Group #34043834 — subgroups

Subgroups — #34043834

Order 19072.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#23750341	C₂	1	1	yes	maximal, characteristic
2	#23749912	C₂	1	2	yes	maximal, complemented
3	#23749374	C₂	1	1	yes	maximal, characteristic, complemented
4	#23749354	C₂	1	2	yes	maximal, complemented
5	#494×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#18011900	—	2	2	no	—
7	#18011886	—	2	2	no	—
8	C₄×D₅₉₆	C₂²	1	2	yes	—
9	#18011869	—	2	2	no	—
10	#18011842	—	2	2	no	—
11	C₄⋊#11924746	C₂²	1	2	yes	—
12	#18011831	—	2	2	no	—
13	C₂×Q₁₆×C₁₄₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
17	C₂×D₅₉₆	—	2	2	no	—
18	C₂×D₅₉₆	D₄	1	2	yes	—
19	C₂×C₄×D₁₄₉	—	2	2	no	—
20	C₂×#11924746	—	2	2	no	—
21	C₄⋊Dic₁₄₉	D₄	1	2	yes	—
22	#14558713	—	2	4	no	—
23	#14558713	—	2	2	no	—
24	C₄×Dic₁₄₉	—	2	2	no	—
25	C₂×Q₈×C₁₄₉	—	2	1	no	—
26	C₂×Q₈×C₁₄₉	D₄	1	2	yes	—
27	Q₁₆×C₁₄₉	—	4	4	no	—
28	C₂×C₈×C₁₄₉	—	2	1	no	—
29	C₄⋊C₄×C₁₄₉	—	2	2	no	—
30	C₄⋊C₄×C₁₄₉	—	2	2	no	—
31	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₄₉	—	2	2	no	—
33	D₅₉₆	—	2	2	no	—
34	D₅₉₆	—	2	2	no	—
35	D₅₉₆	—	8	2	no	—
36	C₂×Dic₁₄₉	—	2	2	no	—
37	C₄×D₁₄₉	—	4	2	no	—
38	C₄×D₁₄₉	—	4	2	no	—
39	#11924746	—	4	2	no	—
40	Q₈×C₁₄₉	—	4	2	no	—
41	Q₈×C₁₄₉	—	4	2	no	—
42	C₂×C₄×C₁₄₉	—	2	2	no	—
43	C₂×C₄×C₁₄₉	—	4	1	no	—
44	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₄₉	—	2	2	no	—
47	C₂²×C₁₄₉	#120	1	1	yes	characteristic
48	C₂×D₁₄₉	—	4	2	no	—
49	C₂×D₁₄₉	—	4	2	no	—
50	C₂×D₁₄₉	—	4	2	no	—
51	C₄×C₁₄₉	—	2	1	no	—
52	C₄×C₁₄₉	—	4	2	no	—
53	C₄×C₁₄₉	—	8	1	no	—
54	C₄×C₁₄₉	#135	1	2	yes	—
55	C₄×C₁₄₉	#137	1	2	yes	—
56	Dic₁₄₉	—	4	2	no	—
57	C₂×C₁₄₉	#452	1	2	yes	—
58	C₂×C₁₄₉	#453	1	1	yes	characteristic
59	D₁₄₉	—	8	2	no	—
60	C₁₄₉	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	149	1	no	maximal, Sylow
62	#555	—	149	1	no	—
63	#494	D₁₄₉	1	1	yes	characteristic, complemented
64	#476	—	149	2	no	—
65	#333	—	149	1	no	—
66	D₄⋊C₈	—	149	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₄₉	1	1	yes	characteristic
70	#124	—	298	2	no	—
71	#121	—	298	2	no	—
72	C₄×D₄	—	149	2	no	—
73	#117	—	298	2	no	—
74	#106	—	298	2	no	—
75	C₄⋊C₈	—	149	2	no	—
76	#102	—	298	2	no	—
77	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₄₉	1	2	yes	—
80	C₂×D₄	—	298	2	no	—
81	C₂×D₄	—	149	2	no	—
82	C₂²×C₄	—	298	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	298	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	149	2	no	—
89	C₂²⋊C₄	—	298	4	no	—
90	C₂²⋊C₄	—	298	2	no	—
91	C₄²	—	298	2	no	—
92	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
93	C₂³	—	298	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	298	2	no	—
97	D₄	—	298	2	no	—
98	D₄	—	1192	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	298	2	no	—
102	C₂×C₄	—	596	2	no	—
103	C₂×C₄	—	596	2	no	—
104	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	596	2	no	—
108	C₂²	—	596	2	no	—
109	C₂²	—	596	2	no	—
110	C₂²	—	596	2	no	—
111	C₂²	#18011879	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	596	2	no	—
116	C₄	#18011934	1	2	yes	—
117	C₄	#18011943	1	2	yes	—
118	C₂	—	1192	2	no	—
119	C₂	#23749786	1	2	yes	—
120	C₂	#23749792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23750341

