Group #34044367 — subgroups

Subgroups — #34044367

Order 19072.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#23750028	C₂	1	1	yes	maximal, characteristic
2	#23750021	C₂	1	1	yes	maximal, characteristic
3	#23750004	C₂	1	1	yes	maximal, characteristic
4	#23749976	C₂	1	1	yes	maximal, characteristic
5	#23749975	C₂	1	1	yes	maximal, characteristic
6	#23749347	C₂	1	1	yes	maximal, characteristic
7	#500×C₁₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14558726	—	2	2	no	—
9	#18011911	C₂²	1	1	yes	characteristic
10	#18011911	C₂²	1	1	yes	characteristic
11	#18011844	—	2	2	no	—
12	#18011842	—	2	2	no	—
13	#18011836	—	2	2	no	—
14	#18011835	—	2	2	no	—
15	C₄×#11924746	C₂²	1	1	yes	characteristic
16	C₄×#11924746	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
19	#107×C₁₄₉	—	2	2	no	—
20	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₉₈	—	2	1	no	—
22	C₂×Dic₂₉₈	—	2	1	no	—
23	#14558726	—	4	4	no	—
24	C₂×#11924746	D₄	1	2	yes	—
25	C₂×#11924746	D₄	1	2	yes	—
26	C₄⋊Dic₁₄₉	—	2	1	no	—
27	C₄⋊Dic₁₄₉	—	2	1	no	—
28	C₄⋊Dic₁₄₉	—	2	1	no	—
29	C₄⋊Dic₁₄₉	—	2	1	no	—
30	C₂×Q₈×C₁₄₉	—	2	1	no	—
31	C₂×Q₈×C₁₄₉	—	2	1	no	—
32	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₄₉	—	2	1	no	—
34	C₄⋊C₄×C₁₄₉	—	2	1	no	—
35	C₄⋊C₄×C₁₄₉	—	2	1	no	—
36	C₄⋊C₄×C₁₄₉	—	2	1	no	—
37	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
38	Dic₂₉₈	—	4	2	no	—
39	Dic₂₉₈	—	4	2	no	—
40	C₂×Dic₁₄₉	—	4	1	no	—
41	C₂×Dic₁₄₉	—	4	1	no	—
42	#11924746	—	2	2	no	—
43	#11924746	Q₁₆	1	4	yes	complemented
44	Q₈×C₁₄₉	—	4	2	no	—
45	Q₈×C₁₄₉	—	4	2	no	—
46	C₂×C₄×C₁₄₉	—	4	1	no	—
47	C₂×C₄×C₁₄₉	—	4	1	no	—
48	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₄₉	—	2	2	no	—
52	C₂²×C₁₄₉	#127	1	1	yes	characteristic
53	C₄×C₁₄₉	—	8	1	no	—
54	C₄×C₁₄₉	—	8	1	no	—
55	C₄×C₁₄₉	C₂×Q₁₆	1	2	yes	—
56	C₄×C₁₄₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₁₄₉	#137	1	2	yes	—
58	Dic₁₄₉	—	8	1	no	—
59	Dic₁₄₉	—	8	1	no	—
60	C₂×C₁₄₉	#494	1	1	yes	characteristic
61	C₂×C₁₄₉	#496	1	1	yes	characteristic
62	C₂×C₁₄₉	#497	1	1	yes	characteristic
63	C₁₄₉	#1723	1	1	yes	characteristic, complemented, Sylow
64	#1723	—	149	1	no	maximal, Sylow
65	#500	—	149	1	no	—
66	#500	D₁₄₉	1	1	yes	characteristic
67	#498	—	149	1	no	—
68	#494	—	149	1	no	—
69	#487	—	149	1	no	—
70	#487	—	149	1	no	—
71	#322	—	149	1	no	—
72	C₂×Q₁₆	—	298	2	no	—
73	C₄⋊Q₈	—	149	1	no	—
74	C₄⋊Q₈	—	149	1	no	—
75	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic
77	#107	—	2	2	no	—
78	#107	—	298	2	no	—
79	#106	—	298	2	no	—
80	#103	—	298	2	no	—
81	#103	—	298	2	no	—
82	C₄×C₈	—	149	1	no	—
83	C₄×C₈	—	149	1	no	—
84	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	298	1	no	—
88	C₂×Q₈	—	298	1	no	—
89	Q₁₆	—	596	4	no	—
90	C₂×C₈	—	149	2	no	—
91	C₂×C₈	—	149	2	no	—
92	C₂×C₈	C₂²⋊D₁₄₉	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	298	1	no	—
98	C₄⋊C₄	—	298	1	no	—
99	C₄⋊C₄	—	298	1	no	—
100	C₄⋊C₄	—	298	1	no	—
101	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	596	2	no	—
105	Q₈	—	596	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	596	1	no	—
109	C₂×C₄	—	596	1	no	—
110	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	298	2	no	—
115	C₈	—	149	4	no	—
116	C₂²	#18011907	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1192	1	no	—
120	C₄	—	1192	1	no	—
121	C₄	Q₁₆×D₁₄₉	1	2	yes	—
122	C₄	#18011935	1	2	yes	—
123	C₄	#18011942	1	2	yes	—
124	C₂	#23750003	1	1	yes	characteristic
125	C₂	#23750013	1	1	yes	characteristic
126	C₂	#23750017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750028

