Group #34034474 — subgroups

Subgroups — #34034474

Order 18688.

166 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22627030	C₂	1	2	yes	maximal
2	C₄⋊#14556681	C₂	1	1	yes	maximal, characteristic
3	#22627014	C₂	1	2	yes	maximal
4	C₄×#14556681	C₂	1	1	yes	maximal, characteristic
5	#22626345	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556681	C₂²	1	1	yes	characteristic
7	C₂×#14556681	C₂²	1	2	yes	—
8	C₂×#14556681	C₂²	1	1	yes	characteristic
9	#17989508	C₄	1	2	yes	—
10	C₄⋊#11924031	C₄	1	1	yes	characteristic
11	#17989473	C₂²	1	2	yes	—
12	C₄×#11924031	C₂²	1	1	yes	characteristic
13	#17988711	C₄	1	1	yes	characteristic
14	#14556681	—	2	1	no	—
15	#14556681	—	2	2	no	—
16	#14556681	Q₈	1	2	yes	complemented
17	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
18	C₂×#11924031	C₂×C₄	1	2	yes	—
19	C₂×#11924031	C₂³	1	1	yes	characteristic
20	C₄×Dic₁₄₆	C₈	1	1	yes	characteristic
21	#14556520	C₈	1	2	yes	—
22	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₄⋊C₈×C₇₃	C₈	1	1	yes	characteristic
25	#11924031	—	2	1	no	—
26	#11924031	—	2	2	no	—
27	#11924031	C₂×Q₈	1	1	yes	characteristic
28	#11924031	#41	1	1	yes	characteristic
29	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
30	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
31	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
32	C₄⋊Dic₇₃	C₁₆	1	1	yes	characteristic
33	#11924007	C₁₆	1	2	yes	—
34	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
36	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
42	Dic₁₄₆	C₃₂	1	4	yes	complemented
43	C₈×C₇₃	—	2	2	no	—
44	#10596831	—	2	1	no	—
45	#10596831	C₄×Q₈	1	1	yes	characteristic
46	#10596831	#131	1	1	yes	characteristic
47	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₇₃	—	2	1	no	—
49	C₄×C₇₃	C₂×C₃₂	1	2	yes	—
50	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
51	C₄×C₇₃	#504	1	1	yes	characteristic
52	Dic₇₃	—	2	2	no	—
53	Dic₇₃	C₂×C₃₂	1	4	yes	—
54	Q₈×C₃₂	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₇₃	C₂²×C₃₂	1	1	yes	characteristic
57	C₂×C₇₃	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	—	73	1	no	—
59	C₄⋊C₃₂	—	73	1	no	—
60	C₄⋊C₃₂	—	73	2	no	—
61	C₄×C₃₂	—	73	1	no	—
62	C₄×C₃₂	—	73	2	no	—
63	C₇₃	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	—	73	1	no	—
65	C₂×C₃₂	—	73	1	no	—
66	C₂×C₃₂	—	73	1	no	—
67	C₂×C₃₂	—	73	2	no	—
68	C₄⋊C₁₆	—	73	1	no	—
69	C₄⋊C₁₆	—	73	2	no	—
70	C₄×C₁₆	—	73	1	no	—
71	C₄×C₁₆	—	73	2	no	—
72	C₄×Q₈	—	73	1	no	—
73	C₂×C₁₆	—	73	1	no	—
74	C₂×C₁₆	—	73	1	no	—
75	C₂×C₁₆	—	73	2	no	—
76	C₄⋊C₈	—	73	2	no	—
77	C₄⋊C₈	#10596842	1	1	yes	characteristic
78	C₄×C₈	—	73	2	no	—
79	C₄×C₈	—	73	1	no	—
80	C₃₂	—	73	2	no	—
81	C₃₂	—	146	1	no	—
82	C₃₂	—	146	2	no	—
83	C₂×Q₈	—	73	1	no	—
84	C₂×C₈	—	73	1	no	—
85	C₂×C₈	—	73	1	no	—
86	C₂×C₈	C₂×#10596842	1	2	yes	—
87	C₄⋊C₄	—	73	2	no	—
88	C₄⋊C₄	—	73	1	no	—
89	C₄²	—	73	2	no	—
90	C₄²	C₂×#10596842	1	1	yes	characteristic
91	C₁₆	—	73	1	no	—
92	C₁₆	—	73	1	no	—
93	C₁₆	—	146	1	no	—
94	C₁₆	—	146	2	no	—
95	Q₈	—	73	4	no	—
96	C₂×C₄	—	73	2	no	—
97	C₂×C₄	—	73	2	no	—
98	C₂×C₄	#14556686	1	1	yes	characteristic
99	C₂×C₄	#14556686	1	1	yes	characteristic
100	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	73	1	no	—
103	C₈	—	73	1	no	—
104	C₈	—	146	1	no	—
105	C₂²	C₂×#14556686	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	73	4	no	—
108	C₄	—	146	2	no	—
109	C₄	#17989694	1	2	yes	—
110	C₄	Q₈×#10596842	1	1	yes	characteristic
111	C₄	#17989708	1	1	yes	characteristic
112	C₂	#22627155	1	1	yes	characteristic
113	C₂	C₂×#17989694	1	1	yes	characteristic
114	C₂	#22627165	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627030

