Finite Group Databasebeta

← #38770523

Subgroups — #38770523

Order 25472.

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

normal subgroups only

This group has 238 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#27164588C211yesmaximal, characteristic
2#27163990C212yesmaximal
3#27163813C211yesmaximal, characteristic, complemented
4#27163812C212yesmaximal, complemented
5#354×C199C211yesmaximal, characteristic, complemented
6C2×#15691885C2211yescharacteristic
7C2×Q8×D199C411yescharacteristic, complemented
8#20263955C411yescharacteristic, complemented
9#20263953C411yescharacteristic, complemented
10#20263942C2211yescharacteristic
11#20263937—22no—
12#20263932—22no—
13#20263912C411yescharacteristic, complemented
14#20263911—22no—
15#20263870C2212yes—
16#20263869—22no—
17#124×C199C2211yescharacteristic
18#99×C199C2212yes—
19#15691885—42no—
20C2×Dic398—21no—
21C2×Dic398C2×C411yescharacteristic
22C2×Dic398C2×C411yescharacteristic
23Q8×D199—42no—
24C2×C22⋊D199D412yes—
25C22×Dic199D412yes—
26C2×C4×D199—21no—
27C2×C4×D199C2×C411yescharacteristic
28C4⋊Dic199—22no—
29Dic199⋊C4—22no—
30Dic199⋊C4—21no—
31Dic199⋊C4—42no—
32C22⋊Dic199—22no—
33C22⋊Dic199—22no—
34(C2×D199)⋊C4—21no—
35C4×Dic199—42no—
36C4×Dic199C2×C411yescharacteristic
37C2×Q8×C199C2×C411yescharacteristic
38C2×D4×C199C2311yescharacteristic
39C22⋊C4×C199—22no—
40C22⋊C4×C199—22no—
41C42×C199C2×C411yescharacteristic
42C22×D199C22⋊C411yescharacteristic
43Dic398—41no—
44Dic398—42no—
45Dic398—41no—
46C22⋊D199—42no—
47C2×Dic199—22no—
48C2×Dic199—22no—
49C2×Dic199—21no—
50C2×Dic199—21no—
51C2×Dic199—42no—
52C2×Dic199C22⋊C412yes—
53C2×Dic199C22⋊C411yescharacteristic
54C4×D199—22no—
55C4×D199—42no—
56C23×C199C2×D412yes—
57Q8×C199—41no—
58D4×C199—41no—
59C2×C4×C199—21no—
60C2×C4×C199—21no—
61C2×C4×C199—42no—
62C2×C4×C199C22×C411yescharacteristic
63C22×C199—22no—
64C22×C199—42no—
65C22×C199C2×C22⋊C411yescharacteristic
66C2×D199—21no—
67C2×D199—41no—
68C4×C199—21no—
69C4×C199—41no—
70C4×C199—41no—
71C4×C199—82no—
72Dic199—22no—
73Dic199—41no—
74Dic199—41no—
75Dic199—82no—
76Dic199#9912yes—
77C2×C199—21no—
78C2×C199—42no—
79C2×C199C2×#9911yescharacteristic
80D199—41no—
81C199#212911yescharacteristic, complemented, Sylow
82#2129—1991nomaximal, Sylow
83#536—1991no—
84#410—1992no—
85#354—1991no—
86#354—1992no—
87#354D19911yescharacteristic, complemented
88C2×#41—1991no—
89C22×Q8—1991no—
90C4⋊Q8—1991no—
91C4⋊Q8—1991no—
92#124—1991no—
93#124C2×D19911yescharacteristic
94#123—3982no—
95C22⋊Q8—3982no—
96#117—3982no—
97#117—1991no—
98#99—3982no—
99#99—1992no—
100#99C2×D19912yes—
101#41—7962no—
102C2×Q8—7962no—
103C2×Q8—3981no—
104C2×Q8—1991no—
105C2×Q8—1991no—
106C2×Q8C4×D19911yescharacteristic
107C2×D4—1992no—
108C2×D4C22×D19911yescharacteristic
109C22×C4—3981no—
110C22×C4—1992no—
111C22×C4—1991no—
112C4⋊C4—7962no—
113C4⋊C4—3981no—
114C4⋊C4—3982no—
115C4⋊C4—3982no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—3981no—
119C22⋊C4—3982no—
120C22⋊C4—3982no—
121C42—7962no—
122C42—1991no—
123C42C4×D19911yescharacteristic
124C23—1991no—
125C23D4×D19912yes—
126Q8—41no—
127Q8—7961no—
128Q8—7961no—
129Q8—7962no—
130D4—41no—
131D4—7962no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—7962no—
136C2×C4—7962no—
137C2×C4—3982no—
138C2×C4—3981no—
139C2×C4—3981no—
140C2×C4—3982no—
141C2×C4—3982no—
142C2×C4—1992no—
143C2×C4—1991no—
144C2×C4C2×C4×D19911yescharacteristic
145C22—22no—
146C22—42no—
147C22—7961no—
148C22—3981no—
149C22C22⋊C4×D19911yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—7961no—
155C4—7961no—
156C4—15922no—
157C4—3982no—
158C4—1992no—
159C2—21no—
160C2—42no—
161C2—7961no—
162C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×D4C22×D199
Fitting#354×C199C2
socleC2×C199C2×#99
prime radicalC2×#15691885C22
cosocleC2×D4×C199C23
hypercenter#354D199
hypocenterC199#2129
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #27163813C2semidirect
4. #27163812C2semidirect
5. #354×C199C2semidirect
7. C2×Q8×D199C4semidirect
8. #20263955C4semidirect
9. #20263953C4semidirect
13. #20263912C4semidirect
81. C199#2129semidirect
87. #354D199semidirect

Maximal subgroups

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

subgroupindex
#271645882
#271639902
#271638132
#271638122
#354×C1992
#2129199