Group #42128577 — subgroups

Subgroups — C₁₀₉×#10583 #42128577

Order 27904.

406 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₁₀₉×#2844	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1847	C₂	1	3	yes	maximal
3	C₁₀₉×#1295	C₂	1	3	yes	maximal
4	C₂×#107×C₁₀₉	—	2	3	no	—
5	C₂×#106×C₁₀₉	—	2	3	no	—
6	C₂×#97×C₁₀₉	C₂²	1	3	yes	—
7	#400×C₁₀₉	—	2	3	no	—
8	#395×C₁₀₉	—	2	3	no	—
9	#384×C₁₀₉	C₂²	1	1	yes	characteristic
10	#382×C₁₀₉	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₀₉	—	2	1	no	—
12	C₂²×C₈×C₁₀₉	—	2	3	no	—
13	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
14	C₂×C₄⋊C₄×C₁₀₉	—	2	3	no	—
15	C₂×C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
17	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
18	#107×C₁₀₉	—	4	6	no	—
19	#106×C₁₀₉	—	4	6	no	—
20	#97×C₁₀₉	—	2	3	no	—
21	#97×C₁₀₉	Q₈	1	6	yes	complemented
22	#95×C₁₀₉	—	2	3	no	—
23	#95×C₁₀₉	—	2	1	no	—
24	#95×C₁₀₉	—	2	6	no	—
25	C₂×Q₈×C₁₀₉	—	8	3	no	—
26	C₂²×C₄×C₁₀₉	—	2	3	no	—
27	C₂²×C₄×C₁₀₉	—	2	3	no	—
28	C₂²×C₄×C₁₀₉	—	4	1	no	—
29	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
30	C₂×C₈×C₁₀₉	—	2	6	no	—
31	C₂×C₈×C₁₀₉	—	2	6	no	—
32	C₂×C₈×C₁₀₉	—	2	6	no	—
33	C₄⋊C₄×C₁₀₉	—	4	3	no	—
34	C₄⋊C₄×C₁₀₉	—	4	3	no	—
35	C₄⋊C₄×C₁₀₉	—	4	3	no	—
36	C₄⋊C₄×C₁₀₉	—	4	3	no	—
37	C₄⋊C₄×C₁₀₉	—	8	3	no	—
38	C₄²×C₁₀₉	C₂×Q₈	1	3	yes	—
39	C₄²×C₁₀₉	#41	1	1	yes	characteristic
40	C₂³×C₁₀₉	#120	1	1	yes	characteristic
41	Q₈×C₁₀₉	—	8	4	no	—
42	C₂×C₄×C₁₀₉	—	2	3	no	—
43	C₂×C₄×C₁₀₉	—	2	3	no	—
44	C₂×C₄×C₁₀₉	—	4	3	no	—
45	C₂×C₄×C₁₀₉	—	4	3	no	—
46	C₂×C₄×C₁₀₉	—	4	6	no	—
47	C₂×C₄×C₁₀₉	—	4	6	no	—
48	C₂×C₄×C₁₀₉	—	8	3	no	—
49	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	3	yes	—
50	C₂×C₄×C₁₀₉	C₄⋊Q₈	1	3	yes	—
51	C₈×C₁₀₉	—	2	12	no	—
52	C₂²×C₁₀₉	#395	1	1	yes	characteristic
53	C₂²×C₁₀₉	#453	1	3	yes	—
54	C₂²×C₁₀₉	#456	1	3	yes	—
55	C₄×C₁₀₉	—	2	3	no	—
56	C₄×C₁₀₉	—	2	3	no	—
57	C₄×C₁₀₉	—	8	3	no	—
58	C₄×C₁₀₉	—	8	3	no	—
59	C₄×C₁₀₉	—	16	1	no	—
60	#10583	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₁₀₉	#1655	1	1	yes	characteristic
62	C₂×C₁₀₉	#1660	1	3	yes	—
63	C₂×C₁₀₉	#2060	1	3	yes	—
64	#2844	C₂×C₁₀₉	1	1	yes	characteristic
65	#1847	C₂×C₁₀₉	1	3	yes	—
66	#1295	C₂×C₁₀₉	1	3	yes	—
67	C₁₀₉	#10583	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×#107	—	2	3	no	—
69	C₂×#106	—	2	3	no	—
70	C₂×#97	C₂²×C₁₀₉	1	3	yes	—
71	#400	—	2	3	no	—
72	#395	—	2	3	no	—
73	#384	C₂²×C₁₀₉	1	1	yes	characteristic
74	#382	C₂²×C₁₀₉	1	3	yes	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	2	3	no	—
77	C₂×C₄⋊C₄	—	2	3	no	—
78	C₂×C₄⋊C₄	—	2	3	no	—
79	C₂×C₄⋊C₄	D₄×C₁₀₉	1	3	yes	—
80	C₂×C₄⋊C₄	D₄×C₁₀₉	1	3	yes	—
81	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
82	#107	—	4	6	no	—
83	#106	—	4	6	no	—
84	#97	—	2	3	no	—
85	#97	Q₈×C₁₀₉	1	6	yes	complemented
86	#95	—	2	3	no	—
87	#95	—	2	1	no	—
88	#95	—	2	6	no	—
89	C₂×Q₈	—	8	3	no	—
90	C₂²×C₄	—	2	3	no	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	4	1	no	—
93	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
94	C₂×C₈	—	2	6	no	—
95	C₂×C₈	—	2	6	no	—
96	C₂×C₈	—	2	6	no	—
97	C₄⋊C₄	—	4	3	no	—
98	C₄⋊C₄	—	4	3	no	—
99	C₄⋊C₄	—	4	3	no	—
100	C₄⋊C₄	—	4	3	no	—
101	C₄⋊C₄	—	8	3	no	—
102	C₄²	C₂×Q₈×C₁₀₉	1	3	yes	—
103	C₄²	#41×C₁₀₉	1	1	yes	characteristic
104	C₂³	#120×C₁₀₉	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	3	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	8	3	no	—
113	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	3	yes	—
114	C₂×C₄	C₄⋊Q₈×C₁₀₉	1	3	yes	—
115	C₈	—	2	12	no	—
116	C₂²	#395×C₁₀₉	1	1	yes	characteristic
117	C₂²	#453×C₁₀₉	1	3	yes	—
118	C₂²	#456×C₁₀₉	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	16	1	no	—
124	C₂	C₁₀₉×#1655	1	1	yes	characteristic
125	C₂	C₁₀₉×#1660	1	3	yes	—
126	C₂	C₁₀₉×#2060	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2844

