Group #34055030 — subgroups

Subgroups — C₂×#23749993 #34055030

Order 19072.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18011903	C₂	1	1	yes	maximal, characteristic
2	#23749993	C₂	1	8	yes	maximal, direct product factor
3	C₂×#18011835	C₂	1	4	yes	maximal
4	C₂×#18011822	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924746	C₂²	1	2	yes	—
7	#18011903	—	2	2	no	—
8	#18011903	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
10	C₂×#14558715	—	2	2	no	—
11	#18011835	C₂²	1	16	yes	—
12	#18011822	C₂²	1	4	yes	—
13	C₂²×Q₈×C₁₄₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
15	C₄⋊Q₈×C₁₄₉	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₄₉	—	2	2	no	—
17	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₄₉	—	2	2	no	—
19	C₂×#11924746	—	2	4	no	—
20	C₂×#11924746	C₂³	1	4	yes	—
21	C₄⋊Dic₁₄₉	—	2	2	no	—
22	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
23	#14558715	—	2	8	no	—
24	C₂×Q₈×C₁₄₉	—	2	2	no	—
25	C₂×Q₈×C₁₄₉	—	2	8	no	—
26	C₂×Q₈×C₁₄₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₄₉	—	2	2	no	—
28	C₂²×C₄×C₁₄₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₄₉	—	2	8	no	—
31	C₄²×C₁₄₉	D₄	1	2	yes	—
32	C₄²×C₁₄₉	C₂³	1	2	yes	—
33	C₂×Dic₁₄₉	—	2	4	no	—
34	C₂×Dic₁₄₉	—	4	4	no	—
35	#11924746	—	2	4	no	—
36	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₄₉	—	2	8	no	—
38	Q₈×C₁₄₉	—	4	4	no	—
39	C₂×C₄×C₁₄₉	—	2	4	no	—
40	C₂×C₄×C₁₄₉	—	2	4	no	—
41	C₂×C₄×C₁₄₉	—	4	4	no	—
42	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₄₉	#41	1	4	yes	—
45	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
46	C₂²×C₁₄₉	#124	1	2	yes	—
47	C₂²×C₁₄₉	#137	1	4	yes	—
48	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₄₉	—	2	4	no	—
50	C₄×C₁₄₉	—	4	4	no	—
51	C₄×C₁₄₉	#124	1	2	yes	—
52	C₄×C₁₄₉	C₂×#41	1	2	yes	—
53	Dic₁₄₉	—	4	4	no	—
54	C₂×C₁₄₉	#491	1	4	yes	complemented
55	C₂×C₁₄₉	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₄₉	C₂×#137	1	2	yes	—
57	C₁₄₉	C₂×#491	1	1	yes	characteristic, complemented, Sylow
58	C₂×#491	—	149	1	no	maximal, Sylow
59	C₂×C₄⋊Q₈	D₁₄₉	1	1	yes	characteristic
60	C₂×#125	—	149	1	no	—
61	#491	—	149	8	no	—
62	C₂×#103	—	149	4	no	—
63	C₂×#97	—	149	1	no	—
64	C₂²×Q₈	C₂×D₁₄₉	1	2	yes	—
65	C₂²×C₈	—	149	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₄₉	1	4	yes	—
68	#125	—	298	2	no	—
69	#125	—	149	4	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	298	2	no	—
72	C₂×C₄⋊C₄	—	149	2	no	—
73	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
74	#103	—	149	16	no	—
75	#97	—	149	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂²×D₁₄₉	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	298	2	no	—
81	C₂²×C₄	C₂²⋊D₁₄₉	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
83	C₂×C₈	—	298	4	no	—
84	C₂×C₈	—	149	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	298	8	no	—
87	C₄⋊C₄	—	298	2	no	—
88	C₄⋊C₄	—	149	4	no	—
89	C₄²	C₂²⋊D₁₄₉	1	2	yes	—
90	C₄²	C₂²×D₁₄₉	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	298	4	no	—
98	C₂×C₄	—	596	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
101	C₂×C₄	#14558735	1	4	yes	—
102	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
103	C₈	—	298	4	no	—
104	C₂²	#18011898	1	2	yes	—
105	C₂²	#18011942	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₄₉	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	596	4	no	—
110	C₄	#18011898	1	2	yes	—
111	C₄	C₂×#14558735	1	2	yes	—
112	C₂	#23749993	1	4	yes	direct product factor
113	C₂	C₂×#18011898	1	1	yes	characteristic
114	C₂	C₂×#18011942	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18011903