C₂

yes

maximal, characteristic

#23750021

C₂

yes

maximal, characteristic

#23750004

C₂

yes

maximal, characteristic

#23749976

C₂

yes

maximal, characteristic

#23749975

C₂

yes

maximal, characteristic

#23749347

C₂

yes

maximal, characteristic

#500×C₁₄₉

C₂

yes

maximal, characteristic

C₂×#14558726

—

#18011911

C₂²

yes

characteristic

#18011911

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11924746

C₂²

yes

characteristic

C₄⋊Q₈×C₁₄₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₄₉

C₂²

yes

characteristic

#107×C₁₄₉

—

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×Dic₂₉₈

—

C₂×Dic₂₉₈

—

#14558726

—

C₂×#11924746

D₄

yes

—

C₂×#11924746

D₄

yes

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₄⋊Dic₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×C₈×C₁₄₉

D₄

yes

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

C₂³

yes

characteristic

Dic₂₉₈

—

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

#11924746

Q₁₆

yes

complemented

Q₈×C₁₄₉

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

#127

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂×Q₁₆

yes

—

C₄×C₁₄₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₄₉

#137

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#1723

—

149

maximal, Sylow

#500

—

149

—

#500

D₁₄₉

yes

characteristic

#498

—

149

—

#494

—

149

—

#487

—

149

—

#487

—

149

—

#322

—

149

—

C₂×Q₁₆

—

298

—

C₄⋊Q₈

—

149

—

C₄⋊Q₈

—

149

—

C₄⋊Q₈

C₂×D₁₄₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₄₉

yes

characteristic

#107

—

#107

—

298

—

#106

—

298

—

#103

—

298

—

#103

—

298

—

C₄×C₈

—

149

—

C₄×C₈

—

149

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

298

—

C₂×Q₈

—

298

—

Q₁₆

—

596

—

C₂×C₈

—

149

—

C₂×C₈

—

149

—

C₂×C₈

C₂²⋊D₁₄₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

298

—

100

C₄⋊C₄

—

298

—

101

C₄²

C₂²×D₁₄₉

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

596

—

105

Q₈

—

596

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

596

—

109

C₂×C₄

—

596

—

110

C₂×C₄

D₄×D₁₄₉

yes

characteristic

111

C₂×C₄

D₄×D₁₄₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

113

C₈

—

114

C₈

—

298

—

115

C₈

—

149

—

116

C₂²

#18011907

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1192

—

120

C₄

—

1192

—

121

C₄

Q₁₆×D₁₄₉

yes

—

122

C₄

#18011935

yes

—

123

C₄

#18011942

yes

—

124

C₂

#23750003

yes

characteristic

125

C₂

#23750013

yes

characteristic

126

C₂

#23750017

yes

characteristic

	subgroup	G/H
center	C₂²	#18011907
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	#500×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	#127
hypercenter	#500	D₁₄₉
hypocenter	C₁₄₉	#1723
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#18011907

commutator, cosocle

C₄²×C₁₄₉

C₂³

Frattini

C₄²

C₂²×D₁₄₉

Fitting

#500×C₁₄₉

C₂

C₂²×C₁₄₉

D₁₄₉

C₁₄₉

G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
43. #11924746	Q₁₆	semidirect
63. C₁₄₉	#1723	semidirect

H ≅ G/N

type

43. #11924746

Q₁₆

semidirect

63. C₁₄₉

#1723

semidirect

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

subgroup	index
#23750028	2
#23750021	2
#23750004	2
#23749976	2
#23749975	2
#23749347	2
#500×C₁₄₉	2
#1723	149

subgroup

index

#500×C₁₄₉

149

Subgroups — #34044367

Conjugacy classes of subgroups

Complements

Maximal subgroups