Finite Group Databasebeta

← #38769268

Subgroups — #38769268

Order 25472.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#27164525C211yesmaximal, characteristic
2#27164002C211yesmaximal, characteristic, complemented
3#27163981C212yesmaximal, complemented
4#27163789C212yesmaximal, complemented
5#568×C199C211yesmaximal, characteristic, complemented
6C22×C4×D199C2211yescharacteristic
7C2×#15691870C2212yes—
8#20263936C412yes—
9#20263931—21no—
10#20263926—21no—
11#20263923C412yes—
12C4×C22⋊D199—22no—
13#20263914C2211yescharacteristic
14#20263874—21no—
15#20263872—21no—
16(C2×D199)⋊C8—24no—
17C2×#41×C199C2211yescharacteristic
18#131×C199—24no—
19C2×M16×C199—22no—
20C2×M16×C199C2212yes—
21C23×D199C2×C411yescharacteristic
22C2×C22⋊D199—22no—
23C22×Dic199C2×C411yescharacteristic
24C2×C4×D199—22no—
25C2×C4×D199—21no—
26C2×C4×D199—21no—
27C2×C4×D199—22no—
28#15691870—22no—
29#15691870—22no—
30C2×C199⋊C8—22no—
31C4⋊Dic199—21no—
32Dic199⋊C4—22no—
33C22⋊Dic199—22no—
34C22⋊Dic199C2×C412yes—
35(C2×D199)⋊C4—22no—
36C4×Dic199—21no—
37#41×C199—22no—
38#41×C199—22no—
39C2×Q8×C199D411yescharacteristic
40C2×D4×C199C2×C412yes—
41C2×D4×C199D411yescharacteristic
42C22×C4×C199D412yes—
43C22×C4×C199C2311yescharacteristic
44M16×C199—24no—
45M16×C199—22no—
46M16×C199—22no—
47C2×C8×C199—24no—
48C2×C8×C199D414yescomplemented
49C22×D199—21no—
50C22×D199—22no—
51C22×D199—42no—
52C22⋊D199—44no—
53C2×Dic199—22no—
54C2×Dic199—21no—
55C2×Dic199—22no—
56C4×D199—41no—
57C4×D199—42no—
58C4×D199—41no—
59C199⋊C8—42no—
60C23×C199C22⋊C412yes—
61C23×C199C22×C411yescharacteristic
62Q8×C199—22no—
63D4×C199—24no—
64D4×C199—22no—
65C2×C4×C199—22no—
66C2×C4×C199—22no—
67C2×C4×C199C22⋊C412yes—
68C2×C4×C199C2×D412yes—
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199#4112yes—
72C8×C199—24no—
73C22×C199—22no—
74C22×C199—41no—
75C22×C199—42no—
76C22×C199C2×C22⋊C411yescharacteristic
77C22×C199C4×D412yes—
78C2×D199—22no—
79C2×D199—24no—
80C2×D199—41no—
81C2×D199—42no—
82C2×D199—41no—
83C4×C199—22no—
84C4×C199#12011yescharacteristic
85C4×C199#12112yes—
86C4×C199#12311yescharacteristic
87Dic199—42no—
88Dic199—41no—
89C2×C199—21no—
90C2×C199—22no—
91C2×C199—42no—
92C2×C199#38611yescharacteristic
93D199—42no—
94C199#189111yescharacteristic, complemented, Sylow
95#1891—1991nomaximal, Sylow
96#568D19911yescharacteristic, complemented
97#525—1991no—
98#413—1991no—
99#407—1992no—
100#344—1992no—
101C2×#41C2×D19911yescharacteristic
102C23×C4—1991no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1992no—
106C2×M16C2×D19912yes—
107#123—1992no—
108C22⋊Q8—3981no—
109#121—3981no—
110#120—1992no—
111C4×D4—3982no—
112#117—1991no—
113#101—3981no—
114#100—3981no—
115C22⋊C8—3984no—
116C24—1991no—
117#41—22no—
118#41—22no—
119C2×Q8D79611yescharacteristic
120C2×D4—3982no—
121C2×D4C4×D19912yes—
122C2×D4D79611yescharacteristic
123C22×C4—3982no—
124C22×C4—3981no—
125C22×C4—3981no—
126C22×C4—3982no—
127C22×C4—1991no—
128C22×C4C22⋊D19912yes—
129C22×C4C22×D19911yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—3982no—
134M16—3982no—
135C2×C8—24no—
136C2×C8—3982no—
137C2×C8C22⋊D19914yescomplemented
138C4⋊C4—3982no—
139C4⋊C4—3981no—
140C22⋊C4—3982no—
141C22⋊C4—3982no—
142C22⋊C4—1992no—
143C42—3981no—
144C23—7962no—
145C23—3981no—
146C23—3982no—
147C23(C2×D199)⋊C412yes—
148C23C2×C4×D19911yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—7964no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—7961no—
156C2×C4—7961no—
157C2×C4—7962no—
158C2×C4—3982no—
159C2×C4—3981no—
160C2×C4—3982no—
161C2×C4(C2×D199)⋊C412yes—
162C2×C4C2×C22⋊D19911yescharacteristic
163C2×C4C2×C22⋊D19912yes—
164C2×C4C2×D79611yescharacteristic
165C2×C4#1569188612yes—
166C8—24no—
167C8—7962no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—7961no—
172C22—7962no—
173C22—7961no—
174C22—3982no—
175C22—3984no—
176C22C2×(C2×D199)⋊C411yescharacteristic
177C22C4×C22⋊D19912yes—
178C4—22no—
179C4—7962no—
180C4—7961no—
181C4#2026392411yescharacteristic
182C4#2026392912yes—
183C4#2026393811yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—7962no—
188C2#2716389711yescharacteristic
subgroupG/H
centerC4#20263924
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#568×C199C2
socleC2×C199#386
prime radical#27164525C2
cosocleC22×C4×C199C23
hypercenter#568D199
hypocenterC199#1891
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27164002C2semidirect
3. #27163981C2semidirect
4. #27163789C2semidirect
5. #568×C199C2semidirect
48. C2×C8×C199D4semidirect
94. C199#1891semidirect
96. #568D199semidirect
137. C2×C8C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645252
#271640022
#271639812
#271637892
#568×C1992
#1891199