Finite Group Databasebeta

← #38770540

Subgroups — #38770540

Order 25472.

262 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164586C211yesmaximal, characteristic
2#27164001C212yesmaximal, complemented
3#27163815C211yesmaximal, characteristic, complemented
4#27163814C212yesmaximal, complemented
5#355×C199C211yesmaximal, characteristic, complemented
6C2×#15691887C2211yescharacteristic
7C2×D4×D199C411yescharacteristic
8M16×D199—22no—
9#20263952C411yescharacteristic
10#20263951C411yescharacteristic
11#20263944C2211yescharacteristic
12#20263928—22no—
13#20263925—22no—
14#20263912C411yescharacteristic
15#20263874C2212yescomplemented
16#20263871—22no—
17#124×C199C2211yescharacteristic
18#101×C199C2212yescomplemented
19C23×D199—21no—
20#15691887—42no—
21C2×D796—22no—
22C2×D796C2×C411yescharacteristic
23C2×D796C2×C411yescharacteristic
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26C2×C22⋊D199—21no—
27D4×D199—42no—
28C2×C4×D199C2×C411yescharacteristic
29C2×C4×D199D412yes—
30#15691870—22no—
31#15691869—42no—
32C8×D199—42no—
33Dic199⋊C4—21no—
34(C2×D199)⋊C4—21no—
35(C2×D199)⋊C4—22no—
36C4×Dic199C2×C411yescharacteristic
37C2×Q8×C199C2311yescharacteristic
38C2×D4×C199C2×C411yescharacteristic
39M16×C199—22no—
40C22⋊C4×C199—22no—
41C42×C199C2×C411yescharacteristic
42C22×D199—21no—
43C22×D199—42no—
44C22×D199—82no—
45C22×D199C22⋊C411yescharacteristic
46C22×D199C22⋊C412yes—
47D796—41no—
48D796—42no—
49D796—41no—
50D796—82no—
51C22⋊D199—42no—
52C22⋊D199—82no—
53C2×Dic199—21no—
54C2×Dic199C22⋊C411yescharacteristic
55C4×D199—22no—
56C4×D199—22no—
57C4×D199—22no—
58C199⋊C8—42no—
59C23×C199—21no—
60Q8×C199—41no—
61D4×C199—41no—
62C2×C4×C199—21no—
63C2×C4×C199C22×C411yescharacteristic
64C2×C4×C199C2×D412yes—
65C8×C199—42no—
66C22×C199—42no—
67C22×C199—41no—
68C22×C199C2×C22⋊C411yescharacteristic
69C2×D199—21no—
70C2×D199—22no—
71C2×D199—42no—
72C2×D199—41no—
73C2×D199—42no—
74C2×D199—41no—
75C2×D199—84no—
76C4×C199—22no—
77C4×C199—21no—
78C4×C199—41no—
79Dic199—41no—
80Dic199#9912yes—
81C2×C199—21no—
82C2×C199—81no—
83C2×C199C2×#9911yescharacteristic
84D199—42no—
85D199—41no—
86D199—81no—
87C199#213411yescharacteristic, complemented, Sylow
88#2134—1991nomaximal, Sylow
89#535—1991no—
90#413—1992no—
91#355—1991no—
92#355—1992no—
93#355D19911yescharacteristic, complemented
94C2×#41—1991no—
95C22×D4—1991no—
96C2×M16—3982no—
97#127—1991no—
98#127—1991no—
99#124—1991no—
100#124C2×D19911yescharacteristic
101#121—3982no—
102#120—3982no—
103#117—1991no—
104#101—1992no—
105#101C2×D19912yescomplemented
106#100—3982no—
107C24—3981no—
108#41—7962no—
109C2×Q8C22×D19911yescharacteristic
110C2×D4—7962no—
111C2×D4—3982no—
112C2×D4—3982no—
113C2×D4—3981no—
114C2×D4—1991no—
115C2×D4—1991no—
116C2×D4—1992no—
117C2×D4C4×D19911yescharacteristic
118C22×C4—1991no—
119C22×C4—1992no—
120M16—22no—
121M16—7962no—
122M16—3982no—
123C2×C8—7962no—
124C4⋊C4—3981no—
125C22⋊C4—22no—
126C22⋊C4—3981no—
127C22⋊C4—3982no—
128C42—1991no—
129C42C4×D19911yescharacteristic
130C23—21no—
131C23—7962no—
132C23—15922no—
133C23—3981no—
134C23—1991no—
135C23—1992no—
136Q8—41no—
137D4—41no—
138D4—7961no—
139D4—7962no—
140D4—7961no—
141D4—7962no—
142D4—15922no—
143D4—15922no—
144C2×C4—21no—
145C2×C4—3982no—
146C2×C4—3982no—
147C2×C4—3982no—
148C2×C4—3981no—
149C2×C4—1991no—
150C2×C4C2×C4×D19911yescharacteristic
151C2×C4D4×D19912yes—
152C8—42no—
153C8—7962no—
154C22—42no—
155C22—41no—
156C22—7962no—
157C22—7961no—
158C22—7962no—
159C22—7961no—
160C22—15924no—
161C22—3981no—
162C22—3982no—
163C22C22⋊C4×D19911yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—7961no—
168C4—1992no—
169C2—21no—
170C2—81no—
171C2—7961no—
172C2—7962no—
173C2—15921no—
174C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×Q8C22×D199
Fitting#355×C199C2
socleC2×C199C2×#99
prime radical#27164586C2
cosocleC2×Q8×C199C23
hypercenter#355D199
hypocenterC199#2134
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #27164001C2semidirect
3. #27163815C2semidirect
4. #27163814C2semidirect
5. #355×C199C2semidirect
15. #20263874C22semidirect
18. #101×C199C22semidirect
87. C199#2134semidirect
93. #355D199semidirect
105. #101C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271645862
#271640012
#271638152
#271638142
#355×C1992
#2134199