Finite Group Databasebeta

← #38768808

Subgroups — #38768808

Order 25472.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20264003C211yesmaximal, characteristic
2C2×#20263883C212yesmaximal, complemented
3#27164017C212yesmaximal, complemented
4C2×#20263874C211yesmaximal, characteristic, complemented
5#391×C199C211yesmaximal, characteristic, complemented
6#20264003—24no—
7#20264003—22no—
8#20264003C414yescomplemented
9C2×#15691887—22no—
10C2×#15691886—22no—
11C2×#15691886C2212yes—
12C2×Q8×D199—22no—
13C2×#15691870C2212yes—
14#20263883—24no—
15#20263878—22no—
16#20263875—22no—
17#20263874—22no—
18C22×Q8×C199C2211yescharacteristic
19C2×C4⋊C4×C199C2211yescharacteristic
20C2×C42×C199C2211yescharacteristic
21#95×C199—22no—
22#15691887—28no—
23#15691887—28no—
24#15691886—28no—
25#15691886—28no—
26#15691886—22no—
27#15691886—22no—
28#15691886C2×C414yes—
29#15691886D414yes—
30C2×Dic398—22no—
31C2×Dic398D412yes—
32Q8×D199—28no—
33Q8×D199—28no—
34C2×D796—22no—
35C2×D796D412yes—
36C2×C22⋊D199—22no—
37C2×C4×D199—24no—
38C2×C4×D199—22no—
39#15691870—22no—
40#15691870—22no—
41C2×C199⋊C8—22no—
42C2×Q8×C199—22no—
43C2×Q8×C199—22no—
44C2×Q8×C199C2×C412yes—
45C2×Q8×C199D412yes—
46C22×C4×C199—21no—
47C22×C4×C199—21no—
48C22×C4×C199D412yes—
49C22×C4×C199C2311yescharacteristic
50C4⋊C4×C199—22no—
51C42×C199—22no—
52C22×D199—22no—
53Dic398—28no—
54Dic398—22no—
55Dic398C22⋊C414yes—
56D796—28no—
57D796—22no—
58D796C22⋊C414yes—
59C22⋊D199—24no—
60C22⋊D199—24no—
61C2×Dic199—22no—
62C4×D199—216no—
63C4×D199—24no—
64C4×D199—24no—
65C199⋊C8—42no—
66C23×C199C2×D411yescharacteristic
67Q8×C199—24no—
68Q8×C199—24no—
69C2×C4×C199—22no—
70C2×C4×C199—22no—
71C2×C4×C199—22no—
72C2×C4×C199—41no—
73C2×C4×C199—41no—
74C2×C4×C199—42no—
75C2×C4×C199—41no—
76C2×C4×C199C22⋊C414yes—
77C2×C4×C199C22×C411yescharacteristic
78C2×C4×C199C2×D412yes—
79C2×C4×C199C2×D412yes—
80C2×C4×C199C2×D411yescharacteristic
81C22×C199—21no—
82C22×C199—21no—
83C22×C199C2×C22⋊C411yescharacteristic
84C22×C199#12011yescharacteristic
85C22×C199#12011yescharacteristic
86C2×D199—24no—
87C2×D199—42no—
88C2×D199—42no—
89C4×C199—24no—
90C4×C199—42no—
91C4×C199—81no—
92C4×C199C2×C22⋊C412yes—
93C4×C199#12012yes—
94Dic199—24no—
95C2×C199—21no—
96C2×C199—21no—
97C2×C199#37911yescharacteristic
98C2×C199#45411yescharacteristic
99C2×C199#45611yescharacteristic
100D199—44no—
101C199#179311yescharacteristic, complemented, Sylow
102#1793—1991nomaximal, Sylow
103C2×2−1+4—1991no—
104C2×#104—1992no—
105#417—1992no—
106C2×#101—1991no—
107#391D19911yescharacteristic, complemented
1082−1+4—3984no—
1092−1+4—3982no—
1102−1+4—1994no—
111C2×#41—3982no—
112C2×#41—3982no—
113C2×#41—1992no—
114C22×Q8—3982no—
115C22×Q8C2×D19911yescharacteristic
116C2×M16—1992no—
117C2×C4⋊C4C2×D19911yescharacteristic
118C2×C42C2×D19911yescharacteristic
119#104—3984no—
120#103—3982no—
121#102—3982no—
122#101—3982no—
123#95—22no—
124#41—3988no—
125#41—3988no—
126#41—3988no—
127#41—3988no—
128#41—3982no—
129#41—3982no—
130#41—1994no—
131#41—1994no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—3982no—
135C2×Q8—3988no—
136C2×Q8—3988no—
137C2×Q8—1992no—
138C2×Q8C4×D19912yes—
139C2×Q8D79612yes—
140C2×D4—3982no—
141C2×D4—3982no—
142C2×D4—1992no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—3984no—
146C22×C4—3982no—
147C22×C4C22⋊D19912yes—
148C22×C4C22×D19911yescharacteristic
149M16—3982no—
150M16—3982no—
151C2×C8—3982no—
152C4⋊C4—22no—
153C42—22no—
154C23—3982no—
155C23C2×C22⋊D19911yescharacteristic
156Q8—24no—
157Q8—24no—
158Q8—3988no—
159Q8—3982no—
160Q8—1994no—
161D4—3988no—
162D4—3984no—
163D4—3984no—
164D4—3982no—
165D4—1994no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—39816no—
174C2×C4—3982no—
175C2×C4—3984no—
176C2×C4—3984no—
177C2×C4(C2×D199)⋊C414yes—
178C2×C4C2×C4×D19911yescharacteristic
179C2×C4D4×D19912yes—
180C2×C4D4×D19912yes—
181C2×C4C2×D79611yescharacteristic
182C8—7962no—
183C22—21no—
184C22—21no—
185C22—7962no—
186C22—7962no—
187C22—3984no—
188C22C2×(C2×D199)⋊C411yescharacteristic
189C22#2026392311yescharacteristic
190C22#2026392311yescharacteristic
191C4—24no—
192C4—42no—
193C4—81no—
194C4—3984no—
195C4C22⋊C4×D19912yes—
196C4#2026392512yes—
197C2—21no—
198C2—21no—
199C2—7964no—
200C2#2716387311yescharacteristic
201C2#2716419011yescharacteristic
202C2#2716420011yescharacteristic
subgroupG/H
centerC22#20263923
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#391×C199C2
socleC22×C199#120
prime radicalC2×#20264003C2
cosocleC22×C4×C199C23
hypercenter#391D199
hypocenterC199#1793
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#20263883C2semidirect
3. #27164017C2semidirect
4. C2×#20263874C2semidirect
5. #391×C199C2semidirect
8. #20264003C4semidirect
101. C199#1793semidirect
107. #391D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202640032
C2×#202638832
#271640172
C2×#202638742
#391×C1992
#1793199