Finite Group Databasebeta

← #38769981

Subgroups — #38769981

Order 25472.

430 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 73 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#27164580C211yesmaximal, characteristic
2C2×#20263938C212yesmaximal, complemented
3C2×#20263869C212yesmaximal, complemented
4#27163785C211yesmaximal, characteristic, complemented
5#381×C199C211yesmaximal, characteristic, complemented
6C22×C22⋊D199C2212yes—
7#20263939—22no—
8#20263938—24no—
9#20263938—24no—
10#20263929—22no—
11#20263925—22no—
12C2×Dic199⋊C4—22no—
13C2×C22⋊Dic199C2212yes—
14C2×(C2×D199)⋊C4—22no—
15#20263869—24no—
16C23×C4×C199C2211yescharacteristic
17C2×C22⋊C4×C199C2211yescharacteristic
18C2×C22⋊C4×C199C2211yescharacteristic
19#95×C199—22no—
20C23×D199D412yes—
21C2×D796—22no—
22C2×C22⋊D199—22no—
23C2×C22⋊D199—22no—
24C2×C22⋊D199—22no—
25C2×C22⋊D199—22no—
26C2×C22⋊D199—22no—
27C2×C22⋊D199—22no—
28C22×Dic199—22no—
29C22×Dic199D412yes—
30C4⋊Dic199—22no—
31Dic199⋊C4—28no—
32C22⋊Dic199—22no—
33C22⋊Dic199D414yescomplemented
34(C2×D199)⋊C4—28no—
35(C2×D199)⋊C4—24no—
36C24×C199C2311yescharacteristic
37C22×C4×C199—22no—
38C22×C4×C199—21no—
39C22×C4×C199—22no—
40C22×C4×C199—21no—
41C22×C4×C199—21no—
42C22×C4×C199D412yes—
43C22×C4×C199D412yescomplemented
44C22⋊C4×C199—22no—
45C22⋊C4×C199—22no—
46C22×D199—22no—
47C22×D199—22no—
48C22×D199—22no—
49C22×D199—44no—
50D796—82no—
51C22⋊D199—42no—
52C22⋊D199—42no—
53C22⋊D199—42no—
54C22⋊D199—42no—
55C2×Dic199—24no—
56C2×Dic199—22no—
57C2×Dic199—22no—
58C2×Dic199—22no—
59C2×Dic199—44no—
60C23×C199—42no—
61C23×C199C2×D412yes—
62C23×C199C2×D412yes—
63C23×C199C2×D411yescharacteristic
64C23×C199C2×D411yescharacteristic
65C23×C199#4111yescharacteristic
66C2×C4×C199—22no—
67C2×C4×C199—24no—
68C2×C4×C199—22no—
69C2×C4×C199—41no—
70C2×C4×C199—42no—
71C2×C4×C199—41no—
72C2×C4×C199—42no—
73C2×C4×C199—42no—
74C2×C4×C199—41no—
75C22×C199—22no—
76C22×C199—24no—
77C22×C199—42no—
78C22×C199—41no—
79C22×C199—41no—
80C22×C199#12011yescharacteristic
81C22×C199#12012yes—
82C22×C199#12112yes—
83C22×C199#12111yescharacteristic
84C22×C199#12711yescharacteristic
85C2×D199—44no—
86C2×D199—42no—
87C2×D199—44no—
88C2×D199—82no—
89C4×C199—42no—
90C4×C199—42no—
91C4×C199—81no—
92Dic199—44no—
93Dic199—42no—
94C2×C199—22no—
95C2×C199—21no—
96C2×C199—22no—
97C2×C199—21no—
98C2×C199#39211yescharacteristic
99C2×C199#45812yes—
100D199—82no—
101C199#202211yescharacteristic, complemented, Sylow
102#2022—1991nomaximal, Sylow
103#534—1991no—
104C2×#123—1992no—
105C2×#99—1992no—
106#381D19911yescharacteristic, complemented
107#342—1991no—
108C22×D4—1992no—
109C23×C4C2×D19911yescharacteristic
110#123—3982no—
111#123—3984no—
112#123—3984no—
113#121—3982no—
114#120—3982no—
115C2×C4⋊C4—3982no—
116C2×C22⋊C4—3982no—
117C2×C22⋊C4—1992no—
118C2×C22⋊C4C2×D19911yescharacteristic
119C2×C22⋊C4C2×D19911yescharacteristic
120#99—3984no—
121#95—22no—
122C24—1992no—
123C24C22×D19911yescharacteristic
124C2×D4—3982no—
125C2×D4—3982no—
126C2×D4—3982no—
127C2×D4—3982no—
128C2×D4—3982no—
129C2×D4—3982no—
130C2×D4—3982no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—3982no—
137C22×C4—1992no—
138C22×C4C22⋊D19912yes—
139C22×C4D79612yescomplemented
140C4⋊C4—3988no—
141C4⋊C4—3982no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—3988no—
145C22⋊C4—3984no—
146C22⋊C4—3982no—
147C22⋊C4—1994no—
148C23—42no—
149C23—7964no—
150C23—3982no—
151C23—3982no—
152C23—3982no—
153C23D4×D19912yes—
154C23D4×D19912yes—
155C23C2×C22⋊D19911yescharacteristic
156C23C2×D79611yescharacteristic
157C23#1569188611yescharacteristic
158D4—7962no—
159D4—7962no—
160D4—7962no—
161D4—7962no—
162D4—15922no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—7964no—
173C2×C4—3982no—
174C2×C4—3984no—
175C2×C4—3982no—
176C2×C4—3982no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—7964no—
183C22—7962no—
184C22—7964no—
185C22—15922no—
186C22#2026392311yescharacteristic
187C22#2026392512yes—
188C22#2026392812yes—
189C22#2026392911yescharacteristic
190C22#2026395111yescharacteristic
191C4—42no—
192C4—42no—
193C4—81no—
194C4—7964no—
195C4—7962no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—15922no—
201C2#2716392711yescharacteristic
202C2#2716420712yes—
subgroupG/H
centerC22#20263923
commutator, cosocleC24×C199C23
FrattiniC24C22×D199
Fitting#381×C199C2
socleC22×C199#120
prime radical#27164580C2
hypercenter#381D199
hypocenterC199#2022
LCSG ▷ C24×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#20263938C2semidirect
3. C2×#20263869C2semidirect
4. #27163785C2semidirect
5. #381×C199C2semidirect
33. C22⋊Dic199D4semidirect
43. C22×C4×C199D4semidirect
101. C199#2022semidirect
106. #381D199semidirect
139. C22×C4D796semidirect

Maximal subgroups

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

subgroupindex
#271645802
C2×#202639382
C2×#202638692
#271637852
#381×C1992
#2022199