Group #34197239 — subgroups

Subgroups — C₈×C₁₆×D₇₉ #34197239

Order 20224.

398 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 251 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×C₁₆×D₇₉	C₂	1	2	yes	maximal
2	C₈²×D₇₉	C₂	1	1	yes	maximal, characteristic
3	C₈×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×C₇₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₇₉	C₄	1	4	yes	—
7	C₄×C₈×D₇₉	C₄	1	2	yes	—
8	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
9	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
10	C₁₆×Dic₇₉	C₄	1	4	yes	—
11	C₈×C₇₉⋊C₈	C₄	1	2	yes	—
12	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
14	C₈²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₉	C₈	1	4	yes	—
16	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
17	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₇₉⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₁₆×D₇₉	C₈	1	16	yes	direct product factor
20	C₄×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₈×Dic₇₉	C₈	1	4	yes	—
23	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
28	C₂×C₄×D₇₉	C₂×C₈	1	2	yes	—
29	C₂×C₇₉⋊C₈	C₄²	1	2	yes	—
30	C₂×C₇₉⋊C₈	C₂×C₈	1	4	yes	—
31	C₈×D₇₉	C₁₆	1	16	yes	direct product factor
32	C₈×D₇₉	C₂×C₈	1	8	yes	—
33	C₄×Dic₇₉	C₄²	1	1	yes	characteristic
34	C₄×Dic₇₉	C₂×C₈	1	2	yes	—
35	C₇₉⋊C₁₆	C₂×C₈	1	8	yes	complemented
36	C₂×C₈×C₇₉	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
38	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
39	C₁₆×C₇₉	C₂×C₈	1	8	yes	complemented
40	C₂²×D₇₉	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₇₉	C₂²×C₈	1	2	yes	—
43	C₂×Dic₇₉	C₄×C₈	1	1	yes	characteristic
44	C₂×Dic₇₉	C₄×C₈	1	2	yes	—
45	C₄×D₇₉	C₄×C₈	1	4	yes	—
46	C₄×D₇₉	C₂×C₁₆	1	8	yes	—
47	C₈×C₇₉	C₂×C₁₆	1	8	yes	complemented
48	C₈×C₇₉	C₂²×C₈	1	4	yes	—
49	C₇₉⋊C₈	C₄×C₈	1	4	yes	—
50	C₇₉⋊C₈	C₂×C₁₆	1	8	yes	complemented
51	C₂²×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
52	C₂×D₇₉	C₈²	1	2	yes	—
53	C₂×D₇₉	C₄×C₁₆	1	4	yes	—
54	C₄×C₇₉	C₂×C₄×C₈	1	2	yes	—
55	C₄×C₇₉	C₂²×C₁₆	1	4	yes	—
56	Dic₇₉	C₈²	1	2	yes	—
57	Dic₇₉	C₄×C₁₆	1	4	yes	—
58	C₂×C₈×C₁₆	—	79	1	no	maximal, Sylow
59	C₂×C₇₉	C₂×C₈²	1	1	yes	characteristic
60	C₂×C₇₉	C₂×C₄×C₁₆	1	2	yes	—
61	D₇₉	C₈×C₁₆	1	4	yes	direct product factor
62	C₂×C₄×C₁₆	—	79	2	no	—
63	C₂×C₈²	—	79	1	no	—
64	C₈×C₁₆	—	79	2	no	—
65	C₈×C₁₆	—	79	1	no	—
66	C₈×C₁₆	D₇₉	1	1	yes	characteristic, direct product factor
67	C₇₉	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	—	79	4	no	—
69	C₂×C₄×C₈	—	79	1	no	—
70	C₂×C₄×C₈	—	79	2	no	—
71	C₄×C₁₆	—	79	4	no	—
72	C₄×C₁₆	—	79	2	no	—
73	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
74	C₈²	—	79	1	no	—
75	C₈²	—	79	2	no	—
76	C₈²	C₂×D₇₉	1	1	yes	characteristic
77	C₂²×C₈	—	79	4	no	—
78	C₂²×C₈	—	79	2	no	—
79	C₂×C₄²	—	79	1	no	—
80	C₂×C₁₆	—	79	4	no	—
81	C₂×C₁₆	—	79	16	no	—
82	C₂×C₁₆	C₄×D₇₉	1	4	yes	—
83	C₄×C₈	—	79	4	no	—
84	C₄×C₈	—	79	2	no	—
85	C₄×C₈	—	79	1	no	—
86	C₄×C₈	—	79	2	no	—
87	C₄×C₈	C₄×D₇₉	1	2	yes	—
88	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
89	C₂²×C₄	—	79	1	no	—
90	C₂²×C₄	—	79	2	no	—
91	C₂×C₈	—	79	2	no	—
92	C₂×C₈	—	79	4	no	—
93	C₂×C₈	—	79	16	no	—
94	C₂×C₈	—	79	8	no	—
95	C₂×C₈	C₈×D₇₉	1	4	yes	—
96	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
97	C₄²	—	79	1	no	—
98	C₄²	—	79	2	no	—
99	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
100	C₁₆	—	79	8	no	—
101	C₁₆	C₈×D₇₉	1	8	yes	direct product factor
102	C₂³	—	79	1	no	—
103	C₂×C₄	—	79	1	no	—
104	C₂×C₄	—	79	2	no	—
105	C₂×C₄	—	79	8	no	—
106	C₂×C₄	—	79	4	no	—
107	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
108	C₂×C₄	C₂×C₈×D₇₉	1	2	yes	—
109	C₈	—	79	8	no	—
110	C₈	—	79	4	no	—
111	C₈	C₁₆×D₇₉	1	8	yes	direct product factor
112	C₈	C₂×C₈×D₇₉	1	4	yes	—
113	C₂²	—	79	2	no	—
114	C₂²	—	79	4	no	—
115	C₂²	C₄×C₈×D₇₉	1	1	yes	characteristic
116	C₄	—	79	2	no	—
117	C₄	—	79	4	no	—
118	C₄	C₄×C₈×D₇₉	1	2	yes	—
119	C₄	C₂×C₁₆×D₇₉	1	4	yes	—
120	C₂	—	79	4	no	—
121	C₂	C₈²×D₇₉	1	1	yes	characteristic
122	C₂	C₄×C₁₆×D₇₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₁₆×D₇₉

