Group #29508281 — subgroups

Subgroups — #29508281

Order 15424.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂²×D₉₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#21440638	C₂	1	2	yes	maximal, complemented
3	#21440631	C₂	1	2	yes	maximal, complemented
4	#21440616	C₂	1	4	yes	maximal, complemented
5	#21440610	C₂	1	4	yes	maximal, complemented
6	#21440601	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₂₄₁	C₂²	1	2	yes	—
9	C₂×D₉₆₄	—	2	4	no	—
10	C₂×D₉₆₄	C₂²	1	2	yes	—
11	C₂×D₉₆₄	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₂₄₁	C₂²	1	8	yes	complemented
13	C₄⋊Dic₂₄₁	C₂²	1	2	yes	complemented
14	C₂²⋊Dic₂₄₁	C₂²	1	2	yes	—
15	#16838747	C₂²	1	8	yes	complemented
16	C₄×Dic₂₄₁	C₂²	1	2	yes	complemented
17	#41×C₂₄₁	—	2	4	no	—
18	C₂×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic, complemented
19	C₂×D₄×C₂₄₁	C₂²	1	2	yes	—
20	C₂×D₄×C₂₄₁	C₂²	1	1	yes	characteristic, complemented
21	C₂²×C₄×C₂₄₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₂₄₁	C₂²	1	2	yes	complemented
23	C₂²×D₂₄₁	—	2	4	no	—
24	C₂²×D₂₄₁	—	4	4	no	—
25	C₂²×D₂₄₁	C₂³	1	4	yes	—
26	D₉₆₄	—	2	4	no	—
27	D₉₆₄	—	2	4	no	—
28	C₂²⋊D₂₄₁	—	4	8	no	—
29	C₂×Dic₂₄₁	C₂³	1	4	yes	—
30	C₂³×C₂₄₁	C₂³	1	2	yes	—
31	C₂³×C₂₄₁	C₂³	1	1	yes	characteristic
32	Q₈×C₂₄₁	—	2	2	no	—
33	D₄×C₂₄₁	—	2	4	no	—
34	D₄×C₂₄₁	—	2	2	no	—
35	C₂×C₄×C₂₄₁	—	2	4	no	—
36	C₂×C₄×C₂₄₁	D₄	1	4	yes	complemented
37	C₂×C₄×C₂₄₁	C₂³	1	2	yes	—
38	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
40	C₂²×C₂₄₁	—	2	2	no	—
41	C₂²×C₂₄₁	—	2	2	no	—
42	C₂²×C₂₄₁	—	2	4	no	—
43	C₂²×C₂₄₁	C₂×D₄	1	2	yes	—
44	C₂²×C₂₄₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₂₄₁	—	2	4	no	—
46	C₂×D₂₄₁	—	2	8	no	—
47	C₂×D₂₄₁	—	4	8	no	—
48	C₄×C₂₄₁	—	2	2	no	—
49	C₄×C₂₄₁	C₂×D₄	1	2	yes	—
50	C₄×C₂₄₁	C₂×D₄	1	2	yes	—
51	Dic₂₄₁	—	2	4	no	—
52	C₂×C₂₄₁	—	2	2	no	—
53	C₂×C₂₄₁	—	4	2	no	—
54	C₂×C₂₄₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₂₄₁	H(5,𝔽₂)	1	2	yes	—
56	D₂₄₁	—	4	4	no	—
57	C₂₄₁	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	241	1	no	maximal, Sylow
59	C₂×#41	D₂₄₁	1	1	yes	characteristic, complemented
60	C₂²×D₄	—	241	1	no	—
61	#127	—	241	2	no	—
62	#124	—	241	2	no	—
63	#121	—	241	4	no	—
64	#120	—	241	4	no	—
65	#117	—	241	1	no	—
66	C₂⁴	—	241	2	no	—
67	#41	—	2	4	no	—
68	C₂×Q₈	C₂×D₂₄₁	1	1	yes	characteristic, complemented
69	C₂×D₄	—	482	4	no	—
70	C₂×D₄	—	241	8	no	—
71	C₂×D₄	—	241	2	no	—
72	C₂×D₄	—	241	2	no	—
73	C₂×D₄	C₂×D₂₄₁	1	2	yes	—
74	C₂×D₄	C₂×D₂₄₁	1	1	yes	characteristic, complemented
75	C₂²×C₄	C₂×D₂₄₁	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₂₄₁	1	2	yes	complemented
77	C₄⋊C₄	—	241	2	no	—
78	C₂²⋊C₄	—	241	2	no	—
79	C₂²⋊C₄	—	241	8	no	—
80	C₄²	—	241	2	no	—
81	C₂³	—	964	4	no	—
82	C₂³	—	482	4	no	—
83	C₂³	—	241	4	no	—
84	C₂³	C₂²×D₂₄₁	1	2	yes	—
85	C₂³	C₂²×D₂₄₁	1	1	yes	characteristic
86	Q₈	—	2	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	2	no	—
89	D₄	—	964	8	no	—
90	D₄	—	482	4	no	—
91	D₄	—	482	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	241	4	no	—
94	C₂×C₄	C₂²⋊D₂₄₁	1	4	yes	complemented
95	C₂×C₄	C₂²×D₂₄₁	1	2	yes	—
96	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	2	4	no	—
101	C₂²	—	964	8	no	—
102	C₂²	—	482	4	no	—
103	C₂²	—	482	8	no	—
104	C₂²	C₂×C₂²⋊D₂₄₁	1	2	yes	—
105	C₂²	C₂³×D₂₄₁	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	482	4	no	—
108	C₄	C₂×C₂²⋊D₂₄₁	1	2	yes	—
109	C₄	C₂×C₂²⋊D₂₄₁	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	2	no	—
112	C₂	—	964	4	no	—
113	C₂	C₂²×C₂²⋊D₂₄₁	1	1	yes	characteristic
114	C₂	#21440688	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₉₆₄

