Finite Group Databasebeta

← #38770537

Subgroups — #38770537

Order 25472.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164592C211yesmaximal, characteristic
2C2×#20263874C211yesmaximal, characteristic, complemented
3#27163816C212yesmaximal, complemented
4#27163814C212yesmaximal, complemented
5C2×#101×C199C211yesmaximal, characteristic, complemented
6C2×#15691886C411yescharacteristic
7C2×#15691870—21no—
8#20263953C411yescharacteristic
9#20263953C2211yescharacteristic
10#20263944C411yescharacteristic
11#20263944C2211yescharacteristic
12#20263938—21no—
13#20263938—21no—
14#20263931—21no—
15#20263931—21no—
16#20263914C411yescharacteristic
17#20263874—21no—
18#20263874C2212yes—
19C22×Q8×C199C2211yescharacteristic
20C2×M16×C199—21no—
21#101×C199—21no—
22#101×C199C2212yes—
23#15691886—41no—
24#15691886—41no—
25C2×Dic398C2×C411yescharacteristic
26C2×D796C2×C411yescharacteristic
27C2×C22⋊D199—21no—
28C2×C4×D199—21no—
29#15691870—22no—
30#15691870—41no—
31C2×C199⋊C8—41no—
32C4⋊Dic199—21no—
33Dic199⋊C4—21no—
34Dic199⋊C4—21no—
35Dic199⋊C4—21no—
36Dic199⋊C4—21no—
37C22⋊Dic199—21no—
38(C2×D199)⋊C4—21no—
39(C2×D199)⋊C4—21no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42C4×Dic199C2×C411yescharacteristic
43C4×Dic199C2×C411yescharacteristic
44C2×Q8×C199—41no—
45C2×Q8×C199—41no—
46C2×Q8×C199C2×C411yescharacteristic
47C2×Q8×C199D411yescharacteristic
48C2×Q8×C199D411yescharacteristic
49C2×Q8×C199C2311yescharacteristic
50C22×C4×C199C2×C411yescharacteristic
51C22×C4×C199D411yescharacteristic
52C22×C4×C199D411yescharacteristic
53M16×C199—22no—
54M16×C199—41no—
55C2×C8×C199—41no—
56C22×D199—21no—
57Dic398—41no—
58D796—41no—
59C22⋊D199—41no—
60C22⋊D199—41no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—21no—
64C4×D199—42no—
65C199⋊C8—42no—
66C23×C199C22⋊C411yescharacteristic
67Q8×C199—41no—
68Q8×C199—41no—
69Q8×C199—41no—
70Q8×C199—41no—
71C2×C4×C199—21no—
72C2×C4×C199—21no—
73C2×C4×C199—21no—
74C2×C4×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199—21no—
77C2×C4×C199C22⋊C411yescharacteristic
78C2×C4×C199C22⋊C411yescharacteristic
79C2×C4×C199C22⋊C411yescharacteristic
80C2×C4×C199C22×C411yescharacteristic
81C2×C4×C199C2×D411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C8×C199—42no—
84C22×C199—41no—
85C22×C199#9911yescharacteristic
86C22×C199#9911yescharacteristic
87C22×C199C2×C22⋊C411yescharacteristic
88C2×D199—42no—
89C2×D199—41no—
90C4×C199—21no—
91C4×C199—21no—
92C4×C199—21no—
93C4×C199—21no—
94C4×C199—21no—
95C4×C199—21no—
96Dic199—41no—
97Dic199—41no—
98Dic199—41no—
99C2×C199—21no—
100C2×C199—21no—
101C2×C199—21no—
102C2×C199C2×#9911yescharacteristic
103D199—81no—
104C199#213311yescharacteristic, complemented, Sylow
105#2133—1991nomaximal, Sylow
106#536—1991no—
107C2×#101—1991no—
108C2×#101D19911yescharacteristic, complemented
109#356—1992no—
110#355—1992no—
111C2×#41—1991no—
112C22×Q8C2×D19911yescharacteristic
113C2×M16—21no—
114C2×M16—3981no—
115C4⋊Q8—1991no—
116C4⋊Q8—1991no—
117#124—1991no—
118#124—1991no—
119#123—3981no—
120#123—3981no—
121C22⋊Q8—3981no—
122C22⋊Q8—3981no—
123#117—1991no—
124#101—21no—
125#101—3981no—
126#101—1992no—
127#101C2×D19912yes—
128#41—7961no—
129#41—7961no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—1991no—
133C2×Q8C4×D19911yescharacteristic
134C2×Q8C22⋊D19911yescharacteristic
135C2×Q8D79611yescharacteristic
136C2×Q8C22×D19911yescharacteristic
137C2×D4—3981no—
138C2×D4—1991no—
139C22×C4—3981no—
140C22×C4C4×D19911yescharacteristic
141C22×C4C22⋊D19911yescharacteristic
142C22×C4D79611yescharacteristic
143M16—22no—
144M16—41no—
145M16—7961no—
146M16—3982no—
147C2×C8—41no—
148C2×C8—7961no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C4⋊C4—3981no—
152C4⋊C4—3981no—
153C4⋊C4—3981no—
154C22⋊C4—3981no—
155C22⋊C4—3981no—
156C22⋊C4—3981no—
157C22⋊C4—3981no—
158C22⋊C4—3981no—
159C42—1991no—
160C42—1991no—
161C23—3981no—
162C23(C2×D199)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—7961no—
168D4—7961no—
169D4—7961no—
170D4—7961no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—7962no—
178C2×C4—3981no—
179C2×C4—3981no—
180C2×C4—3981no—
181C2×C4(C2×D199)⋊C411yescharacteristic
182C2×C4(C2×D199)⋊C411yescharacteristic
183C2×C4(C2×D199)⋊C411yescharacteristic
184C2×C4C2×C4×D19911yescharacteristic
185C2×C4C2×C22⋊D19911yescharacteristic
186C2×C4C2×D79611yescharacteristic
187C8—42no—
188C8—7962no—
189C22—41no—
190C22—7962no—
191C22—7961no—
192C22#2026386911yescharacteristic
193C22#2026386911yescharacteristic
194C22C2×(C2×D199)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—7961no—
202C4—7961no—
203C4—7961no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—15921no—
208C2C2×#2026386911yescharacteristic
subgroupG/H
centerC2C2×#20263869
commutatorC2×C4×C199C22×C4
FrattiniC2×Q8C22×D199
FittingC2×#101×C199C2
socleC2×C199C2×#99
prime radicalC2×#15691886C4
cosocleC2×Q8×C199C23
hypercenterC2×#101D199
hypocenterC199#2133
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#20263874C2semidirect
3. #27163816C2semidirect
4. #27163814C2semidirect
5. C2×#101×C199C2semidirect
104. C199#2133semidirect
108. C2×#101D199semidirect

Maximal subgroups

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

subgroupindex
#271645922
C2×#202638742
#271638162
#271638142
C2×#101×C1992
#2133199