Finite Group Databasebeta

← #38768869

Subgroups — #38768869

Order 25472.

312 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 87 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#27164486C211yesmaximal, characteristic
2#27164041C211yesmaximal, characteristic, complemented
3C2×#20263883C211yesmaximal, characteristic
4C2×#20263881C211yesmaximal, characteristic
5C4×Dic199⋊C4C211yesmaximal, characteristic, complemented
6#27163777C211yesmaximal, characteristic, complemented
7#339×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9C2×#15691870C2211yescharacteristic
10#20263920C411yescharacteristic, complemented
11#20263920C411yescharacteristic, complemented
12#20263917C411yescharacteristic, complemented
13#20263917C411yescharacteristic, complemented
14(C22⋊D199)⋊C4—21no—
15(C22⋊D199)⋊C4—21no—
16#20263912—21no—
17C2×Dic199⋊C4—21no—
18C2×C4×Dic199C2211yescharacteristic
19C2×C4×Dic199C2211yescharacteristic
20C42×D199—21no—
21Dic199⋊C8—21no—
22Dic199⋊C8—21no—
23#20263883—22no—
24#20263881—22no—
25C199⋊#95—21no—
26C2×M16×C199C2211yescharacteristic
27#117×C199C2211yescharacteristic
28C2×C42×C199C2211yescharacteristic
29#15691886—21no—
30#15691886—21no—
31#15691886D412yes—
32#15691886Q812yes—
33C2×Dic398C2×C411yescharacteristic
34C2×D796C2×C411yescharacteristic
35C2×C22⋊D199—21no—
36C22×Dic199—21no—
37C22×Dic199D411yescharacteristic
38C22×Dic199D411yescharacteristic
39C2×C4×D199—21no—
40C2×C4×D199—21no—
41C2×C4×D199—21no—
42#15691870—21no—
43#15691870—21no—
44C2×C199⋊C8—21no—
45Dic199⋊C4—24no—
46Dic199⋊C4—21no—
47Dic199⋊C4—21no—
48(C2×D199)⋊C4—21no—
49(C2×D199)⋊C4—21no—
50C4×Dic199—21no—
51C4×Dic199—21no—
52C4×Dic199—21no—
53C4×Dic199—21no—
54C4×Dic199C2×C411yescharacteristic
55C4×Dic199C2×C411yescharacteristic
56C4×Dic199D411yescharacteristic
57C4×Dic199D411yescharacteristic
58C22×C4×C199—21no—
59C22×C4×C199C2311yescharacteristic
60M16×C199—21no—
61M16×C199—21no—
62C2×C8×C199—21no—
63C4⋊C4×C199C2×C411yescharacteristic
64C4⋊C4×C199C2×C411yescharacteristic
65C22⋊C4×C199—21no—
66C42×C199—21no—
67C42×C199—21no—
68C42×C199—21no—
69C22×D199—21no—
70Dic398—21no—
71Dic398C4⋊C412yescomplemented
72D796—21no—
73D796C4⋊C412yescomplemented
74C22⋊D199—22no—
75C22⋊D199—22no—
76C2×Dic199—21no—
77C2×Dic199—22no—
78C2×Dic199—22no—
79C2×Dic199—41no—
80C2×Dic199—41no—
81C2×Dic199—41no—
82C2×Dic199C22⋊C411yescharacteristic
83C2×Dic199C22⋊C411yescharacteristic
84C2×Dic199C22⋊C411yescharacteristic
85C2×Dic199C22⋊C411yescharacteristic
86C4×D199—22no—
87C4×D199—22no—
88C4×D199—42no—
89C4×D199—42no—
90C199⋊C8—41no—
91C23×C199C2×D411yescharacteristic
92C2×C4×C199—22no—
93C2×C4×C199—21no—
94C2×C4×C199—21no—
95C2×C4×C199—41no—
96C2×C4×C199—41no—
97C2×C4×C199C22×C411yescharacteristic
98C2×C4×C199C2×D411yescharacteristic
99C2×C4×C199C2×D411yescharacteristic
100C2×C4×C199C2×Q811yescharacteristic
101C2×C4×C199#4111yescharacteristic
102C2×C4×C199#4111yescharacteristic
103C8×C199—41no—
104C22×C199—21no—
105C22×C199—21no—
106C22×C199C2×C22⋊C411yescharacteristic
107C22×C199#12111yescharacteristic
108C22×C199C22⋊Q811yescharacteristic
109C2×D199—22no—
110C2×D199—41no—
111C2×D199—41no—
112C4×C199—42no—
113C4×C199—41no—
114C4×C199—41no—
115C4×C199C2×C4⋊C411yescharacteristic
116C4×C199#11711yescharacteristic
117C4×C199#12111yescharacteristic
118C4×C199C22⋊Q811yescharacteristic
119Dic199—22no—
120Dic199—21no—
121Dic199—21no—
122Dic199—21no—
123Dic199—42no—
124Dic199#10412yescomplemented
125C2×C199—21no—
126C2×C199—21no—
127C2×C199#38011yescharacteristic
