Finite Group Databasebeta

← #32203875

Subgroups — #32203875

Order 18176.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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
1#22596604C213yesmaximal
2#22596600C2112yesmaximal, complemented
3#22591607C2112yesmaximal, complemented
4#22590718C213yesmaximal
5C71×#2840C211yesmaximal, characteristic
6C24×Dic71C2211yescharacteristic
7C2×Q8×Dic71C2214yescomplemented
8C2×#14556084C2216yes—
9#17985939C22112yescomplemented
10#17985935C22112yes—
11#17985933C22124yescomplemented
12#17985922C22124yes—
13#17985918C22112yescomplemented
14#17985860C2216yes—
15#17985857C22112yes—
16#17985855C2213yes—
17#17985850C22112yescomplemented
18C22⋊C4×Dic71C22112yes—
19#393×C71C22112yes—
20#379×C71C2213yes—
21C23×Dic71—44no—
22C23×Dic71Q818yescomplemented
23C23×Dic71C2313yes—
24C23×Dic71C2313yes—
25Q8×Dic71—48no—
26#14556084—412no—
27C2×C4⋊Dic71C23112yes—
28C2×Dic71⋊C4C23124yes—
29C2×Dic71⋊C4C23112yes—
30C2×C22⋊Dic71C23112yes—
31C2×C22⋊Dic71C2316yes—
32C2×C4×Dic71C23112yes—
33C71⋊#95C23124yes—
34C71⋊#95C23112yes—
35C25×C71C2311yescharacteristic
36C22×Q8×C71C2314yes—
37C2×C22⋊C4×C71C23112yes—
38C2×C22⋊C4×C71C2316yes—
39#95×C71C23112yes—
40C22×Dic71—212no—
41C22×Dic71—416no—
42C22×Dic71—46no—
43C22×Dic71C2×Q8112yes—
44C22×Dic71C2×Q8112yes—
45C22×Dic71C24112yes—
46C22×Dic71C2413yes—
47C22×Dic71C2411yescharacteristic
48C4⋊Dic71—412no—
49Dic71⋊C4—424no—
50Dic71⋊C4—412no—
51C22⋊Dic71—412no—
52C22⋊Dic71—46no—
53C4×Dic71—224no—
54C24×C71—42no—
55C24×C71—44no—
56C24×C71C2×Q814yes—
57C24×C71C2413yes—
58C2×Q8×C71—48no—
59C2×Q8×C71—44no—
60C22×C4×C71C24112yes—
61C22⋊C4×C71—412no—
62C22⋊C4×C71—46no—
63C23×C71—26no—
64C23×C71—212no—
65C23×C71—48no—
66C23×C71—44no—
67C23×C71—416no—
68C23×C71C22×Q816yes—
69C23×C71C2511yescharacteristic
70Q8×C71—416no—
71C2×C4×C71—212no—
72C2×C4×C71—224no—
73C2×Dic71—26no—
74C2×Dic71—224no—
75C2×Dic71—212no—
76C2×Dic71—224no—
77C2×Dic71—412no—
78C2×Dic71C22×Q816yes—
79C2×Dic71C22×Q812yes—
80C2×Dic71H(5,𝔽2)14yes—
81C22×C71—26no—
82C22×C71—212no—
83C22×C71—416no—
84C22×C71—412no—
85C22×C71C23×Q811yescharacteristic
86C22×C71C2×H(5,𝔽2)12yes—
87C22×C71#58514yes—
88C4×C71—412no—
89Dic71—26no—
90Dic71—412no—
91Dic71#54314yes—
92#36674—711nomaximal, Sylow
93C2×C71—46no—
94C2×C71C2×#54311yescharacteristic
95C2×C71#345812yes—
96C2×C71#352914yes—
97#2842—713no—
98#2841—7112no—
99#2840D7111yescharacteristic
100#2396—7112no—
101#2280—713no—
102C71#3667411yescharacteristic, complemented, Sylow
103C24×C4—711no—
104C2×C4×Q8—714no—
105C2×#117—716no—
106#400—7112no—
107#399—7112no—
108#398—7124no—
109#396—7124no—
110#395—7112no—
111#393C2×D71112yes—
112#381—716no—
113#380—7112no—
114#379—713no—
115#379C2×D7113yes—
116C4×C4⋊C4—7112no—
117C4×C22⋊C4—7112no—
118C25C22×D7111yescharacteristic
119C22×Q8C22×D7114yes—
120C23×C4—2844no—
121C23×C4—718no—
122C23×C4—713no—
123C23×C4—713no—
124C4×Q8—2848no—
125#117—28412no—
126C2×C4⋊C4—7112no—
127C2×C4⋊C4—7112no—
128C2×C4⋊C4—7124no—
129C2×C22⋊C4—716no—
130C2×C22⋊C4—7112no—
131C2×C22⋊C4C22×D7116yes—
132C2×C22⋊C4C22×D71112yes—
133C2×C42—7112no—
134#95—7124no—
135#95—7112no—
136#95C22×D71112yes—
137C24—42no—
138C24—44no—
139C24Q8×D7114yes—
140C24C23×D7113yes—
141C2×Q8—44no—
142C2×Q8—48no—
143C22×C4—2846no—
144C22×C4—28416no—
145C22×C4—7112no—
146C22×C4—711no—
147C22×C4—7112no—
148C22×C4—713no—
149C22×C4—7112no—
150C22×C4—14212no—
151C22×C4C23×D71112yes—
152C4⋊C4—28424no—
153C4⋊C4—28412no—
154C4⋊C4—28412no—
155C22⋊C4—412no—
156C22⋊C4—46no—
157C22⋊C4—28412no—
158C22⋊C4—2846no—
159C42—14224no—
160C23—26no—
161C23—212no—
162C23—44no—
163C23—48no—
164C23—416no—
165C23C2×Q8×D7116yes—
166C23C24×D7111yescharacteristic
167Q8—416no—
168C2×C4—212no—
169C2×C4—224no—
170C2×C4—28412no—
171C2×C4—712no—
172C2×C4—714no—
173C2×C4—716no—
174C2×C4—14212no—
175C2×C4—14224no—
176C2×C4—14224no—
177C2×C4—1426no—
178C22—26no—
179C22—212no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D7111yescharacteristic
183C22H(5,𝔽2)×D7112yes—
184C22#1798683414yes—
185C4—412no—
186C4—28412no—
187C4—714no—
188C4—1426no—
189C2—46no—
190C2#543×D7111yescharacteristic
191C2#2260174212yes—
192C2#2260248014yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2840C2
prime radicalC25×C71C23
hypercenter#2840D71
hypocenterC71#36674
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #22596600C2semidirect
3. #22591607C2semidirect
7. C2×Q8×Dic71C22semidirect
9. #17985939C22semidirect
11. #17985933C22semidirect
13. #17985918C22semidirect
17. #17985850C22semidirect
22. C23×Dic71Q8semidirect
102. C71#36674semidirect

Maximal subgroups

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

subgroupindex
#225966042
#225966002
#225916072
#225907182
C71×#28402
#3667471