Finite Group Databasebeta

← #38767187

Subgroups — #38767187

Order 25472.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×#20263917C211yesmaximal, characteristic
2#27163980C212yesmaximal, complemented
3#27163744C211yesmaximal, characteristic, complemented
4#27163741C212yesmaximal, complemented
5#324×C199C211yesmaximal, characteristic, complemented
6C22×D796C411yescharacteristic
7C22×C4×D199C411yescharacteristic
8C22×C4×D199C2211yescharacteristic
9C2×#15691869—22no—
10#20263917—22no—
11#20263917—22no—
12C2×(C2×D199)⋊C4C411yescharacteristic
13C2×(C2×D199)⋊C4C2211yescharacteristic
14C2×C4×Dic199C411yescharacteristic
15C22⋊(C199⋊C8)C2212yes—
16(C2×D199)⋊C8—22no—
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C23×D199C2×C411yescharacteristic
20C23×D199C2×C411yescharacteristic
21C2×D796—22no—
22C2×D796—22no—
23C2×D796—22no—
24C22×Dic199C2×C411yescharacteristic
25C22×Dic199C2×C411yescharacteristic
26C2×C4×D199—21no—
27C2×C4×D199—21no—
28C2×C4×D199—42no—
29C2×C4×D199—42no—
30C2×C4×D199D412yes—
31C2×C4×D199D412yes—
32#15691869—44no—
33C2×C199⋊C8—22no—
34(C2×D199)⋊C4—22no—
35(C2×D199)⋊C4—22no—
36C4×Dic199—21no—
37C4×Dic199—21no—
38C22×C4×C199C2×C411yescharacteristic
39C22×C4×C199C2×C411yescharacteristic
40C22×C4×C199C2311yescharacteristic
41C2×C8×C199—22no—
42C4⋊C4×C199—22no—
43C22×D199—21no—
44C22×D199—21no—
45C22×D199—21no—
46C22×D199—21no—
47C22×D199—41no—
48C22×D199—41no—
49C22×D199—41no—
50C22×D199—41no—
51C22×D199C22⋊C412yes—
52C22×D199M1612yescomplemented
53D796—28no—
54C2×Dic199—22no—
55C2×Dic199—21no—
56C2×Dic199—41no—
57C2×Dic199C22⋊C411yescharacteristic
58C2×Dic199C22⋊C411yescharacteristic
59C4×D199—24no—
60C4×D199—42no—
61C4×D199—42no—
62C4×D199—42no—
63C199⋊C8—42no—
64C23×C199C22×C411yescharacteristic
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—21no—
68C2×C4×C199—41no—
69C2×C4×C199—41no—
70C2×C4×C199M1612yescomplemented
71C2×C4×C199C2×D412yes—
72C8×C199—42no—
73C22×C199—21no—
74C22×C199—21no—
75C22×C199C2×C22⋊C411yescharacteristic
76C22×C199C2×M1611yescharacteristic
77C22×C199C2×M1611yescharacteristic
78C2×D199—22no—
79C2×D199—22no—
80C2×D199—22no—
81C2×D199—22no—
82C2×D199—42no—
83C2×D199—42no—
84C2×D199—42no—
85C2×D199—41no—
86C2×D199—41no—
87C2×D199—42no—
88C4×C199—22no—
89C4×C199—22no—
90C4×C199—41no—
91Dic199—21no—
92Dic199—41no—
93Dic199#9912yes—
94C2×C199—21no—
95C2×C199—21no—
96C2×C199#40711yescharacteristic
97C2×C199C2×#9911yescharacteristic
98C2×C199#41311yescharacteristic
99D199—42no—
100D199—42no—
101C199#146511yescharacteristic, complemented, Sylow
102#1465—1991nomaximal, Sylow
103C2×C4×D4—1991no—
104#407—1992no—
105#324—1991no—
106#324D19911yescharacteristic, complemented
107#323—1992no—
108C22×D4—1991no—
109C23×C4—1991no—
110C23×C4—1991no—
111C2×M16—3982no—
112C4×D4—3982no—
113C4×D4—3982no—
114C2×C4⋊C4C2×D19911yescharacteristic
115C2×C22⋊C4—1991no—
116C2×C22⋊C4—1991no—
117C2×C42—1991no—
118C22⋊C8—3982no—
119C22⋊C8—1992no—
120C22⋊C8C2×D19912yes—
121C24—1991no—
122C24—1991no—
123C2×D4—3982no—
124C2×D4—3982no—
125C2×D4—3982no—
126C22×C4—7962no—
127C22×C4—7962no—
128C22×C4—3981no—
129C22×C4—3981no—
130C22×C4—1991no—
131C22×C4—1992no—
132C22×C4—1991no—
133C22×C4—1992no—
134C22×C4C4×D19911yescharacteristic
135C22×C4C4×D19911yescharacteristic
136C22×C4C22×D19911yescharacteristic
137M16—7964no—
138C2×C8—22no—
139C2×C8—3982no—
140C4⋊C4—22no—
141C22⋊C4—3982no—
142C22⋊C4—3982no—
143C42—3981no—
144C42—3981no—
145C23—7961no—
146C23—7961no—
147C23—7961no—
148C23—7961no—
149C23—3981no—
150C23—3981no—
151C23—3981no—
152C23—3981no—
153C23—1992no—
154C23—1992no—
155C23C2×C4×D19911yescharacteristic
156D4—3988no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—7962no—
163C2×C4—7962no—
164C2×C4—7962no—
165C2×C4—7961no—
166C2×C4—3984no—
167C2×C4—3981no—
168C2×C4—3982no—
169C2×C4—1991no—
170C2×C4—1991no—
171C2×C4#1569186912yescomplemented
172C2×C4D4×D19912yes—
173C8—42no—
174C8—7962no—
175C22—21no—
176C22—21no—
177C22—7962no—
178C22—7962no—
179C22—7962no—
180C22—7962no—
181C22—7961no—
182C22—7961no—
183C22—3982no—
184C22—3982no—
185C22—3982no—
186C22—3982no—
187C22C22⋊C4×D19911yescharacteristic
188C22M16×D19911yescharacteristic
189C22C2×#1569186911yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—7961no—
194C4—3981no—
195C4—1992no—
196C2—21no—
197C2—21no—
198C2—7962no—
199C2—7962no—
200C2#2716398011yescharacteristic
201C2#99×D19911yescharacteristic
202C2#2716400111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC22×D796C4
cosocleC22×C4×C199C23
hypercenter#324D199
hypocenterC199#1465
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27163980C2semidirect
3. #27163744C2semidirect
4. #27163741C2semidirect
5. #324×C199C2semidirect
52. C22×D199M16semidirect
70. C2×C4×C199M16semidirect
101. C199#1465semidirect
106. #324D199semidirect
171. C2×C4#15691869semidirect

Maximal subgroups

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

subgroupindex
C2×#202639172
#271639802
#271637442
#271637412
#324×C1992
#1465199