C₂

yes

maximal, characteristic

#23749912

C₂

yes

maximal, complemented

#23749374

C₂

yes

maximal, characteristic, complemented

#23749354

C₂

yes

maximal, complemented

#494×C₁₄₉

C₂

yes

maximal, characteristic, complemented

#18011900

—

#18011886

—

C₄×D₅₉₆

C₂²

yes

—

#18011869

—

#18011842

—

C₄⋊#11924746

C₂²

yes

—

#18011831

—

C₂×Q₁₆×C₁₄₉

—

C₄⋊Q₈×C₁₄₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₄₉

C₂²

yes

characteristic

C₄×C₈×C₁₄₉

C₂²

yes

characteristic

C₂×D₅₉₆

—

C₂×D₅₉₆

D₄

yes

—

C₂×C₄×D₁₄₉

—

C₂×#11924746

—

C₄⋊Dic₁₄₉

D₄

yes

—

#14558713

—

#14558713

—

C₄×Dic₁₄₉

—

C₂×Q₈×C₁₄₉

—

C₂×Q₈×C₁₄₉

D₄

yes

—

Q₁₆×C₁₄₉

—

C₂×C₈×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄⋊C₄×C₁₄₉

—

C₄²×C₁₄₉

C₂³

yes

characteristic

C₂²×D₁₄₉

—

D₅₉₆

—

D₅₉₆

—

D₅₉₆

—

C₂×Dic₁₄₉

—

C₄×D₁₄₉

—

C₄×D₁₄₉

—

#11924746

—

Q₈×C₁₄₉

—

Q₈×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

—

C₂×C₄×C₁₄₉

C₂×D₄

yes

characteristic

C₈×C₁₄₉

—

C₂²×C₁₄₉

#120

yes

characteristic

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₂×D₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

—

C₄×C₁₄₉

#135

yes

—

C₄×C₁₄₉

#137

yes

—

Dic₁₄₉

—

C₂×C₁₄₉

#452

yes

—

C₂×C₁₄₉

#453

yes

characteristic

D₁₄₉

—

C₁₄₉

#1634

yes

characteristic, complemented, Sylow

#1634

—

149

maximal, Sylow

#555

—

149

—

#494

D₁₄₉

yes

characteristic, complemented

#476

—

149

—

#333

—

149

—

D₄⋊C₈

—

149

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₄₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₄₉

yes

characteristic

#124

—

298

—

#121

—

298

—

C₄×D₄

—

149

—

#117

—

298

—

#106

—

298

—

C₄⋊C₈

—

149

—

#102

—

298

—

C₄×C₈

C₂×D₁₄₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₄₉

yes

—

C₂×D₄

—

298

—

C₂×D₄

—

149

—

C₂²×C₄

—

298

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

298

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

149

—

C₂²⋊C₄

—

298

—

C₂²⋊C₄

—

298

—

C₄²

—

298

—

C₄²

C₂²×D₁₄₉

yes

characteristic

C₂³

—

298

—

Q₈

—

Q₈

—

D₄

—

298

—

D₄

—

298

—

D₄

—

1192

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

298

—

102

C₂×C₄

—

596

—

103

C₂×C₄

—

596

—

104

C₂×C₄

D₄×D₁₄₉

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₄₉

yes

characteristic

106

C₈

—

107

C₈

—

596

—

108

C₂²

—

596

—

109

C₂²

—

596

—

110

C₂²

—

596

—

111

C₂²

#18011879

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

596

—

116

C₄

#18011934

yes

—

117

C₄

#18011943

yes

—

118

C₂

—

1192

—

119

C₂

#23749786

yes

—

120

C₂

#23749792

yes

characteristic

	subgroup	G/H
center	C₂²	#18011879
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	#494×C₁₄₉	C₂
socle	C₂²×C₁₄₉	#120
prime radical	#23750341	C₂
hypercenter	#494	D₁₄₉
hypocenter	C₁₄₉	#1634
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#18011879

commutator, cosocle

C₄²×C₁₄₉

C₂³

Frattini

C₄²

C₂²×D₁₄₉

Fitting

#494×C₁₄₉

C₂

C₂²×C₁₄₉

C₂

D₁₄₉

C₁₄₉

G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #23749912	C₂	semidirect
3. #23749374	C₂	semidirect
4. #23749354	C₂	semidirect
5. #494×C₁₄₉	C₂	semidirect
60. C₁₄₉	#1634	semidirect
63. #494	D₁₄₉	semidirect

H ≅ G/N

type

2. #23749912

C₂

semidirect

3. #23749374

C₂

semidirect

4. #23749354

C₂

semidirect

5. #494×C₁₄₉

C₂

semidirect

60. C₁₄₉

#1634

semidirect

63. #494

D₁₄₉

semidirect

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

subgroup	index
#23750341	2
#23749912	2
#23749374	2
#23749354	2
#494×C₁₄₉	2
#1634	149

subgroup

index

#494×C₁₄₉

149

Subgroups — #34043834

Conjugacy classes of subgroups

Complements

Maximal subgroups