Finite Group Databasebeta

← #38769298

Subgroups — #38769298

Order 25472.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#27164529C211yesmaximal, characteristic
2C2×#20263883C212yesmaximal, complemented
3#27163982C211yesmaximal, characteristic, complemented
4#27163778C212yesmaximal, complemented
5C2×#117×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C2211yescharacteristic
7C2×#15691870C2212yes—
8#20263938—21no—
9#20263934C412yescomplemented
10#20263929C412yescomplemented
11#20263924—21no—
12C4×C22⋊D199—22no—
13#20263914C2211yescharacteristic
14#20263883—24no—
15C22⋊(C199⋊C8)—22no—
16C23×C4×C199C2211yescharacteristic
17#117×C199—24no—
18C2×C4⋊C4×C199—21no—
19C2×C22⋊C4×C199—21no—
20C2×C42×C199C2212yes—
21#15691886—22no—
22#15691886—22no—
23C2×Dic398C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—22no—
26C2×C22⋊D199—21no—
27C2×C4×D199—21no—
28#15691870—22no—
29#15691870—22no—
30C2×C199⋊C8—22no—
31C4⋊Dic199C2×C412yes—
32Dic199⋊C4—22no—
33C22⋊Dic199—22no—
34C22⋊Dic199—21no—
35(C2×D199)⋊C4—22no—
36C4×Dic199—21no—
37C24×C199D411yescharacteristic
38C22×C4×C199—22no—
39C22×C4×C199—22no—
40C22×C4×C199—22no—
41C22×C4×C199C2×C412yes—
42C22×C4×C199D411yescharacteristic
43C22×C4×C199D412yes—
44C22×C4×C199C2311yescharacteristic
45C4⋊C4×C199—24no—
46C22⋊C4×C199—24no—
47C42×C199—22no—
48C42×C199D414yescomplemented
49C22×D199—21no—
50Dic398—22no—
51D796—22no—
52C22⋊D199—44no—
53C22⋊D199—41no—
54C22⋊D199—41no—
55C2×Dic199—22no—
56C2×Dic199—21no—
57C4×D199—42no—
58C199⋊C8—42no—
59C23×C199—21no—
60C23×C199—21no—
61C23×C199—42no—
62C23×C199C22⋊C412yes—
63C23×C199C2×D411yescharacteristic
64C2×C4×C199—21no—
65C2×C4×C199—21no—
66C2×C4×C199—24no—
67C2×C4×C199—24no—
68C2×C4×C199—24no—
69C2×C4×C199—44no—
70C2×C4×C199C22⋊C412yes—
71C2×C4×C199C22×C411yescharacteristic
72C2×C4×C199C2×D412yes—
73C2×C4×C199C2×D411yescharacteristic
74C2×C4×C199#4112yes—
75C22×C199—22no—
76C22×C199—22no—
77C22×C199—22no—
78C22×C199—22no—
79C22×C199—42no—
80C22×C199—42no—
81C22×C199C2×C22⋊C411yescharacteristic
82C22×C199#12011yescharacteristic
83C22×C199#12311yescharacteristic
84C2×D199—41no—
85C2×D199—42no—
86C4×C199—22no—
87C4×C199—44no—
88C4×C199C4×D412yes—
89C4×C199#12112yes—
90Dic199—41no—
91Dic199—42no—
92C2×C199—21no—
93C2×C199—21no—
94C2×C199—42no—
95C2×C199#38611yescharacteristic
96C2×C199#42112yes—
97D199—81no—
98C199#189711yescharacteristic, complemented, Sylow
99#1897—1991nomaximal, Sylow
100#525—1991no—
101C2×#117D19911yescharacteristic, complemented
102C2×#104—1992no—
103#407—1991no—
104#339—1992no—
105C2×#41—1991no—
106C23×C4C2×D19911yescharacteristic
107C2×M16—1992no—
108#123—3981no—
109C22⋊Q8—1992no—
110#121—1992no—
111#120—3981no—
112C4×D4—3982no—
113#117—24no—
114#117—1991no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D19912yes—
118#104—3984no—
119C22⋊C8—3982no—
120C24C22⋊D19911yescharacteristic
121#41—3982no—
122#41—3982no—
123C2×Q8—1991no—
124C2×D4—3982no—
125C2×D4—3981no—
126C2×D4—1991no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—3981no—
131C22×C4C4×D19912yes—
132C22×C4C22⋊D19911yescharacteristic
133C22×C4D79612yes—
134C22×C4C22×D19911yescharacteristic
135M16—3982no—
136M16—3982no—
137C2×C8—3982no—
138C4⋊C4—24no—
139C4⋊C4—3982no—
140C4⋊C4—1992no—
141C22⋊C4—24no—
142C22⋊C4—3982no—
143C22⋊C4—3981no—
144C22⋊C4—3982no—
145C42—22no—
146C42—3981no—
147C42C22⋊D19914yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—3981no—
152C23(C2×D199)⋊C412yes—
153C23C2×C22⋊D19911yescharacteristic
154Q8—3982no—
155D4—7964no—
156D4—7961no—
157D4—7961no—
158D4—3982no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—7962no—
166C2×C4—3982no—
167C2×C4—3981no—
168C2×C4(C2×D199)⋊C412yes—
169C2×C4C2×C4×D19911yescharacteristic
170C2×C4C2×C22⋊D19912yes—
171C2×C4C2×D79611yescharacteristic
172C2×C4#1569188612yes—
173C8—7962no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—7962no—
181C22—7961no—
182C22C2×(C2×D199)⋊C411yescharacteristic
183C22#2026392411yescharacteristic
184C22#2026393811yescharacteristic
185C4—22no—
186C4—44no—
187C4—7962no—
188C4—7961no—
189C4C4×C22⋊D19912yes—
190C4#2026392912yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—15921no—
195C2#2716389711yescharacteristic
196C2#2716403612yes—
subgroupG/H
centerC2×C4C2×D796
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×#117×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164529C2
cosocleC22×C4×C199C23
hypercenterC2×#117D199
hypocenterC199#1897
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#20263883C2semidirect
3. #27163982C2semidirect
4. #27163778C2semidirect
5. C2×#117×C199C2semidirect
9. #20263934C4semidirect
10. #20263929C4semidirect
48. C42×C199D4semidirect
98. C199#1897semidirect
101. C2×#117D199semidirect
147. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645292
C2×#202638832
#271639822
#271637782
C2×#117×C1992
#1897199