Finite Group Databasebeta

← #38770543

Subgroups — #38770543

Order 25472.

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

normal subgroups only

This group has 230 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#27164593C211yesmaximal, characteristic
2#27164001C212yesmaximal
3#27163817C211yesmaximal, characteristic, complemented
4#27163816C212yesmaximal, complemented
5#356×C199C211yesmaximal, characteristic, complemented
6C2×#15691887C2211yescharacteristic
7C2×Q8×D199C411yescharacteristic
8M16×D199—22no—
9#20263953C2211yescharacteristic
10#20263944C411yescharacteristic
11#20263943C411yescharacteristic
12#20263940—22no—
13#20263933—22no—
14#20263912C411yescharacteristic
15#20263874C2212yes—
16#20263871—22no—
17C4⋊Q8×C199C2211yescharacteristic
18#101×C199C2212yes—
19#15691887—42no—
20C2×Dic398—21no—
21C2×Dic398C2×C411yescharacteristic
22Q8×D199—42no—
23C2×D796C2×C411yescharacteristic
24C2×D796D412yes—
25C2×C4×D199—21no—
26C2×C4×D199C2×C411yescharacteristic
27C2×C4×D199D412yes—
28#15691870—22no—
29#15691869—42no—
30C8×D199—42no—
31Dic199⋊C4—21no—
32Dic199⋊C4—22no—
33(C2×D199)⋊C4—21no—
34(C2×D199)⋊C4—22no—
35(C2×D199)⋊C4—22no—
36C4×Dic199C2×C411yescharacteristic
37C2×Q8×C199C2×C411yescharacteristic
38C2×Q8×C199C2311yescharacteristic
39M16×C199—22no—
40C4⋊C4×C199—22no—
41C42×C199C2×C411yescharacteristic
42C22×D199C22⋊C412yes—
43C22×D199C22⋊C411yescharacteristic
44Dic398—42no—
45Dic398—41no—
46D796—41no—
47D796—42no—
48C2×Dic199—21no—
49C2×Dic199—21no—
50C2×Dic199C22⋊C411yescharacteristic
51C4×D199—22no—
52C4×D199—22no—
53C4×D199—22no—
54C4×D199—42no—
55C199⋊C8—42no—
56Q8×C199—41no—
57Q8×C199—41no—
58C2×C4×C199—21no—
59C2×C4×C199—21no—
60C2×C4×C199C22×C411yescharacteristic
61C2×C4×C199C2×D412yes—
62C8×C199—42no—
63C22×C199C2×C22⋊C411yescharacteristic
64C2×D199—21no—
65C2×D199—22no—
66C2×D199—42no—
67C2×D199—41no—
68C4×C199—22no—
69C4×C199—21no—
70C4×C199—41no—
71C4×C199—41no—
72Dic199—41no—
73Dic199—41no—
74Dic199#9912yes—
75C2×C199—21no—
76C2×C199C2×#9911yescharacteristic
77D199—41no—
78D199—42no—
79C199#213511yescharacteristic, complemented, Sylow
80#2135—1991nomaximal, Sylow
81#536—1991no—
82#413—1992no—
83#356—1991no—
84#356—1992no—
85#356D19911yescharacteristic, complemented
86C2×#41—1991no—
87C22×Q8—1991no—
88C2×M16—3982no—
89C4⋊Q8—1991no—
90C4⋊Q8C2×D19911yescharacteristic
91#124—1991no—
92#124—1991no—
93#123—3982no—
94C22⋊Q8—3982no—
95#117—1991no—
96#101—1992no—
97#101C2×D19912yes—
98#100—3982no—
99#41—7962no—
100C2×Q8—7962no—
101C2×Q8—3981no—
102C2×Q8—1991no—
103C2×Q8C4×D19911yescharacteristic
104C2×Q8C22×D19911yescharacteristic
105C2×D4—1991no—
106C2×D4—1992no—
107C22×C4—3981no—
108C22×C4—1991no—
109C22×C4—1992no—
110M16—22no—
111M16—7962no—
112M16—3982no—
113C2×C8—7962no—
114C4⋊C4—22no—
115C4⋊C4—3981no—
116C4⋊C4—3982no—
117C22⋊C4—3981no—
118C22⋊C4—3982no—
119C22⋊C4—3982no—
120C42—1991no—
121C42C4×D19911yescharacteristic
122C23—1992no—
123C23—1991no—
124Q8—41no—
125Q8—41no—
126Q8—7961no—
127Q8—7962no—
128D4—7962no—
129D4—7961no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—7962no—
133C2×C4—3982no—
134C2×C4—3982no—
135C2×C4—3982no—
136C2×C4—3981no—
137C2×C4—3981no—
138C2×C4—1991no—
139C2×C4C2×C4×D19911yescharacteristic
140C2×C4D4×D19912yes—
141C8—42no—
142C8—7962no—
143C22—7962no—
144C22—7961no—
145C22—3981no—
146C22—3982no—
147C22C22⋊C4×D19911yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—7961no—
153C4—7961no—
154C4—1992no—
155C2—21no—
156C2—7961no—
157C2—7962no—
158C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×Q8C22×D199
Fitting#356×C199C2
socleC2×C199C2×#99
prime radicalC2×#15691887C22
cosocleC2×Q8×C199C23
hypercenter#356D199
hypocenterC199#2135
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #27163817C2semidirect
4. #27163816C2semidirect
5. #356×C199C2semidirect
79. C199#2135semidirect
85. #356D199semidirect

Maximal subgroups

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

subgroupindex
#271645932
#271640012
#271638172
#271638162
#356×C1992
#2135199