C₂

yes

maximal, characteristic, complemented

#21440638

C₂

yes

maximal, complemented

#21440631

C₂

yes

maximal, complemented

#21440616

C₂

yes

maximal, complemented

#21440610

C₂

yes

maximal, complemented

#21440601

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₂₄₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₄₁

C₂²

yes

—

C₂×D₉₆₄

—

C₂×D₉₆₄

C₂²

yes

—

C₂×D₉₆₄

C₂²

yes

—

C₂×C₂²⋊D₂₄₁

C₂²

yes

complemented

C₄⋊Dic₂₄₁

C₂²

yes

complemented

C₂²⋊Dic₂₄₁

C₂²

yes

—

#16838747

C₂²

yes

complemented

C₄×Dic₂₄₁

C₂²

yes

complemented

#41×C₂₄₁

—

C₂×Q₈×C₂₄₁

C₂²

yes

characteristic, complemented

C₂×D₄×C₂₄₁

C₂²

yes

—

C₂×D₄×C₂₄₁

C₂²

yes

characteristic, complemented

C₂²×C₄×C₂₄₁

C₂²

yes

characteristic

C₂²×C₄×C₂₄₁

C₂²

yes

complemented

C₂²×D₂₄₁

—

C₂²×D₂₄₁

—

C₂²×D₂₄₁

C₂³

yes

—

D₉₆₄

—

D₉₆₄

—

C₂²⋊D₂₄₁

—

C₂×Dic₂₄₁

C₂³

yes

—

C₂³×C₂₄₁

C₂³

yes

—

C₂³×C₂₄₁

C₂³

yes

characteristic

Q₈×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

D₄

yes

complemented

C₂×C₄×C₂₄₁

C₂³

yes

—

C₂×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₄×C₂₄₁

C₂³

yes

characteristic

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₂×D₄

yes

—

C₂²×C₂₄₁

C₂⁴

yes

characteristic

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂×D₄

yes

—

C₄×C₂₄₁

C₂×D₄

yes

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

C₂²×D₄

yes

characteristic

C₂×C₂₄₁

H(5,𝔽₂)