C₂

yes

maximal

C₈²×D₇₉

C₂

yes

maximal, characteristic

C₈×C₇₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₁₆×C₇₉⋊C₈

C₂

yes

maximal, complemented

C₈×C₁₆×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₉

C₄

yes

—

C₄×C₈×D₇₉

C₄

yes

—

C₄×C₈×D₇₉

C₂²

yes

characteristic

C₄×C₇₉⋊C₁₆

C₂²

yes

—

C₁₆×Dic₇₉

C₄

yes

—

C₈×C₇₉⋊C₈

C₄

yes

—

C₈×C₇₉⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₇₉

C₂²

yes

—

C₈²×C₇₉

C₂²

yes

characteristic

C₂×C₈×D₇₉

C₈

yes

—

C₂×C₈×D₇₉

C₂×C₄

yes

—

C₄²×D₇₉

C₂×C₄

yes

characteristic

C₂×C₇₉⋊C₁₆

C₂×C₄

yes

—

C₁₆×D₇₉

C₈

yes

direct product factor

C₄×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄×C₇₉⋊C₈

C₂×C₄

yes

—

C₈×Dic₇₉

C₈

yes

—

C₈×Dic₇₉

C₂×C₄

yes

—

C₂×C₁₆×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄×D₇₉

C₄²

yes

characteristic

C₂×C₄×D₇₉

C₂×C₈

yes

—

C₂×C₇₉⋊C₈

C₄²

yes

—

C₂×C₇₉⋊C₈

C₂×C₈

yes

—

C₈×D₇₉

C₁₆

yes

direct product factor

C₈×D₇₉

C₂×C₈

yes

—

C₄×Dic₇₉

C₄²

yes

characteristic

C₄×Dic₇₉

C₂×C₈

yes

—

C₇₉⋊C₁₆

C₂×C₈

yes

complemented

C₂×C₈×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₁₆×C₇₉

C₂×C₈

yes

complemented

C₂²×D₇₉

C₄×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₉

C₂²×C₈

yes

—

C₂×Dic₇₉

C₄×C₈

yes

characteristic

C₂×Dic₇₉

C₄×C₈

yes

—

C₄×D₇₉

C₄×C₈

yes

—

C₄×D₇₉

C₂×C₁₆

yes

—

C₈×C₇₉

C₂×C₁₆

yes

complemented

C₈×C₇₉

C₂²×C₈

yes

—

C₇₉⋊C₈

C₄×C₈

yes

—

C₇₉⋊C₈

C₂×C₁₆

yes

complemented

C₂²×C₇₉

C₂×C₄×C₈

yes

characteristic

C₂×D₇₉

C₈²

yes

—

C₂×D₇₉

C₄×C₁₆

yes

—

C₄×C₇₉

C₂×C₄×C₈

yes

—

C₄×C₇₉

C₂²×C₁₆

yes

—

Dic₇₉

C₈²

yes

—

Dic₇₉

C₄×C₁₆

yes

—

C₂×C₈×C₁₆

—

maximal, Sylow

C₂×C₇₉

C₂×C₈²

yes

characteristic

C₂×C₇₉

C₂×C₄×C₁₆

yes

—

D₇₉

C₈×C₁₆

yes

direct product factor

C₂×C₄×C₁₆

—

C₂×C₈²

—

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

D₇₉

yes

characteristic, direct product factor

C₇₉

C₂×C₈×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₉

yes

—

C₈²

—

C₈²

—

C₈²

C₂×D₇₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₉

yes

—

C₄×C₈

C₂²×D₇₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇₉

yes

—

C₂×C₈

C₂×C₄×D₇₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₇₉

yes

characteristic

100

C₁₆

—

