Group #34034475 — subgroups

Subgroups — #34034475

Order 18688.

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

normal subgroups only

This group has 158 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	C₄⋊#14556681	C₂	1	2	yes	maximal
2	C₄⋊#14556681	C₂	1	1	yes	maximal, characteristic
3	#22627016	C₂	1	2	yes	maximal
4	C₄×#14556681	C₂	1	1	yes	maximal, characteristic
5	Q₈×#11924031	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	C₄⋊#11924031	C₄	1	1	yes	characteristic
10	C₄⋊#11924031	C₄	1	2	yes	—
11	C₄×#11924031	C₂²	1	1	yes	characteristic
12	C₄×#11924031	C₂²	1	2	yes	—
13	Q₈×#10596831	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⋊#10596831	C₈	1	2	yes	—
21	C₄⋊#10596831	C₈	1	1	yes	characteristic
22	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₂×C₄	1	2	yes	—
24	C₄×Q₈×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×#10596831	C₂×C₈	1	1	yes	characteristic
30	C₂×#10596831	C₂×C₈	1	2	yes	—
31	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
32	C₂×Q₈×C₇₃	C₁₆	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	C₁₆	1	2	yes	—
34	C₄⋊C₄×C₇₃	C₁₆	1	1	yes	characteristic
35	C₄²×C₇₃	C₂×C₈	1	2	yes	—
36	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
37	Q₈×C₇₃	—	2	2	no	—
38	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
39	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
41	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
43	#10596831	—	2	1	no	—
44	#10596831	—	2	2	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	2	no	—
49	C₄×C₇₃	—	2	2	no	—
50	C₄×C₇₃	—	2	1	no	—
51	C₄×C₇₃	M₆₄	1	2	yes	—
52	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
53	C₄×C₇₃	#504	1	1	yes	characteristic
54	#13660	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₇₃	C₂×M₆₄	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	#1397	—	73	2	no	—
62	C₄×C₃₂	—	73	1	no	—
63	C₇₃	#13660	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	#10596842	1	1	yes	characteristic
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	—	73	1	no	—
78	C₄×C₈	—	73	1	no	—
79	C₄×C₈	—	73	2	no	—
80	C₃₂	—	73	2	no	—
81	C₃₂	—	146	1	no	—
82	C₃₂	—	146	2	no	—
83	C₂×Q₈	#11924040	1	1	yes	characteristic
84	C₂×C₈	—	73	1	no	—
85	C₂×C₈	—	73	1	no	—
86	C₂×C₈	—	73	2	no	—
87	C₄⋊C₄	#11924040	1	2	yes	—
88	C₄⋊C₄	#11924040	1	1	yes	characteristic
89	C₄²	C₂×#10596842	1	2	yes	—
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	—	2	2	no	—
96	C₂×C₄	C₂×#11924040	1	2	yes	—
97	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
98	C₂×C₄	C₂×#11924040	1	2	yes	—
99	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
100	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
101	C₈	—	73	1	no	—
102	C₈	—	73	1	no	—
103	C₈	—	146	1	no	—
104	C₈	—	146	2	no	—
105	C₂²	C₂²×#11924040	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	#17989695	1	2	yes	—
110	C₄	Q₈×#10596842	1	1	yes	characteristic
111	C₄	#17989709	1	1	yes	characteristic
112	C₂	Q₈×#11924040	1	1	yes	characteristic
113	C₂	C₂×#17989695	1	1	yes	characteristic
114	C₂	#22627164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#14556681

C₂

yes

maximal

C₄⋊#14556681

C₂

yes

maximal, characteristic

#22627016

C₂

yes

maximal

C₄×#14556681

C₂

yes

maximal, characteristic

Q₈×#11924031

C₂

yes

maximal, characteristic

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

characteristic

C₄⋊#11924031

C₄

yes

characteristic

C₄⋊#11924031

C₄

yes

—

C₄×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

—

Q₈×#10596831

C₄

yes

—

—

Q₈

yes

complemented

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂³

yes

characteristic

C₄⋊#10596831

C₈

yes

—

C₄⋊#10596831

C₈

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₄×Q₈×C₇₃

C₈

yes

—

—

C₂×Q₈

yes

yes

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×Q₈×C₇₃

C₁₆

yes

characteristic

C₄⋊C₄×C₇₃

C₁₆

yes

—

C₄⋊C₄×C₇₃

C₁₆

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

—

C₄×Q₈

yes

yes

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

M₆₄

yes

—

C₄×C₇₃

C₈×Q₈

yes

C₄×C₇₃

yes

—

C₂×C₇₃

Q₈×C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₇₃

#13660

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×Q₈

#10596842

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂×Q₈

#11924040

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

#11924040

yes

—

C₄⋊C₄

#11924040

yes

characteristic

C₄²

C₂×#10596842

yes

—

C₄²

C₂×#10596842

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

146

—

C₁₆

—

146

—

Q₈

—

C₂×C₄

C₂×#11924040

yes

—

C₂×C₄

C₂×#11924040

yes

characteristic

C₂×C₄

C₂×#11924040

yes

—

C₂×C₄

C₂×#11924040

yes

characteristic

100

C₂×C₄

C₂²×#10596842

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

146

—

104

C₈

—

146

—

105

C₂²

C₂²×#11924040

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#17989695

yes

—

110

C₄

Q₈×#10596842

yes

characteristic

111

C₄

#17989709

yes

characteristic

112

C₂

Q₈×#11924040

yes

characteristic

113

C₂

C₂×#17989695

yes

characteristic

114

C₂

#22627164

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle, prime radical	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄×Q₈×C₇₃	C₈
cosocle	C₂×#11924031	C₂³
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	#13660
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle, prime radical

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₄×Q₈×C₇₃

C₈

C₂³

C₄×Q₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×Q₈

N	H ≅ G/N	type
16. #14556681	Q₈	semidirect
63. C₇₃	#13660	semidirect

H ≅ G/N

type

16. #14556681

Q₈

semidirect

63. C₇₃

#13660

semidirect

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

subgroup	index
C₄⋊#14556681	2
#22627016	2
C₄×#14556681	2
Q₈×#11924031	2
#13660	73

subgroup

index

Subgroups — #34034475

Conjugacy classes of subgroups

Complements

Maximal subgroups