Finite Group Databasebeta

← #38768581

Subgroups — #38768581

Order 25472.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#20263911C212yesmaximal
2C2×#20263911C211yesmaximal, characteristic
3#27163785C211yesmaximal, characteristic
4#27163784C212yesmaximal
5#342×C199C211yesmaximal, characteristic
6C23×Dic199C2211yescharacteristic
7#20263911—24no—
8#20263911—22no—
9#20263911—22no—
10#20263911C418yescomplemented
11C2×Dic199⋊C4—22no—
12C2×Dic199⋊C4C412yescomplemented
13C2×C22⋊Dic199C2211yescharacteristic
14C2×C22⋊Dic199C2212yes—
15C2×C4×Dic199—22no—
16C2×C4×Dic199C412yescomplemented
17C2×C22⋊C4×C199C2211yescharacteristic
18C2×C22⋊C4×C199C2212yes—
19C22×Dic199—22no—
20C22×Dic199C2×C412yes—
21C22×Dic199C2×C411yescharacteristic
22C22×Dic199C2×C411yescharacteristic
23C22×Dic199C2×C414yes—
24C22×Dic199D412yes—
25C22×Dic199D412yes—
26C22×Dic199D412yes—
27C22×Dic199Q812yes—
28Dic199⋊C4—28no—
29Dic199⋊C4—24no—
30C22⋊Dic199—22no—
31C22⋊Dic199—22no—
32C22⋊Dic199C2×C414yes—
33C4×Dic199—28no—
34C4×Dic199—22no—
35C4×Dic199—22no—
36C24×C199C2311yescharacteristic
37C22×C4×C199—22no—
38C22×C4×C199C2×C412yes—
39C22⋊C4×C199—22no—
40C22⋊C4×C199—22no—
41C22⋊C4×C199C2×C414yes—
42C2×Dic199—22no—
43C2×Dic199—24no—
44C2×Dic199—44no—
45C2×Dic199—42no—
46C2×Dic199C4212yes—
47C2×Dic199C4212yes—
48C2×Dic199C22⋊C414yes—
49C2×Dic199C22⋊C411yescharacteristic
50C2×Dic199C22⋊C411yescharacteristic
51C2×Dic199C22⋊C414yes—
52C2×Dic199C22⋊C412yes—
53C2×Dic199C4⋊C412yes—
54C2×Dic199C4⋊C412yes—
55C2×Dic199C4⋊C414yes—
56C2×Dic199C4⋊C414yes—
57C23×C199—42no—
58C23×C199C22×C411yescharacteristic
59C23×C199C22×C412yes—
60C23×C199C2×D412yes—
61C23×C199C2×D411yescharacteristic
62C23×C199C2×Q811yescharacteristic
63C2×C4×C199—22no—
64C2×C4×C199—24no—
65C2×C4×C199—44no—
66C2×C4×C199—42no—
67C22×C199—22no—
68C22×C199—24no—
69C22×C199—42no—
70C22×C199—41no—
71C22×C199—41no—
72C22×C199C2×C4211yescharacteristic
73C22×C199C2×C22⋊C412yes—
74C22×C199C2×C22⋊C411yescharacteristic
75C22×C199C2×C4⋊C411yescharacteristic
76C22×C199C2×C4⋊C412yes—
77C4×C199—44no—
78C4×C199—42no—
79Dic199—44no—
80Dic199—42no—
81Dic199#9512yes—
82Dic199#9514yes—
83Dic199#9512yes—
84C2×C199—22no—
85C2×C199—21no—
86C2×C199—21no—
87C2×C199—22no—
88C2×C199C2×#9511yescharacteristic
89C2×C199#41012yes—
90C199#174011yescharacteristic, complemented, Sylow
91#1740—1991nomaximal, Sylow
92C2×#117—1992no—
93C2×#117—1991no—
94#342—1992no—
95#342—1991no—
96#342D19911yescharacteristic
97C23×C4—1991no—
98#117—3984no—
99#117—3982no—
100#117—3982no—
101#117—1998no—
102C2×C4⋊C4—3982no—
103C2×C4⋊C4—1992no—
104C2×C22⋊C4—1991no—
105C2×C22⋊C4—1992no—
106C2×C22⋊C4C2×D19911yescharacteristic
107C2×C22⋊C4C2×D19912yes—
108C2×C42—3982no—
109C2×C42—1992no—
110C24C22×D19911yescharacteristic
111C22×C4—22no—
112C22×C4—3982no—
113C22×C4—1992no—
114C22×C4—1991no—
115C22×C4—1991no—
116C22×C4—1994no—
117C22×C4—1992no—
118C22×C4—1992no—
119C22×C4—1992no—
120C22×C4—1992no—
121C22×C4C4×D19912yes—
122C4⋊C4—3988no—
123C4⋊C4—3984no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—3982no—
127C22⋊C4—3982no—
128C22⋊C4—1994no—
129C22⋊C4C4×D19914yes—
130C42—3988no—
131C42—3982no—
132C42—3982no—
133C23—42no—
134C23C2×C4×D19911yescharacteristic
135C23C2×C4×D19912yes—
136C23D4×D19912yes—
137C23D4×D19911yescharacteristic
138C23Q8×D19911yescharacteristic
139C2×C4—24no—
140C2×C4—22no—
141C2×C4—44no—
142C2×C4—42no—
143C2×C4—7964no—
144C2×C4—7962no—
145C2×C4—3984no—
146C2×C4—3982no—
147C2×C4—1992no—
148C2×C4—1992no—
149C2×C4—1994no—
150C2×C4—1994no—
151C2×C4—1991no—
152C2×C4—1991no—
153C2×C4—1992no—
154C2×C4—1992no—
155C2×C4—1992no—
156C2×C4—1994no—
157C2×C4—1994no—
158C22—22no—
159C22—24no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22C42×D19911yescharacteristic
164C22C22⋊C4×D19912yes—
165C22C22⋊C4×D19911yescharacteristic
166C22C4⋊C4×D19911yescharacteristic
167C22C4⋊C4×D19912yes—
168C4—44no—
169C4—42no—
170C4—7964no—
171C4—7962no—
172C4—1992no—
173C4—1994no—
174C4—1992no—
175C2—22no—
176C2—21no—
177C2—21no—
178C2—22no—
179C2#95×D19911yescharacteristic
180C2#2716399012yes—
subgroupG/H
centerC22C22⋊C4×D199
commutatorC22×C199C2×C42
FrattiniC24C22×D199
Fitting#342×C199C2
socleC22×C199C2×C22⋊C4
prime radical, cosocleC24×C199C23
hypercenter#342D199
hypocenterC199#1740
LCSG ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
10. #20263911C4semidirect
12. C2×Dic199⋊C4C4semidirect
16. C2×C4×Dic199C4semidirect
90. C199#1740semidirect

Maximal subgroups

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

subgroupindex
C2×#202639112
#271637852
#271637842
#342×C1992
#1740199