C₂

yes

maximal, characteristic

C₁₀₉×#1847

C₂

yes

maximal

C₁₀₉×#1295

C₂

yes

maximal

C₂×#107×C₁₀₉

—

C₂×#106×C₁₀₉

—

C₂×#97×C₁₀₉

C₂²

yes

—

#400×C₁₀₉

—

#395×C₁₀₉

—

#384×C₁₀₉

C₂²

yes

characteristic

#382×C₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#107×C₁₀₉

—

#106×C₁₀₉

—

#97×C₁₀₉

—

#97×C₁₀₉

Q₈

yes

complemented

#95×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×Q₈

yes

—

C₄²×C₁₀₉

yes

C₂³×C₁₀₉

yes

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#395

yes

characteristic

C₂²×C₁₀₉

#453

yes

—

C₂²×C₁₀₉

#456

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

#10583

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

#1655

yes

characteristic

C₂×C₁₀₉

#1660

yes

—

C₂×C₁₀₉

#2060

yes

—

#2844

C₂×C₁₀₉

yes

characteristic

#1847

C₂×C₁₀₉

yes

—

#1295

C₂×C₁₀₉

yes

—

C₁₀₉

#10583

yes

characteristic, direct product factor, Sylow

C₂×#107

—

C₂×#106

—

C₂×#97

C₂²×C₁₀₉

yes

—

#400

—

#395

—

#384

C₂²×C₁₀₉

yes

characteristic

#382

C₂²×C₁₀₉

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

C₂×C₄²

C₂³×C₁₀₉

yes

—

—

—

Q₈×C₁₀₉

yes

—

—

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

C₂×Q₈×C₁₀₉

yes

—

103

C₄²

#41×C₁₀₉

yes

characteristic

104

C₂³

#120×C₁₀₉

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

114

C₂×C₄

C₄⋊Q₈×C₁₀₉

yes

—

115

C₈

—

116

C₂²

#395×C₁₀₉

yes

characteristic

117

C₂²

#453×C₁₀₉

yes

—

118

C₂²

#456×C₁₀₉

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

C₁₀₉×#1655

yes

characteristic

125

C₂

C₁₀₉×#1660

yes

—

126

C₂

C₁₀₉×#2060

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
21. #97×C₁₀₉	Q₈	semidirect
60. #10583	C₁₀₉	direct (N × H)
67. C₁₀₉	#10583	direct (N × H)
85. #97	Q₈×C₁₀₉	semidirect

H ≅ G/N

type

21. #97×C₁₀₉

Q₈

semidirect

60. #10583

C₁₀₉

direct (N × H)

67. C₁₀₉

#10583

direct (N × H)

85. #97

Q₈×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#2844	2
C₁₀₉×#1847	2
C₁₀₉×#1295	2
#10583	109

subgroup

index

C₁₀₉×#2844

C₁₀₉×#1847

C₁₀₉×#1295

#10583

109

Subgroups — C₁₀₉×#10583 #42128577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10583 #42128577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10583 #42128577