yes

—

D₂₄₁

—

C₂₄₁

#535

yes

characteristic, complemented, Sylow

#535

—

241

maximal, Sylow

C₂×#41

D₂₄₁

yes

characteristic, complemented

C₂²×D₄

—

241

—

#127

—

241

—

#124

—

241

—

#121

—

241

—

#120

—

241

—

#117

—

241

—

C₂⁴

—

241

—

#41

—

C₂×Q₈

C₂×D₂₄₁

yes

characteristic, complemented

C₂×D₄

—

482

—

C₂×D₄

—

241

—

C₂×D₄

—

241

—

C₂×D₄

—

241

—

C₂×D₄

C₂×D₂₄₁

yes

—

C₂×D₄

C₂×D₂₄₁

yes

characteristic, complemented

C₂²×C₄

C₂×D₂₄₁

yes

characteristic

C₂²×C₄

C₂×D₂₄₁

yes

complemented

C₄⋊C₄

—

241

—

C₂²⋊C₄

—

241

—

C₂²⋊C₄

—

241

—

C₄²

—

241

—

C₂³

—

964

—

C₂³

—

482

—

C₂³

—

241

—

C₂³

C₂²×D₂₄₁

yes

—

C₂³

C₂²×D₂₄₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

964

—

D₄

—

482

—

D₄

—

482

—

C₂×C₄

—

C₂×C₄

—

241

—

C₂×C₄

C₂²⋊D₂₄₁

yes

complemented

C₂×C₄

C₂²×D₂₄₁

yes

—

C₂×C₄

C₂²×D₂₄₁

yes

characteristic

C₂×C₄

C₂²×D₂₄₁

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

964

—

102

C₂²

—

482

—

103

C₂²

—

482

—

104

C₂²

C₂×C₂²⋊D₂₄₁

yes

—

105

C₂²

C₂³×D₂₄₁

yes

characteristic

106

C₄

—

107

C₄

—

482

—

108

C₄

C₂×C₂²⋊D₂₄₁

yes

—

109

C₄

C₂×C₂²⋊D₂₄₁

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

964

—

113

C₂

C₂²×C₂²⋊D₂₄₁

yes

characteristic

114

C₂

#21440688

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₄₁
commutator, socle, cosocle	C₂²×C₂₄₁	C₂⁴
Fitting	C₂×#41×C₂₄₁	C₂
hypercenter	C₂×#41	D₂₄₁
hypocenter	C₂₄₁	#535
LCS	G ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₄₁

commutator, socle, cosocle

C₂²×C₂₄₁

C₂⁴

Fitting

C₂×#41×C₂₄₁

C₂

hypercenter

C₂×#41

D₂₄₁

C₂₄₁

G ▷ C₂²×C₂₄₁ ▷ C₂₄₁

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×D₉₆₄	C₂	semidirect
2. #21440638	C₂	semidirect
3. #21440631	C₂	semidirect
4. #21440616	C₂	semidirect
5. #21440610	C₂	semidirect
6. #21440601	C₂	semidirect
7. C₂×#41×C₂₄₁	C₂	semidirect
12. C₂×C₂²⋊D₂₄₁	C₂²	semidirect
13. C₄⋊Dic₂₄₁	C₂²	semidirect
15. #16838747	C₂²	semidirect
16. C₄×Dic₂₄₁	C₂²	semidirect
18. C₂×Q₈×C₂₄₁	C₂²	semidirect
20. C₂×D₄×C₂₄₁	C₂²	semidirect
22. C₂²×C₄×C₂₄₁	C₂²	semidirect
36. C₂×C₄×C₂₄₁	D₄	semidirect
57. C₂₄₁	#535	semidirect
59. C₂×#41	D₂₄₁	semidirect
68. C₂×Q₈	C₂×D₂₄₁	semidirect
74. C₂×D₄	C₂×D₂₄₁	semidirect
76. C₂²×C₄	C₂×D₂₄₁	semidirect
94. C₂×C₄	C₂²⋊D₂₄₁	semidirect

H ≅ G/N

type

1. C₂²×D₉₆₄

C₂

semidirect

2. #21440638

C₂

semidirect

3. #21440631

C₂

semidirect

4. #21440616

C₂

semidirect

5. #21440610

C₂

semidirect

6. #21440601

C₂

semidirect

7. C₂×#41×C₂₄₁

C₂

semidirect

12. C₂×C₂²⋊D₂₄₁

C₂²

semidirect

13. C₄⋊Dic₂₄₁

C₂²

semidirect

15. #16838747

C₂²

semidirect

16. C₄×Dic₂₄₁

C₂²

semidirect

18. C₂×Q₈×C₂₄₁

C₂²

semidirect

20. C₂×D₄×C₂₄₁

C₂²

semidirect

22. C₂²×C₄×C₂₄₁

C₂²

semidirect

36. C₂×C₄×C₂₄₁

D₄

semidirect

57. C₂₄₁

#535

semidirect

59. C₂×#41

D₂₄₁

semidirect

68. C₂×Q₈

C₂×D₂₄₁

semidirect

74. C₂×D₄

C₂×D₂₄₁

semidirect

76. C₂²×C₄

C₂×D₂₄₁

semidirect

94. C₂×C₄

C₂²⋊D₂₄₁

semidirect

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

subgroup	index
C₂²×D₉₆₄	2
#21440638	2
#21440631	2
#21440616	2
#21440610	2
#21440601	2
C₂×#41×C₂₄₁	2
#535	241

subgroup

index

C₂²×D₉₆₄

C₂×#41×C₂₄₁

241

Subgroups — #29508281

Conjugacy classes of subgroups

Complements

Maximal subgroups