101

C₁₆

C₈×D₇₉

yes

direct product factor

102

C₂³

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄²×D₇₉

yes

characteristic

108

C₂×C₄

C₂×C₈×D₇₉

yes

—

109

C₈

—

110

C₈

—

111

C₈

C₁₆×D₇₉

yes

direct product factor

112

C₈

C₂×C₈×D₇₉

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₄×C₈×D₇₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₄×C₈×D₇₉

yes

—

119

C₄

C₂×C₁₆×D₇₉

yes

—

120

C₂

—

121

C₂

C₈²×D₇₉

yes

characteristic

122

C₂

C₄×C₁₆×D₇₉

yes

—

	subgroup	G/H
center, hypercenter	C₈×C₁₆	D₇₉
commutator, hypocenter	C₇₉	C₂×C₈×C₁₆
Frattini	C₄×C₈	C₂²×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄×C₈
prime radical	C₂²×D₇₉	C₄×C₈
cosocle	C₄×C₈×C₇₉	C₂³
LCS	G ▷ C₇₉
UCS	C₁ ◁ C₈×C₁₆

subgroup

G/H

center, hypercenter

C₈×C₁₆

D₇₉

commutator, hypocenter

C₇₉

C₂×C₈×C₁₆

Frattini

C₄×C₈

C₂²×D₇₉

Fitting

C₈×C₁₆×C₇₉

C₂

socle

C₂²×C₇₉

C₂×C₄×C₈

prime radical

C₂²×D₇₉

C₄×C₈

cosocle

C₄×C₈×C₇₉

C₂³

LCS

G ▷ C₇₉

UCS

C₁ ◁ C₈×C₁₆

N	H ≅ G/N	type
3. C₈×C₇₉⋊C₁₆	C₂	semidirect
4. C₁₆×C₇₉⋊C₈	C₂	semidirect
5. C₈×C₁₆×C₇₉	C₂	semidirect
19. C₁₆×D₇₉	C₈	direct (N × H)
31. C₈×D₇₉	C₁₆	direct (N × H)
35. C₇₉⋊C₁₆	C₂×C₈	semidirect
39. C₁₆×C₇₉	C₂×C₈	semidirect
47. C₈×C₇₉	C₂×C₁₆	semidirect
50. C₇₉⋊C₈	C₂×C₁₆	semidirect
61. D₇₉	C₈×C₁₆	direct (N × H)
66. C₈×C₁₆	D₇₉	direct (N × H)
67. C₇₉	C₂×C₈×C₁₆	semidirect
101. C₁₆	C₈×D₇₉	direct (N × H)
111. C₈	C₁₆×D₇₉	direct (N × H)

H ≅ G/N

type

3. C₈×C₇₉⋊C₁₆

C₂

semidirect

4. C₁₆×C₇₉⋊C₈

C₂

semidirect

5. C₈×C₁₆×C₇₉

C₂

semidirect

19. C₁₆×D₇₉

C₈

direct (N × H)

31. C₈×D₇₉

C₁₆

direct (N × H)

35. C₇₉⋊C₁₆

C₂×C₈

semidirect

39. C₁₆×C₇₉

C₂×C₈

semidirect

47. C₈×C₇₉

C₂×C₁₆

semidirect

50. C₇₉⋊C₈

C₂×C₁₆

semidirect

61. D₇₉

C₈×C₁₆

direct (N × H)

66. C₈×C₁₆

D₇₉

direct (N × H)

67. C₇₉

C₂×C₈×C₁₆

semidirect

101. C₁₆

C₈×D₇₉

direct (N × H)

111. C₈

C₁₆×D₇₉

direct (N × H)

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

subgroup	index
C₄×C₁₆×D₇₉	2
C₈²×D₇₉	2
C₈×C₇₉⋊C₁₆	2
C₁₆×C₇₉⋊C₈	2
C₈×C₁₆×C₇₉	2
C₂×C₈×C₁₆	79

subgroup

index

C₄×C₁₆×D₇₉

C₈²×D₇₉

C₈×C₇₉⋊C₁₆

C₁₆×C₇₉⋊C₈

C₈×C₁₆×C₇₉

C₂×C₈×C₁₆

Subgroups — C₈×C₁₆×D₇₉ #34197239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×C16×D79 #34197239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₁₆×D₇₉ #34197239