Finite Group Databasebeta

← #38767161

Subgroups — #38767161

Order 25472.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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×#20263911C211yesmaximal, characteristic
2#27164068C212yesmaximal, complemented
3#27163743C212yesmaximal, complemented
4#27163742C211yesmaximal, characteristic
5#323×C199C211yesmaximal, characteristic
6C23×Dic199C411yescharacteristic
7#20263911—22no—
8#20263911—22no—
9C2×Dic199⋊C4C411yescharacteristic
10C2×Dic199⋊C4C2211yescharacteristic
11C2×C22⋊Dic199C411yescharacteristic
12C2×C4×Dic199C411yescharacteristic
13C2×C4×Dic199C2211yescharacteristic
14Dic199⋊C8—22no—
15C22⋊(C199⋊C8)C2212yes—
16C8×Dic199—22no—
17C2×C22⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C22×Dic199—21no—
20C22×Dic199—21no—
21C22×Dic199—22no—
22C22×Dic199C814yescomplemented
23C22×Dic199C2×C411yescharacteristic
24C22×Dic199C2×C411yescharacteristic
25C22×Dic199C2×C411yescharacteristic
26C22×Dic199C2×C411yescharacteristic
27C2×C199⋊C8—22no—
28Dic199⋊C4—22no—
29Dic199⋊C4—22no—
30C22⋊Dic199—22no—
31C4×Dic199—21no—
32C4×Dic199—21no—
33C4×Dic199D412yes—
34C4×Dic199D412yes—
35C24×C199C2×C411yescharacteristic
36C22×C4×C199C2×C411yescharacteristic
37C22×C4×C199C2311yescharacteristic
38C2×C8×C199—22no—
39C22⋊C4×C199—22no—
40C2×Dic199—21no—
41C2×Dic199—21no—
42C2×Dic199—24no—
43C2×Dic199—24no—
44C2×Dic199—21no—
45C2×Dic199—41no—
46C2×Dic199—41no—
47C2×Dic199C22⋊C412yes—
48C2×Dic199C22⋊C411yescharacteristic
49C2×Dic199C22⋊C411yescharacteristic
50C2×Dic199C2×C812yes—
51C2×Dic199C2×C812yes—
52C2×Dic199M1612yes—
53C199⋊C8—42no—
54C23×C199—21no—
55C23×C199—21no—
56C23×C199—41no—
57C23×C199—41no—
58C23×C199C2×C812yes—
59C23×C199C22×C411yescharacteristic
60C2×C4×C199—21no—
61C2×C4×C199—41no—
62C2×C4×C199—41no—
63C2×C4×C199C2×D412yes—
64C8×C199—42no—
65C22×C199—21no—
66C22×C199—21no—
67C22×C199—22no—
68C22×C199—42no—
69C22×C199—42no—
70C22×C199—41no—
71C22×C199—41no—
72C22×C199C2×C22⋊C411yescharacteristic
73C22×C199C22×C811yescharacteristic
74C22×C199C2×M1611yescharacteristic
75C4×C199—22no—
76C4×C199—41no—
77Dic199—22no—
78Dic199—22no—
79Dic199—41no—
80Dic199C22⋊C812yes—
81Dic199C22⋊C812yes—
82C2×C199—21no—
83C2×C199—21no—
84C2×C199—42no—
85C2×C199C2×C22⋊C811yescharacteristic
86C2×C199#41011yescharacteristic
87C2×C199#41311yescharacteristic
88C199#145811yescharacteristic, complemented, Sylow
89#1458—1991nomaximal, Sylow
90C2×#117—1991no—
91#431—1992no—
92#324—1992no—
93#323—1991no—
94#323D19911yescharacteristic
95C23×C4—1991no—
96#117—3982no—
97#117—3982no—
98C2×C4⋊C4—1991no—
99C2×C4⋊C4—1991no—
100C2×C22⋊C4—1991no—
101C2×C22⋊C4C2×D19911yescharacteristic
102C2×C42—1991no—
103C2×C42—1991no—
104C4⋊C8—3982no—
105C22⋊C8—1992no—
106C22⋊C8C2×D19912yes—
107C4×C8—3982no—
108C24C4×D19911yescharacteristic
109C22×C4—3981no—
110C22×C4—3981no—
111C22×C4—3982no—
112C22×C4—1991no—
113C22×C4—1991no—
114C22×C4—1994no—
115C22×C4—1991no—
116C22×C4—1991no—
117C22×C4C4×D19911yescharacteristic
118C22×C4C22×D19911yescharacteristic
119C2×C8—22no—
120C2×C8—3982no—
121C4⋊C4—3982no—
122C4⋊C4—3982no—
123C22⋊C4—22no—
124C22⋊C4—3982no—
125C42—3981no—
126C42—3981no—
127C42—1992no—
128C42—1992no—
129C23—21no—
130C23—21no—
131C23—41no—
132C23—41no—
133C23C8×D19912yes—
134C23C2×C4×D19911yescharacteristic
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—7961no—
139C2×C4—7961no—
140C2×C4—3981no—
141C2×C4—3981no—
142C2×C4—3981no—
143C2×C4—3984no—
144C2×C4—3984no—
145C2×C4—1992no—
146C2×C4—1992no—
147C2×C4—1992no—
148C2×C4—1992no—
149C2×C4—1991no—
150C2×C4—1991no—
151C2×C4D4×D19912yes—
152C8—42no—
153C8—7962no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22C22⋊C4×D19911yescharacteristic
162C22C2×C8×D19911yescharacteristic
163C22M16×D19911yescharacteristic
164C4—22no—
165C4—41no—
166C4—7961no—
167C4—3982no—
168C4—3982no—
169C4—1992no—
170C4—1992no—
171C2—21no—
172C2—21no—
173C2—42no—
174C2C22⋊C8×D19911yescharacteristic
175C2#2716399011yescharacteristic
176C2#2716399911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC22×C199C22×C8
FrattiniC22×C4C22×D199
Fitting#323×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC24×C199C2×C4
cosocleC22×C4×C199C23
hypercenter#323D199
hypocenterC199#1458
LCSG ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27164068C2semidirect
3. #27163743C2semidirect
22. C22×Dic199C8semidirect
88. C199#1458semidirect

Maximal subgroups

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

subgroupindex
C2×#202639112
#271640682
#271637432
#271637422
#323×C1992
#1458199