C₂

yes

maximal, characteristic

#23749993

C₂

yes

maximal, direct product factor

C₂×#18011835

C₂

yes

maximal

C₂×#18011822

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₄₉

C₂

yes

maximal, characteristic

C₂²×#11924746

C₂²

yes

—

#18011903

—

#18011903

C₂²

yes

—

C₂×C₄⋊Dic₁₄₉

C₂²

yes

—

C₂×#14558715

—

#18011835

C₂²

yes

—

#18011822

C₂²

yes

—

C₂²×Q₈×C₁₄₉

C₂²

yes

—

C₄⋊Q₈×C₁₄₉

—

C₄⋊Q₈×C₁₄₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₄₉

—

C₂×C₄²×C₁₄₉

C₂²

yes

characteristic

C₂²×Dic₁₄₉

—

C₂×#11924746

—

C₂×#11924746

C₂³

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

C₂³

yes

—

#14558715

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

C₂³

yes

—

C₂²×C₄×C₁₄₉

—

C₂²×C₄×C₁₄₉

D₄

yes

—

C₂²×C₄×C₁₄₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

D₄

yes

—

C₄²×C₁₄₉

C₂³

yes

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

C₂×D₄

yes

characteristic

Q₈×C₁₄₉

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

#41

yes

—

C₂×C₄×C₁₄₉

C₂⁴

yes

characteristic

C₂²×C₁₄₉

#124

yes

—

C₂²×C₁₄₉

#137

yes

—

C₂²×C₁₄₉

C₂²×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#124

yes

—

C₄×C₁₄₉

C₂×#41

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

—

C₁₄₉

C₂×#491

yes

characteristic, complemented, Sylow

C₂×#491

—

149

maximal, Sylow

C₂×C₄⋊Q₈

D₁₄₉

yes

characteristic

C₂×#125

—

149

—

#491

—

149

—

C₂×#103

—

149

—

C₂×#97

—

149

—

C₂²×Q₈

C₂×D₁₄₉

yes

—

C₂²×C₈

—

149

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₄₉

yes

—

#125

—

298

—

#125

—

149

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

298

—

C₂×C₄⋊C₄

—

149

—

C₂×C₄²

C₂×D₁₄₉

yes

characteristic

#103

—

149

—

#97

—

149

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₄₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

298

—

C₂²×C₄

C₂²⋊D₁₄₉

yes

—

C₂²×C₄

C₂²×D₁₄₉

yes

characteristic

C₂×C₈

—

298

—

C₂×C₈

—

149

—

C₄⋊C₄

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

149

—

C₄²

C₂²⋊D₁₄₉

yes

—

C₄²

C₂²×D₁₄₉

yes

—

C₂³

C₂×C₂²⋊D₁₄₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

596

—

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

101

C₂×C₄

#14558735

yes

—

102

C₂×C₄

C₂³×D₁₄₉

yes

characteristic

103

C₈

—

298

—

104

C₂²

#18011898

yes

—

105

C₂²

#18011942

yes

—

106

C₂²

C₂²×C₂²⋊D₁₄₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

596

—

110

C₄

#18011898

yes

—

111

C₄

C₂×#14558735

yes

—

112

C₂

#23749993

yes

direct product factor

113

C₂

C₂×#18011898

yes

characteristic

114

C₂

C₂×#18011942

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂×C₄×C₁₄₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	C₂×C₄⋊Q₈×C₁₄₉	C₂
socle, prime radical	C₂³×C₁₄₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₄₉
hypocenter	C₁₄₉	C₂×#491
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂×C₄×C₁₄₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₄₉

Fitting

C₂×C₄⋊Q₈×C₁₄₉

C₂

socle, prime radical

C₂³×C₁₄₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₄₉

hypocenter

C₁₄₉

C₂×#491

LCS

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #23749993	C₂	direct (N × H)
54. C₂×C₁₄₉	#491	semidirect
57. C₁₄₉	C₂×#491	semidirect
112. C₂	#23749993	direct (N × H)

H ≅ G/N

type

2. #23749993

C₂

direct (N × H)

54. C₂×C₁₄₉

#491

semidirect

57. C₁₄₉

C₂×#491

semidirect

112. C₂

#23749993

direct (N × H)

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

subgroup	index
C₂×#18011903	2
#23749993	2
C₂×#18011835	2
C₂×#18011822	2
C₂×C₄⋊Q₈×C₁₄₉	2
C₂×#491	149

subgroup

index

C₂×C₄⋊Q₈×C₁₄₉

149

Subgroups — C₂×#23749993 #34055030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23749993 #34055030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23749993 #34055030