Group #34009638 — subgroups

Subgroups — C₈×#14556664 #34009638

Order 18688.

398 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 236 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×#14556664	C₂	1	2	yes	maximal
2	C₈×#11924031	C₂	1	4	yes	maximal, complemented
3	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556664	C₄	1	4	yes	—
5	#17989473	C₄	1	4	yes	—
6	C₄×#11924031	C₂²	1	4	yes	—
7	C₄×C₈×D₇₃	C₄	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₈×#10596831	C₄	1	1	yes	characteristic
10	C₈×#10596831	C₂²	1	2	yes	—
11	C₈²×C₇₃	C₄	1	1	yes	characteristic
12	#14556664	C₈	1	16	yes	direct product factor
13	C₂×#11924031	C₂×C₄	1	8	yes	—
14	C₂×C₈×D₇₃	C₈	1	4	yes	—
15	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
16	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
17	C₄×#10596831	C₂×C₄	1	2	yes	—
18	C₄×#10596831	C₂³	1	1	yes	characteristic
19	C₈×Dic₇₃	C₈	1	4	yes	—
20	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	#11924031	C₂×C₈	1	16	yes	complemented
24	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
25	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
26	C₂×#10596831	C₂×C₈	1	4	yes	—
27	C₂×#10596831	C₂²×C₄	1	2	yes	—
28	C₈×D₇₃	C₁₆	1	16	yes	complemented
29	C₈×D₇₃	C₂×C₈	1	8	yes	—
30	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
31	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₇₃	C₄²	1	2	yes	—
33	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
34	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
38	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
39	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
40	C₄×D₇₃	C₄×C₈	1	4	yes	—
41	C₄×D₇₃	C₂×C₁₆	1	8	yes	—
42	C₈×C₇₃	C₄×C₈	1	4	yes	—
43	C₈×C₇₃	C₂×C₁₆	1	8	yes	complemented
44	#10596831	C₂×C₁₆	1	8	yes	complemented
45	#10596831	C₂²×C₈	1	4	yes	—
46	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
47	C₂×D₇₃	C₈²	1	2	yes	—
48	C₂×D₇₃	C₄×C₁₆	1	4	yes	—
49	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
50	C₄×C₇₃	C₂²×C₁₆	1	4	yes	—
51	Dic₇₃	C₈²	1	2	yes	—
52	Dic₇₃	C₄×C₁₆	1	4	yes	—
53	C₂×C₈×C₁₆	—	73	1	no	maximal, Sylow
54	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
55	C₂×C₇₃	C₂×C₄×C₁₆	1	2	yes	—
56	D₇₃	C₈×C₁₆	1	4	yes	complemented
57	C₂×C₄×C₁₆	—	73	2	no	—
58	C₂×C₈²	—	73	1	no	—
59	C₈×C₁₆	—	73	4	no	—
60	C₇₃	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₁₆	—	73	4	no	—
62	C₂×C₄×C₈	—	73	1	no	—
63	C₂×C₄×C₈	—	73	2	no	—
64	C₄×C₁₆	—	73	4	no	—
65	C₄×C₁₆	—	73	4	no	—
66	C₈²	—	73	2	no	—
67	C₈²	—	73	1	no	—
68	C₈²	#64408	1	1	yes	characteristic
69	C₂²×C₈	—	73	2	no	—
70	C₂²×C₈	—	73	4	no	—
71	C₂×C₄²	—	73	1	no	—
72	C₂×C₁₆	—	73	8	no	—
73	C₂×C₁₆	—	73	16	no	—
74	C₄×C₈	—	73	2	no	—
75	C₄×C₈	—	73	4	no	—
76	C₄×C₈	—	73	2	no	—
77	C₄×C₈	—	73	1	no	—
78	C₄×C₈	C₂×#64408	1	1	yes	characteristic
79	C₄×C₈	C₂×#64408	1	2	yes	—
80	C₂²×C₄	—	73	2	no	—
81	C₂²×C₄	—	73	1	no	—
82	C₂×C₈	—	73	4	no	—
83	C₂×C₈	—	73	2	no	—
84	C₂×C₈	—	73	8	no	—
85	C₂×C₈	—	73	16	no	—
86	C₂×C₈	C₄×#64408	1	2	yes	—
87	C₂×C₈	#11924036	1	4	yes	—
88	C₄²	—	73	2	no	—
89	C₄²	—	73	1	no	—
90	C₄²	C₂²×#64408	1	1	yes	characteristic
91	C₁₆	—	73	16	no	—
92	C₂³	—	73	1	no	—
93	C₂×C₄	—	73	2	no	—
94	C₂×C₄	—	73	1	no	—
95	C₂×C₄	—	73	8	no	—
96	C₂×C₄	—	73	4	no	—
97	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
98	C₂×C₄	C₂×#11924036	1	2	yes	—
99	C₈	—	73	8	no	—
100	C₈	—	73	4	no	—
101	C₈	C₈×#64408	1	4	yes	—
102	C₈	#14556664	1	8	yes	direct product factor
103	C₂²	—	73	4	no	—
104	C₂²	—	73	2	no	—
105	C₂²	C₄×#11924036	1	1	yes	characteristic
106	C₄	—	73	4	no	—
107	C₄	—	73	2	no	—
108	C₄	C₂×C₈×#64408	1	2	yes	—
109	C₄	C₂×#14556664	1	4	yes	—
110	C₂	—	73	4	no	—
111	C₂	C₈×#11924036	1	1	yes	characteristic
112	C₂	C₄×#14556664	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556664

