Finite Group Databasebeta

← #38770512

Subgroups — #38770512

Order 25472.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164566C211yesmaximal, characteristic
2#27163990C212yesmaximal, complemented
3#27163812C212yesmaximal, complemented
4#27163811C211yesmaximal, characteristic, complemented
5#353×C199C211yesmaximal, characteristic, complemented
6C2×#15691885C411yescharacteristic, complemented
7C2×#15691885C2211yescharacteristic
8#20263955C411yescharacteristic, complemented
9#20263942C411yescharacteristic, complemented
10#20263942C2211yescharacteristic
11#20263926—22no—
12D4×Dic199—22no—
13#20263911—22no—
14C42×D199C411yescharacteristic, complemented
15#20263870C2212yes—
16#20263869—22no—
17#127×C199C2211yescharacteristic
18#99×C199C2212yes—
19#15691885—41no—
20#15691885—41no—
21#15691885—41no—
22#15691885—41no—
23C2×Dic398C2×C411yescharacteristic
24C2×Dic398C2×C411yescharacteristic
25C2×C22⋊D199—21no—
26C2×C22⋊D199D412yes—
27C22×Dic199—21no—
28C22×Dic199D412yes—
29C2×C4×D199—21no—
30C2×C4×D199C2×C411yescharacteristic
31C4⋊Dic199—22no—
32Dic199⋊C4—42no—
33C22⋊Dic199—22no—
34C22⋊Dic199—22no—
35C22⋊Dic199—22no—
36C4×Dic199—21no—
37C4×Dic199—42no—
38C4×Dic199C2×C411yescharacteristic
39C2×D4×C199—22no—
40C2×D4×C199C2×C411yescharacteristic
41C2×D4×C199C2311yescharacteristic
42C22⋊C4×C199—22no—
43C42×C199C2×C411yescharacteristic
44C22×D199C22⋊C411yescharacteristic
45Dic398—41no—
46Dic398—41no—
47C22⋊D199—42no—
48C22⋊D199—42no—
49C2×Dic199—22no—
50C2×Dic199—22no—
51C2×Dic199—21no—
52C2×Dic199—21no—
53C2×Dic199—41no—
54C2×Dic199—41no—
55C2×Dic199—42no—
56C2×Dic199C22⋊C412yes—
57C2×Dic199C22⋊C411yescharacteristic
58C4×D199—21no—
59C4×D199—21no—
60C4×D199—42no—
61C23×C199—21no—
62C23×C199C2×D412yes—
63D4×C199—41no—
64D4×C199—41no—
65D4×C199—44no—
66C2×C4×C199—21no—
67C2×C4×C199—42no—
68C2×C4×C199C22×C411yescharacteristic
69C22×C199—22no—
70C22×C199—42no—
71C22×C199—42no—
72C22×C199—41no—
73C22×C199C2×C22⋊C411yescharacteristic
74C2×D199—21no—
75C2×D199—41no—
76C4×C199—22no—
77C4×C199—21no—
78C4×C199—82no—
79Dic199—22no—
80Dic199—41no—
81Dic199—41no—
82Dic199—82no—
83Dic199#9911yescharacteristic
84Dic199#9911yescharacteristic
85C2×C199—21no—
86C2×C199—42no—
87C2×C199—81no—
88C2×C199C2×#9911yescharacteristic
89D199—41no—
90C199#212611yescharacteristic, complemented, Sylow
91#2126—1991nomaximal, Sylow
92#532—1991no—
93#410—1992no—
94#354—1992no—
95#353—1991no—
96#353D19911yescharacteristic, complemented
97C2×#41—1991no—
98C2×#41—1991no—
99C4⋊Q8—1991no—
100#127C2×D19911yescharacteristic
101#124—1991no—
102#124—1991no—
103#121—3982no—
104C4×D4—3982no—
105#117—3982no—
106C2×C42—1991no—
107#99—3982no—
108#99—1992no—
109#99C2×D19912yes—
110#41—7961no—
111#41—7961no—
112#41—7961no—
113#41—7961no—
114C2×Q8—1991no—
115C2×Q8—1991no—
116C2×D4—22no—
117C2×D4—3981no—
118C2×D4—1992no—
119C2×D4C4×D19911yescharacteristic
120C2×D4C22×D19911yescharacteristic
121C22×C4—3981no—
122C22×C4—3981no—
123C22×C4—1992no—
124C22×C4—1991no—
125C4⋊C4—7962no—
126C4⋊C4—3982no—
127C22⋊C4—22no—
128C22⋊C4—3982no—
129C22⋊C4—3982no—
130C22⋊C4—3982no—
131C42—7962no—
132C42—3981no—
133C42—1991no—
134C42C4×D19911yescharacteristic
135C23—21no—
136C23—1991no—
137C23D4×D19912yes—
138Q8—7961no—
139Q8—7961no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—7962no—
144D4—7962no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—7962no—
148C2×C4—7961no—
149C2×C4—7961no—
150C2×C4—7962no—
151C2×C4—3981no—
152C2×C4—3981no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—3982no—
156C2×C4—3982no—
157C2×C4—1992no—
158C2×C4—1991no—
159C2×C4C2×C4×D19911yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—7961no—
165C22—3981no—
166C22C22⋊C4×D19911yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—7961no—
171C4—7961no—
172C4—15922no—
173C4—3982no—
174C4—1991no—
175C4—1991no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—7961no—
180C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#353×C199C2
socleC2×C199C2×#99
prime radical#27164566C2
cosocleC2×D4×C199C23
hypercenter#353D199
hypocenterC199#2126
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #27163990C2semidirect
3. #27163812C2semidirect
4. #27163811C2semidirect
5. #353×C199C2semidirect
6. C2×#15691885C4semidirect
8. #20263955C4semidirect
9. #20263942C4semidirect
14. C42×D199C4semidirect
90. C199#2126semidirect
96. #353D199semidirect

Maximal subgroups

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

subgroupindex
#271645662
#271639902
#271638122
#271638112
#353×C1992
#2126199