Finite Group Databasebeta

← #38770534

Subgroups — #38770534

Order 25472.

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

normal subgroups only

This group has 250 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#27164569C211yesmaximal, characteristic
2#27164001C212yesmaximal, complemented
3#27163817C211yesmaximal, characteristic, complemented
4#27163814C212yesmaximal, complemented
5#356×C199C211yesmaximal, characteristic, complemented
6C2×#15691887C411yescharacteristic
7C2×#15691887C2211yescharacteristic
8M16×D199—22no—
9#20263952C411yescharacteristic
10#20263944C411yescharacteristic
11#20263944C2211yescharacteristic
12#20263930—22no—
13#20263917—22no—
14C42×D199C411yescharacteristic
15#20263874C2212yes—
16#20263871—22no—
17C4⋊Q8×C199C2211yescharacteristic
18#101×C199C2212yes—
19#15691887—41no—
20#15691887—41no—
21#15691887—41no—
22#15691887—41no—
23C2×D796—22no—
24C2×D796—21no—
25C2×D796C2×C411yescharacteristic
26C2×D796C2×C411yescharacteristic
27C2×D796D412yes—
28C2×C4×D199—21no—
29C2×C4×D199—21no—
30C2×C4×D199C2×C411yescharacteristic
31C2×C4×D199D412yes—
32#15691870—22no—
33#15691869—42no—
34C8×D199—42no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37C4×Dic199—21no—
38C4×Dic199C2×C411yescharacteristic
39C2×Q8×C199C2×C411yescharacteristic
40C2×Q8×C199C2311yescharacteristic
41M16×C199—22no—
42C4⋊C4×C199—22no—
43C42×C199C2×C411yescharacteristic
44C22×D199—21no—
45C22×D199C22⋊C412yes—
46C22×D199C22⋊C411yescharacteristic
47D796—41no—
48D796—42no—
49D796—41no—
50D796—44no—
51D796—42no—
52C2×Dic199—21no—
53C2×Dic199C22⋊C411yescharacteristic
54C4×D199—22no—
55C4×D199—22no—
56C4×D199—21no—
57C4×D199—21no—
58C4×D199—42no—
59C4×D199—41no—
60C4×D199—41no—
61C199⋊C8—42no—
62Q8×C199—41no—
63Q8×C199—41no—
64C2×C4×C199—21no—
65C2×C4×C199—21no—
66C2×C4×C199C22×C411yescharacteristic
67C2×C4×C199C2×D412yes—
68C8×C199—42no—
69C22×C199C2×C22⋊C411yescharacteristic
70C2×D199—21no—
71C2×D199—22no—
72C2×D199—41no—
73C2×D199—42no—
74C2×D199—42no—
75C2×D199—41no—
76C4×C199—22no—
77C4×C199—22no—
78C4×C199—21no—
79C4×C199—41no—
80Dic199—41no—
81Dic199#9911yescharacteristic
82Dic199#9911yescharacteristic
83C2×C199—21no—
84C2×C199C2×#9911yescharacteristic
85D199—41no—
86D199—42no—
87D199—81no—
88C199#213211yescharacteristic, complemented, Sylow
89#2132—1991nomaximal, Sylow
90#532—1991no—
91#413—1992no—
92#356—1991no—
93#356D19911yescharacteristic, complemented
94#355—1992no—
95C2×#41—1991no—
96C2×#41—1991no—
97C2×M16—3982no—
98C4⋊Q8C2×D19911yescharacteristic
99#127—1991no—
100#124—1991no—
101#124—1991no—
102#121—3982no—
103C4×D4—3982no—
104C2×C42—1991no—
105#101—1992no—
106#101C2×D19912yes—
107#100—3982no—
108#41—7961no—
109#41—7961no—
110#41—7961no—
111#41—7961no—
112C2×Q8C4×D19911yescharacteristic
113C2×Q8C22×D19911yescharacteristic
114C2×D4—3982no—
115C2×D4—3981no—
116C2×D4—1991no—
117C2×D4—1991no—
118C2×D4—1992no—
119C22×C4—3981no—
120C22×C4—3981no—
121C22×C4—1991no—
122C22×C4—1992no—
123M16—22no—
124M16—7962no—
125M16—3982no—
126C2×C8—7962no—
127C4⋊C4—22no—
128C22⋊C4—3982no—
129C22⋊C4—3982no—
130C42—3981no—
131C42—1991no—
132C42C4×D19911yescharacteristic
133C23—3981no—
134C23—1992no—
135C23—1991no—
136Q8—41no—
137Q8—41no—
138D4—7961no—
139D4—7964no—
140D4—7962no—
141D4—7961no—
142D4—7962no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—7962no—
146C2×C4—7961no—
147C2×C4—7961no—
148C2×C4—3982no—
149C2×C4—3982no—
150C2×C4—3981no—
151C2×C4—3981no—
152C2×C4—3981no—
153C2×C4—1991no—
154C2×C4C2×C4×D19911yescharacteristic
155C2×C4D4×D19912yes—
156C8—42no—
157C8—7962no—
158C22—7961no—
159C22—7962no—
160C22—7962no—
161C22—7961no—
162C22—3981no—
163C22—3982no—
164C22C22⋊C4×D19911yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—7961no—
170C4—1991no—
171C4—1991no—
172C2—21no—
173C2—7961no—
174C2—7962no—
175C2—15921no—
176C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×Q8C22×D199
Fitting#356×C199C2
socleC2×C199C2×#99
prime radical#27164569C2
cosocleC2×Q8×C199C23
hypercenter#356D199
hypocenterC199#2132
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
2. #27164001C2semidirect
3. #27163817C2semidirect
4. #27163814C2semidirect
5. #356×C199C2semidirect
88. C199#2132semidirect
93. #356D199semidirect

Maximal subgroups

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

subgroupindex
#271645692
#271640012
#271638172
#271638142
#356×C1992
#2132199