C₂

yes

maximal

C₈×#11924031

C₂

yes

maximal, complemented

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

C₄

yes

—

#17989473

C₄

yes

—

C₄×#11924031

C₂²

yes

—

C₄×C₈×D₇₃

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₂²

yes

—

C₈²×C₇₃

C₄

yes

characteristic

#14556664

C₈

yes

direct product factor

C₂×#11924031

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

#11924031

C₂×C₈

yes

complemented

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₈×D₇₃

C₁₆

yes

complemented

C₈×D₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₄²

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

—

C₄×D₇₃

C₄×C₈

yes

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

C₄×C₈

yes

—

C₈×C₇₃

C₂×C₁₆

yes

complemented

#10596831

C₂×C₁₆

yes

complemented

#10596831

C₂²×C₈

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

C₈²

yes

—

C₂×D₇₃

C₄×C₁₆

yes

—

C₄×C₇₃

C₂×C₄×C₈

yes

—

C₄×C₇₃

C₂²×C₁₆

yes

—

Dic₇₃

C₈²

yes

—

Dic₇₃

C₄×C₁₆

yes

—

C₂×C₈×C₁₆

—

maximal, Sylow

C₂×C₇₃

C₂×C₈²

yes

characteristic

C₂×C₇₃

C₂×C₄×C₁₆

yes

—

D₇₃

C₈×C₁₆

yes

complemented

C₂×C₄×C₁₆

—

C₂×C₈²

—

C₈×C₁₆

—

C₇₃

C₂×C₈×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₄×C₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×#64408

yes

—

C₂×C₈

#11924036

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#64408

yes

characteristic

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×#64408

yes

characteristic

C₂×C₄

C₂×#11924036

yes

—

C₈

—

100

C₈

—

101

C₈

C₈×#64408

yes

—

102

C₈

#14556664

yes

direct product factor

103

C₂²

—

104

C₂²

—

105

C₂²

C₄×#11924036

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×C₈×#64408

yes

—

109

C₄

C₂×#14556664

yes

—

110

C₂

—

111

C₂

C₈×#11924036

yes

characteristic

112

C₂

C₄×#14556664

yes

—

	subgroup	G/H
center, hypercenter	C₈²	#64408
commutator, hypocenter	C₇₃	C₂×C₈×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₄×C₈
cosocle	C₄×#10596831	C₂³
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₈²

subgroup

G/H

center, hypercenter

C₈²

#64408

commutator, hypocenter

C₇₃

C₂×C₈×C₁₆

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄×C₈

prime radical

C₂²×D₇₃

C₄×C₈

C₂³

G ▷ C₇₃

C₁ ◁ C₈²

N	H ≅ G/N	type
2. C₈×#11924031	C₂	semidirect
12. #14556664	C₈	direct (N × H)
23. #11924031	C₂×C₈	semidirect
28. C₈×D₇₃	C₁₆	semidirect
43. C₈×C₇₃	C₂×C₁₆	semidirect
44. #10596831	C₂×C₁₆	semidirect
56. D₇₃	C₈×C₁₆	semidirect
60. C₇₃	C₂×C₈×C₁₆	semidirect
102. C₈	#14556664	direct (N × H)

H ≅ G/N

type

2. C₈×#11924031

C₂

semidirect

12. #14556664

C₈

direct (N × H)

23. #11924031

C₂×C₈

semidirect

28. C₈×D₇₃

C₁₆

semidirect

43. C₈×C₇₃

C₂×C₁₆

semidirect

44. #10596831

C₂×C₁₆

semidirect

56. D₇₃

C₈×C₁₆

semidirect

60. C₇₃

C₂×C₈×C₁₆

semidirect

102. C₈

#14556664

direct (N × H)

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

subgroup	index
C₄×#14556664	2
C₈×#11924031	2
C₈²×D₇₃	2
C₂×C₈×C₁₆	73

subgroup

index

C₄×#14556664

C₈×#11924031

C₈²×D₇₃

C₂×C₈×C₁₆

Subgroups — C₈×#14556664 #34009638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#14556664 #34009638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14556664 #34009638