Finite Group Databasebeta

← #38769170

Subgroups — #38769170

Order 25472.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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×#20263929C211yesmaximal, characteristic
2C2×#20263914C211yesmaximal, characteristic, complemented
3C2×#20263877C212yesmaximal, complemented
4#27163781C212yesmaximal, complemented
5C2×C22⋊C8×C199C211yesmaximal, characteristic, complemented
6C22×D796C2211yescharacteristic
7C22×C22⋊D199—21no—
8#20263929—22no—
9#20263929C414yescomplemented
10#20263914—24no—
11C2×C4⋊Dic199C2211yescharacteristic
12C2×C4⋊Dic199C2212yes—
13C2×C22⋊Dic199—21no—
14C2×(C2×D199)⋊C4—21no—
15C2×C4×Dic199—21no—
16#20263877—24no—
17C23×C4×C199C2211yescharacteristic
18C22×C8×C199C2212yes—
19C22⋊C8×C199—22no—
20C23×D199—21no—
21C2×D796—22no—
22C2×D796—21no—
23C2×D796—22no—
24C2×D796C2×C412yes—
25C2×C22⋊D199—24no—
26C2×C22⋊D199—44no—
27C22×Dic199—21no—
28C22×Dic199—22no—
29C4⋊Dic199—22no—
30C4⋊Dic199—21no—
31C4⋊Dic199C2×C412yes—
32C4⋊Dic199D414yescomplemented
33C22⋊Dic199—24no—
34(C2×D199)⋊C4—24no—
35C4×Dic199—24no—
36C24×C199D411yescharacteristic
37C22×C4×C199—22no—
38C22×C4×C199—22no—
39C22×C4×C199C2×C412yes—
40C22×C4×C199D412yes—
41C22×C4×C199D411yescharacteristic
42C22×C4×C199C2311yescharacteristic
43C2×C8×C199—22no—
44C2×C8×C199—22no—
45C2×C8×C199—22no—
46C22×D199—22no—
47C22×D199—41no—
48C22×D199—41no—
49C22×D199—42no—
50C22×D199—42no—
51D796—24no—
52D796—42no—
53C22⋊D199—48no—
54C2×Dic199—22no—
55C2×Dic199—24no—
56C2×Dic199—42no—
57C2×Dic199—42no—
58C2×Dic199—42no—
59C23×C199—21no—
60C23×C199—21no—
61C23×C199—22no—
62C23×C199—22no—
63C23×C199C22⋊C412yes—
64C23×C199C2×D411yescharacteristic
65C2×C4×C199—21no—
66C2×C4×C199—21no—
67C2×C4×C199—24no—
68C2×C4×C199—24no—
69C2×C4×C199C22⋊C412yes—
70C2×C4×C199C22×C411yescharacteristic
71C2×C4×C199C2×D412yes—
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199#4112yes—
74C8×C199—42no—
75C22×C199—22no—
76C22×C199—22no—
77C22×C199—22no—
78C22×C199—42no—
79C22×C199—42no—
80C22×C199C2×C22⋊C411yescharacteristic
81C22×C199#12011yescharacteristic
82C22×C199#12311yescharacteristic
83C22×C199#13512yes—
84C22×C199AGL(1,ℤ/8)12yes—
85C2×D199—42no—
86C2×D199—42no—
87C2×D199—42no—
88C2×D199—82no—
89C2×D199—82no—
90C4×C199—22no—
91C4×C199C4×D412yes—
92C4×C199#12112yes—
93Dic199—42no—
94Dic199—44no—
95C2×C199—42no—
96C2×C199#38611yescharacteristic
97C2×C199#41611yescharacteristic
98C2×C199#41811yescharacteristic
99C2×C199#44912yes—
100C2×C199#48212yes—
101D199—82no—
102C199#187511yescharacteristic, complemented, Sylow
103#1875—1991nomaximal, Sylow
104C2×#121—1991no—
105C2×#117—1991no—
106C2×#102—1992no—
107C2×C22⋊C8D19911yescharacteristic, complemented
108#340—1992no—
109C22×D4—3981no—
110C22×D4—1991no—
111C23×C4C2×D19911yescharacteristic
112C22×C8C2×D19912yes—
113#121—3982no—
114#121—1994no—
115#117—3984no—
116C2×C4⋊C4—1992no—
117C2×C4⋊C4—1991no—
118C2×C22⋊C4—3981no—
119C2×C22⋊C4—3981no—
120C2×C42—3981no—
121#102—3984no—
122C22⋊C8—22no—
123C24—3981no—
124C24D79611yescharacteristic
125C2×D4—7964no—
126C2×D4—3984no—
127C2×D4—3982no—
128C2×D4—3981no—
129C2×D4—3982no—
130C2×D4—1992no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—3982no—
134C22×C4—3981no—
135C22×C4C4×D19912yes—
136C22×C4C22⋊D19912yes—
137C22×C4D79611yescharacteristic
138C22×C4C22×D19911yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—3982no—
143C4⋊C4—3981no—
144C4⋊C4—1992no—
145C4⋊C4—1994no—
146C22⋊C4—3984no—
147C22⋊C4—3984no—
148C42—3984no—
149C23—21no—
150C23—22no—
151C23—22no—
152C23—21no—
153C23—7961no—
154C23—7962no—
155C23—7962no—
156C23—7961no—
157C23—3982no—
158C23(C2×D199)⋊C412yes—
159C23C2×D79611yescharacteristic
160D4—7968no—
161D4—7962no—
162D4—3984no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—7962no—
168C2×C4—7962no—
169C2×C4—7962no—
170C2×C4—3982no—
171C2×C4—3984no—
172C2×C4(C2×D199)⋊C412yes—
173C2×C4C2×C4×D19911yescharacteristic
174C2×C4D4×D19912yes—
175C2×C4C2×C22⋊D19911yescharacteristic
176C2×C4#1569188512yes—
177C8—42no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—7962no—
184C22—7962no—
185C22—7962no—
186C22—15922no—
187C22—15922no—
188C22C2×(C2×D199)⋊C411yescharacteristic
189C22#2026392511yescharacteristic
190C22#2026393911yescharacteristic
191C22#2026397712yes—
192C22#2026398312yes—
193C4—22no—
194C4—7964no—
195C4—7962no—
196C4(C22⋊D199)⋊C412yes—
197C4#2026392712yes—
198C2—42no—
199C2—15922no—
200C2#2716389511yescharacteristic
201C2#2716401511yescharacteristic
202C2#2716402411yescharacteristic
203C2#2716416512yes—
204C2#2716434312yes—
subgroupG/H
centerC23C2×D796
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×C22⋊C8×C199C2
socleC23×C199C2×D4
prime radicalC2×#20263929C2
cosocleC22×C4×C199C23
hypercenterC2×C22⋊C8D199
hypocenterC199#1875
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#20263914C2semidirect
3. C2×#20263877C2semidirect
4. #27163781C2semidirect
5. C2×C22⋊C8×C199C2semidirect
9. #20263929C4semidirect
32. C4⋊Dic199D4semidirect
102. C199#1875semidirect
107. C2×C22⋊C8D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639292
C2×#202639142
C2×#202638772
#271637812
C2×C22⋊C8×C1992
#1875199