128C2×C199C2×#10411yescharacteristic
129C2×C199#42111yescharacteristic
130D199—42no—
131C199#180611yescharacteristic, complemented, Sylow
132#1806—1991nomaximal, Sylow
133C4×#41—1991no—
134#423—1991no—
135C2×#104—1991no—
136C2×#104—1991no—
137C4×C4⋊C4—1991no—
138#339—1991no—
139#339D19911yescharacteristic, complemented
140C2×#41—1991no—
141C2×M16—1991no—
142C2×M16C2×D19911yescharacteristic
143C4×Q8—1991no—
144C4×Q8—1991no—
145C4×D4—3981no—
146C4×D4—3981no—
147C4×D4—1991no—
148C4×D4—1991no—
149#117—3981no—
150#117C2×D19911yescharacteristic
151C2×C4⋊C4—3981no—
152C2×C42—3981no—
153C2×C42—1991no—
154C2×C42—1991no—
155C2×C42C2×D19911yescharacteristic
156C4⋊C8—3981no—
157C4⋊C8—3981no—
158#104—3982no—
159#104—3982no—
160#95—3981no—
161#41—3981no—
162#41—3981no—
163#41—1992no—
164#41—1992no—
165C2×Q8—1991no—
166C2×D4—3981no—
167C2×D4—1991no—
168C22×C4—21no—
169C22×C4—3981no—
170C22×C4—3981no—
171C22×C4—3981no—
172C22×C4—3981no—
173C22×C4—1991no—
174C22×C4—1991no—
175C22×C4C22×D19911yescharacteristic
176M16—21no—
177M16—21no—
178M16—3981no—
179M16—3981no—
180C2×C8—21no—
181C2×C8—3981no—
182C4⋊C4—3984no—
183C4⋊C4—3981no—
184C4⋊C4—3981no—
185C4⋊C4C4×D19911yescharacteristic
186C4⋊C4C4×D19911yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—3981no—
189C22⋊C4—3981no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—3981no—
194C42—3981no—
195C42—3981no—
196C42—3981no—
197C42—1991no—
198C42—1991no—
199C42—1991no—
200C42—1991no—
201C23—3981no—
202C23D4×D19911yescharacteristic
203Q8—3981no—
204Q8—1992no—
205D4—3982no—
206D4—3982no—
207D4—3981no—
208D4—1992no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—7962no—
215C2×C4—7962no—
216C2×C4—7961no—
217C2×C4—7961no—
218C2×C4—7961no—
219C2×C4—3982no—
220C2×C4—3981no—
221C2×C4—3982no—
222C2×C4—3982no—
223C2×C4—3982no—
224C2×C4—1991no—
225C2×C4—1991no—
226C2×C4—1991no—
227C2×C4—1991no—
228C2×C4C2×C4×D19911yescharacteristic
229C2×C4D4×D19911yescharacteristic
230C2×C4D4×D19911yescharacteristic
231C2×C4Q8×D19911yescharacteristic
232C2×C4#1569188611yescharacteristic
233C2×C4#1569188611yescharacteristic
234C8—41no—
235C8—7961no—
236C22—21no—
237C22—21no—
238C22—7961no—
239C22—7961no—
240C22—3982no—
241C22C22⋊C4×D19911yescharacteristic
242C22#2026392811yescharacteristic
243C22#2026393311yescharacteristic
244C4—42no—
245C4—41no—
246C4—41no—
247C4—7962no—
248C4—3981no—
249C4—3981no—
250C4—3981no—
251C4—3982no—
252C4—1992no—
253C4C4⋊C4×D19911yescharacteristic
254C4#2026391211yescharacteristic
255C4#2026392811yescharacteristic
256C4#2026393311yescharacteristic
257C2—21no—
258C2—21no—
259C2—7962no—
260C2#2716387711yescharacteristic
261C2#104×D19911yescharacteristic
262C2#2716403311yescharacteristic
subgroupG/H
centerC2×C4D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#339×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC2×#15691886C22
cosocleC22×C4×C199C23
hypercenter#339D199
hypocenterC199#1806
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #27164041C2semidirect
5. C4×Dic199⋊C4C2semidirect
6. #27163777C2semidirect
7. #339×C199C2semidirect
10. #20263920C4semidirect
11. #20263920C4semidirect
12. #20263917C4semidirect
13. #20263917C4semidirect
71. Dic398C4⋊C4semidirect
73. D796C4⋊C4semidirect
124. Dic199#104semidirect
131. C199#1806semidirect
139. #339D199semidirect

Maximal subgroups

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

subgroupindex
#271644862
#271640412
C2×#202638832
C2×#202638812
C4×Dic199⋊C42
#271637772
#339×C1992
#1806199