Finite Group Databasebeta

← C22×#379×C71 #31854875

Subgroups — C22×#379×C71 #31854875

Order 18176.

33214 conjugacy classes of subgroups; only the 3614 normal subgroups are shown.

normal subgroups only

This group has 33214 conjugacy classes of subgroups — too many to draw. Only its 3614 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C27×C71C211yesmaximal, characteristic
2C23×C22⋊C4×C71C216yesmaximal, complemented
3C2×#379×C71C2124yesmaximal, direct product factor
4C26×C71—248no—
5C26×C71C4116yescomplemented
6C26×C71C22112yes—
7C26×C71C2213yes—
8C24×C4×C71C2214yescomplemented
9C22×C22⋊C4×C71—248no—
10C22×C22⋊C4×C71C22172yescomplemented
11#379×C71C22164yesdirect product factor
12C25×C71—2512no—
13C25×C71—2384no—
14C25×C71—2288no—
15C25×C71—248no—
16C25×C71C2×C4124yes—
17C25×C71C2×C4196yescomplemented
18C25×C71D4148yes—
19C25×C71C23116yes—
20C25×C71C23118yes—
21C25×C71C2311yescharacteristic
22C23×C4×C71—248no—
23C23×C4×C71C23124yescomplemented
24C2×C22⋊C4×C71—2288no—
25C2×C22⋊C4×C71C23196yescomplemented
26C24×C71—21536no—
27C24×C71—22048no—
28C24×C71—2192no—
29C24×C71—21152no—
30C24×C71—2384no—
31C24×C71—2192no—
32C24×C71—2144no—
33C24×C71C22⋊C41192yescomplemented
34C24×C71C22×C4172yes—
35C24×C71C22×C414yes—
36C24×C71C22×C4164yescomplemented
37C24×C71C2×D41144yes—
38C24×C71C2×D4124yes—
39C24×C71C24112yes—
40C24×C71C2413yes—
41C22×C4×C71—2144no—
42C22×C4×C71—464no—
43C22×C4×C71C24116yescomplemented
44C22⋊C4×C71—2192no—
45C23×C71—2512no—
46C23×C71—23072no—
47C23×C71—2288no—
48C23×C71—2384no—
49C23×C71—2576no—
50C23×C71—248no—
51C23×C71—2768no—
52C23×C71C2×C22⋊C4148yes—
53C23×C71C2×C22⋊C41288yescomplemented
54C23×C71#120164yes—
55C23×C71C23×C4124yes—
56C23×C71C23×C416yes—
57C23×C71C22×D4148yes—
58C23×C71C22×D4136yes—
59C23×C71C2511yescharacteristic
60C2×C4×C71—248no—
61C2×C4×C71—496no—
62C22×C71—2512no—
63C22×C71—248no—
64C22×C71—296no—
65C22×C71—2576no—
66C22×C71#379164yesdirect product factor
67C22×C71C22×C22⋊C4136yes—
68C22×C71C22×C22⋊C4148yescomplemented
69C22×C71C2×#120148yes—
70C22×C71C24×C411yescharacteristic
71C22×C71C23×D416yes—
72C4×C71—416no—
73C22×#379C7111yesmaximal, characteristic, direct product factor, Sylow
74C2×C71—248no—
75C2×C71C2×#379124yesdirect product factor
76C2×C71C23×C22⋊C413yes—
77C2×C71C22×#12014yes—
78C27C2×C7111yescharacteristic
79C23×C22⋊C4C2×C7116yescomplemented
80C2×#379C2×C71124yesdirect product factor
81C71C22×#37911yescharacteristic, direct product factor, Sylow
82C26—248no—
83C26C4×C71116yescomplemented
84C26C22×C71112yes—
85C26C22×C7113yes—
86C24×C4C22×C7114yescomplemented
87C22×C22⋊C4—248no—
88C22×C22⋊C4C22×C71172yescomplemented
89#379C22×C71164yesdirect product factor
90C25—2512no—
91C25—2384no—
92C25—2288no—
