteristic

C₂²×D₂₉

—

C₂²×D₂₉

—

C₂²×D₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

D₁₁₆

—

D₁₁₆

—

Dic₅₈

—

#2858

—

maximal, Sylow

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₂²×C₂₉

C₂²×D₄

yes

—

C₂²×C₂₉

C₂×#41

yes

—

C₂×D₂₉

—

C₂×D₂₉

—

C₂×D₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#124

yes

complemented

C₄×C₂₉

#127

yes

—

Dic₂₉

—

C₂×C₄⋊Q₈

—

C₂×#127

—

C₂×#124

—

#390

—

C₄³

D₂₉

yes

characteristic, complemented

C₂×C₂₉

C₂×#124

yes

—

C₂×C₂₉

C₂×#127

yes

characteristic

C₂×C₂₉

#532

yes

—

D₂₉

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

#127

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₂×C₄²

C₂×D₂₉

yes

—

C₂₉

#2858

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₁₁₆

yes

—

C₄²

D₁₁₆

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₂₉

yes

characteristic

100

Q₈

—

116

—

101

D₄

—

116

—

102

D₄

—

116

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×D₁₁₆

yes

—

107

C₂×C₄

C₂×D₁₁₆

yes

—

108

C₂×C₄

C₂×D₁₁₆

yes

—

109

C₂×C₄

C₂×D₁₁₆

yes

—

110

C₂×C₄

#90805

yes

—

111

C₂²

—

116

—

112

C₂²

—

116

—

113

C₂²

—

116

—

114

C₂²

C₂²×D₁₁₆

yes

characteristic

115

C₂²

C₂²×D₁₁₆

yes

—

116

C₂²

C₂×#90805

yes

—

117

C₄

—

118

C₄

—

116

—

119

C₄

#11745236

yes

—

120

C₄

#11745237

yes

complemented

121

C₂

—

232

—

122

C₂

C₂×#11745237

yes

—

123

C₂

C₂×#11745236

yes

characteristic

124

C₂

#14271713

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator, socle, cosocle	C₂³×C₂₉	C₂⁴
Fitting	C₄³×C₂₉	C₂
prime radical	C₂×#11745236	C₂
hypercenter	C₄³	D₂₉
hypocenter	C₂₉	#2858
LCS	G ▷ C₂³×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₉

commutator, socle, cosocle

C₂³×C₂₉

C₂⁴

Fitting

C₄³×C₂₉

C₂

prime radical

C₂×#11745236

C₂

hypercenter

C₄³

D₂₉

C₂₉

G ▷ C₂³×C₂₉ ▷ C₂₉

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
1. C₂×#11745231	C₂	semidirect
3. C₂×#11745237	C₂	semidirect
4. #14271065	C₂	semidirect
5. C₄³×C₂₉	C₂	semidirect
30. C₄²×C₂₉	D₄	semidirect
56. C₄×C₂₉	#124	semidirect
63. C₄³	D₂₉	semidirect
79. C₂₉	#2858	semidirect
95. C₄²	D₁₁₆	semidirect
120. C₄	#11745237	semidirect

H ≅ G/N

type

1. C₂×#11745231

C₂

semidirect

3. C₂×#11745237

C₂

semidirect

4. #14271065

C₂

semidirect

5. C₄³×C₂₉

C₂

semidirect

30. C₄²×C₂₉

D₄

semidirect

56. C₄×C₂₉

#124

semidirect

63. C₄³

D₂₉

semidirect

79. C₂₉

#2858

semidirect

95. C₄²

D₁₁₆

semidirect

120. C₄

#11745237

semidirect

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

subgroup	index
C₂×#11745231	2
C₂×#11745236	2
C₂×#11745237	2
#14271065	2
C₄³×C₂₉	2
#2858	29

subgroup

index

C₄³×C₂₉

Subgroups — #16827849

Conjugacy classes of subgroups

Complements

Maximal subgroups