Finite Group Databasebeta

← #38769266

Subgroups — #38769266

Order 25472.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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#27164785C211yesmaximal, characteristic, complemented
2#27164525C211yesmaximal, characteristic
3#27163981C212yesmaximal, complemented
4#27163790C212yesmaximal, complemented
5#413×C199C211yesmaximal, characteristic, complemented
6C22×C4×D199C2211yescharacteristic
7#20263962—24no—
8C2×#15691870—22no—
9C2×#15691870C2212yes—
10#20263936C412yes—
11#20263931—21no—
12#20263926—21no—
13#20263923C412yes—
14C4×C22⋊D199—22no—
15#20263914C2211yescharacteristic
16(C2×D199)⋊C8—24no—
17C2×#41×C199C2211yescharacteristic
18C2×M16×C199C2212yes—
19#101×C199—21no—
20#100×C199—21no—
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—24no—
29#15691870—22no—
30#15691870—22no—
31C2×C199⋊C8—24no—
32C2×C199⋊C8D414yescomplemented
33C4⋊Dic199—21no—
34Dic199⋊C4—22no—
35C22⋊Dic199—22no—
36C22⋊Dic199C2×C412yes—
37(C2×D199)⋊C4—22no—
38C4×Dic199—21no—
39#41×C199—22no—
40#41×C199—22no—
41C2×Q8×C199D411yescharacteristic
42C2×D4×C199C2×C412yes—
43C2×D4×C199D411yescharacteristic
44C22×C4×C199D412yes—
45C22×C4×C199C2311yescharacteristic
46M16×C199—22no—
47M16×C199—22no—
48C2×C8×C199—22no—
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—24no—
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—42no—
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#568—1991no—
97#525—1991no—
98#413D19911yescharacteristic, complemented
99#407—1992no—
100#344—1992no—
101C2×#41C2×D19911yescharacteristic
102C23×C4—1991no—
103#131—3984no—
104C2×M16—3982no—
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—21no—
114#100—21no—
115C22⋊C8—3984no—
116C24—1991no—
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D19911yescharacteristic
120C2×D4—3982no—
121C2×D4C4×D19912yes—
122C2×D4C22⋊D19911yescharacteristic
123C22×C4—3982no—
124C22×C4—3981no—
125C22×C4—3981no—
126C22×C4—3982no—
127C22×C4—1991no—
128C22×C4D79612yes—
129C22×C4C22×D19911yescharacteristic
130M16—22no—
131M16—22no—
132M16—3984no—
133M16—3982no—
134M16—3982no—
135C2×C8—22no—
136C2×C8—3984no—
137C2×C8—1994no—
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×C4D4×D19912yes—
163C2×C4C2×C22⋊D19911yescharacteristic
164C2×C4C2×D79611yescharacteristic
165C2×C4#1569188512yes—
166C8—42no—
167C8—3984no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—7961no—
172C22—7962no—
173C22—7961no—
174C22—3982no—
175C22—3984no—
176C22C2×(C2×D199)⋊C411yescharacteristic
177C22(C22⋊D199)⋊C412yes—
178C4—22no—
179C4—7962no—
180C4—7961no—
181C4#2026392511yescharacteristic
182C4#2026392712yes—
183C4#2026393911yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—7962no—
188C2#2716389511yescharacteristic
subgroupG/H
centerC4#20263925
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#413×C199C2
socleC2×C199#386
prime radical#27164525C2
cosocleC22×C4×C199C23
hypercenter#413D199
hypocenterC199#1891
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #27164785C2semidirect
3. #27163981C2semidirect
4. #27163790C2semidirect
5. #413×C199C2semidirect
32. C2×C199⋊C8D4semidirect
94. C199#1891semidirect
98. #413D199semidirect

Maximal subgroups

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

subgroupindex
#271647852
#271645252
#271639812
#271637902
#413×C1992
#1891199