93C25—248no—
94C25C2×C4×C71124yes—
95C25C2×C4×C71196yescomplemented
96C25D4×C71148yes—
97C25C23×C71116yes—
98C25C23×C71118yes—
99C25C23×C7111yescharacteristic
100C23×C4—248no—
101C23×C4C23×C71124yescomplemented
102C2×C22⋊C4—2288no—
103C2×C22⋊C4C23×C71196yescomplemented
104C24—21536no—
105C24—22048no—
106C24—2192no—
107C24—21152no—
108C24—2384no—
109C24—2192no—
110C24—2144no—
111C24C22⋊C4×C711192yescomplemented
112C24C22×C4×C71172yes—
113C24C22×C4×C7114yes—
114C24C22×C4×C71164yescomplemented
115C24C2×D4×C711144yes—
116C24C2×D4×C71124yes—
117C24C24×C71112yes—
118C24C24×C7113yes—
119C22×C4—2144no—
120C22×C4—464no—
121C22×C4C24×C71116yescomplemented
122C22⋊C4—2192no—
123C23—23072no—
124C23—2512no—
125C23—2288no—
126C23—2384no—
127C23—2576no—
128C23—248no—
129C23—2768no—
130C23C2×C22⋊C4×C71148yes—
131C23C2×C22⋊C4×C711288yescomplemented
132C23#120×C71164yes—
133C23C23×C4×C71124yes—
134C23C23×C4×C7116yes—
135C23C22×D4×C71148yes—
136C23C22×D4×C71136yes—
137C23C25×C7111yescharacteristic
138C2×C4—248no—
139C2×C4—496no—
140C22—2512no—
141C22—2576no—
142C22—248no—
143C22—296no—
144C22#379×C71164yesdirect product factor
145C22C22×C22⋊C4×C71136yes—
146C22C22×C22⋊C4×C71148yescomplemented
147C22C2×#120×C71148yes—
148C22C24×C4×C7111yescharacteristic
149C22C23×D4×C7116yes—
150C4—416no—
151C2—248no—
152C2C2×#379×C71124yesdirect product factor
153C2C23×C22⋊C4×C7113yes—
154C2C22×#120×C7114yes—
subgroupG/H
center, socleC25×C71C23
commutatorC22C24×C4×C71
Frattini, cosocleC23C25×C71
prime radicalC27×C71C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C71 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×C22⋊C4×C71C2semidirect
3. C2×#379×C71C2direct (N × H)
5. C26×C71C4semidirect
8. C24×C4×C71C22semidirect
10. C22×C22⋊C4×C71C22semidirect
11. #379×C71C22direct (N × H)
17. C25×C71C2×C4semidirect
23. C23×C4×C71C23semidirect
25. C2×C22⋊C4×C71C23semidirect
33. C24×C71C22⋊C4semidirect
36. C24×C71C22×C4semidirect
43. C22×C4×C71C24semidirect
53. C23×C71C2×C22⋊C4semidirect
66. C22×C71#379direct (N × H)
68. C22×C71C22×C22⋊C4semidirect
73. C22×#379C71direct (N × H)
75. C2×C71C2×#379direct (N × H)
79. C23×C22⋊C4C2×C71semidirect
80. C2×#379C2×C71direct (N × H)
81. C71C22×#379direct (N × H)
83. C26C4×C71semidirect
86. C24×C4C22×C71semidirect
88. C22×C22⋊C4C22×C71semidirect
89. #379C22×C71direct (N × H)
95. C25C2×C4×C71semidirect
101. C23×C4C23×C71semidirect
103. C2×C22⋊C4C23×C71semidirect
111. C24C22⋊C4×C71semidirect
114. C24C22×C4×C71semidirect
121. C22×C4C24×C71semidirect
131. C23C2×C22⋊C4×C71semidirect
144. C22#379×C71direct (N × H)
146. C22C22×C22⋊C4×C71semidirect
152. C2C2×#379×C71direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C712
C23×C22⋊C4×C712
C2×#379×C712
C22×#37971