C₂

yes

maximal

C₄⋊#14556681

C₂

yes

maximal, characteristic

#22627014

C₂

yes

maximal

C₄×#14556681

C₂

yes

maximal, characteristic

#22626345

C₂

yes

maximal, characteristic

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

characteristic

#17989508

C₄

yes

—

C₄⋊#11924031

C₄

yes

characteristic

#17989473

C₂²

yes

—

C₄×#11924031

C₂²

yes

characteristic

#17988711

C₄

yes

—

—

Q₈

yes

complemented

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂³

yes

characteristic

C₄×Dic₁₄₆

C₈

yes

characteristic

#14556520

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

—

C₄⋊C₈×C₇₃

C₈

yes

—

—

C₂×Q₈

yes

yes

C₂×Dic₁₄₆

C₁₆

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₄⋊Dic₇₃

C₁₆

yes

characteristic

#11924007

C₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

Dic₁₄₆

C₃₂

yes

complemented

C₈×C₇₃

—

#10596831

—

#10596831

C₄×Q₈

yes

yes

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×C₃₂

yes

—

C₄×C₇₃

C₈×Q₈

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

C₂×C₃₂

yes

—

Q₈×C₃₂

—

maximal, Sylow

C₂×C₇₃

Q₈×C₁₆

yes

characteristic

C₂×C₇₃

C₂²×C₃₂

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

C₄×C₃₂

—

C₄×C₃₂

—

C₇₃

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

#10596842

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#10596842

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×#10596842

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

146

—

C₁₆

—

146

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²×#10596842

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

146

—

105

C₂²

C₂×#14556686

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

146

—

109

C₄

#17989694

yes

—

110

C₄

Q₈×#10596842

yes

characteristic

111

C₄

#17989708

yes

characteristic

112

C₂

#22627155

yes

characteristic

113

C₂

C₂×#17989694

yes

characteristic

114

C₂

#22627165

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator	C₂×C₇₃	C₂²×C₃₂
Fitting	C₄⋊C₈×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	C₂²×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	Q₈×C₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator

C₂×C₇₃

C₂²×C₃₂

Fitting

C₄⋊C₈×C₇₃

C₈

socle, prime radical

C₂²×C₇₃

C₂²×C₁₆

C₂³

C₄⋊C₈

C₇₃

Q₈×C₃₂

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
16. #14556681	Q₈	semidirect
42. Dic₁₄₆	C₃₂	semidirect
63. C₇₃	Q₈×C₃₂	semidirect

H ≅ G/N

type

16. #14556681

Q₈

semidirect

42. Dic₁₄₆

C₃₂

semidirect

63. C₇₃

Q₈×C₃₂

semidirect

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

subgroup	index
#22627030	2
C₄⋊#14556681	2
#22627014	2
C₄×#14556681	2
#22626345	2
Q₈×C₃₂	73

subgroup

index

Q₈×C₃₂

Subgroups — #34034474

Conjugacy classes of subgroups

Complements

Maximal subgroups