Finite Group Databasebeta

← #38769132

Subgroups — #38769132

Order 25472.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#27164491C211yesmaximal, characteristic
2C2×#20263869C211yesmaximal, characteristic
3#27163895C211yesmaximal, characteristic
4#27163893C211yesmaximal, characteristic, complemented
5#27163890C211yesmaximal, characteristic, complemented
6#27163784C211yesmaximal, characteristic, complemented
7#342×C199C211yesmaximal, characteristic, complemented
8C22×C22⋊D199C2211yescharacteristic
9C23×Dic199C2211yescharacteristic
10C4×C22⋊D199—21no—
11C4×C22⋊D199—21no—
12(C22⋊D199)⋊C4—21no—
13(C22⋊D199)⋊C4—21no—
14#20263911—21no—
15C2×C4⋊Dic199—21no—
16C2×Dic199⋊C4—21no—
17C2×C22⋊Dic199C2211yescharacteristic
18C2×C22⋊Dic199C2211yescharacteristic
19C2×(C2×D199)⋊C4—21no—
20C22⋊C4×D199—21no—
21#20263869C414yescomplemented
22C199⋊#95—21no—
23C199⋊#95—21no—
24C199⋊#95—21no—
25C2×C22⋊C4×C199C2211yescharacteristic
26C2×C22⋊C4×C199C2211yescharacteristic
27C2×C22⋊C4×C199C2211yescharacteristic
28C23×D199D411yescharacteristic
29C2×C22⋊D199—21no—
30C2×C22⋊D199—21no—
31C2×C22⋊D199—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199C2×C412yes—
34C2×C22⋊D199Q812yes—
35C22×Dic199—21no—
36C22×Dic199—21no—
37C22×Dic199—21no—
38C22×Dic199—21no—
39C22×Dic199—42no—
40C22×Dic199D411yescharacteristic
41C22×Dic199D411yescharacteristic
42C22×Dic199D411yescharacteristic
43C2×C4×D199—21no—
44C2×C4×D199—21no—
45C4⋊Dic199—42no—
46Dic199⋊C4—21no—
47Dic199⋊C4—21no—
48Dic199⋊C4—42no—
49C22⋊Dic199—21no—
50C22⋊Dic199—21no—
51C22⋊Dic199—21no—
52C22⋊Dic199C2×C412yescomplemented
53(C2×D199)⋊C4—21no—
54(C2×D199)⋊C4—21no—
55(C2×D199)⋊C4—42no—
56C4×Dic199—21no—
57C4×Dic199—21no—
58C24×C199C2311yescharacteristic
59C22×C4×C199—21no—
60C22×C4×C199—21no—
61C22×C4×C199D411yescharacteristic
62C22×C4×C199D411yescharacteristic
63C22⋊C4×C199—21no—
64C22⋊C4×C199—21no—
65C22⋊C4×C199—21no—
66C22⋊C4×C199—21no—
67C22⋊C4×C199—21no—
68C22⋊C4×C199C2×C412yescomplemented
69C22×D199—21no—
70C22×D199—21no—
71C22×D199—41no—
72C22×D199—41no—
73C22×D199C4⋊C412yescomplemented
74C22⋊D199—22no—
75C22⋊D199—22no—
76C22⋊D199—22no—
77C22⋊D199—22no—
78C2×Dic199—21no—
79C2×Dic199—21no—
80C2×Dic199—22no—
81C2×Dic199—22no—
82C2×Dic199—22no—
83C2×Dic199—22no—
84C2×Dic199—21no—
85C2×Dic199—21no—
86C2×Dic199—41no—
87C2×Dic199—41no—
88C2×Dic199—41no—
89C2×Dic199—41no—
90C2×Dic199—41no—
91C2×Dic199—41no—
92C2×Dic199—41no—
93C2×Dic199—41no—
94C2×Dic199C4⋊C412yescomplemented
95C4×D199—81no—
96C4×D199—81no—
97C23×C199—41no—
98C23×C199—41no—
99C23×C199C22×C411yescharacteristic
100C23×C199C2×D411yescharacteristic
101C23×C199C2×D411yescharacteristic
102C23×C199C2×D411yescharacteristic
103C23×C199C2×Q811yescharacteristic
104C23×C199#4111yescharacteristic
105C23×C199#4111yescharacteristic
106C2×C4×C199—22no—
107C2×C4×C199—21no—
108C2×C4×C199—21no—
109C2×C4×C199—41no—
110C2×C4×C199—41no—
111C2×C4×C199—41no—
112C2×C4×C199—41no—
113C2×C4×C199—41no—
114C2×C4×C199—41no—
115C2×C4×C199—41no—
116C22×C199—21no—
117C22×C199—21no—
118C22×C199—21no—
119C22×C199—21no—
120C22×C199—21no—
121C22×C199—21no—
122C22×C199—41no—
123C22×C199—41no—
124C22×C199—41no—
125C22×C199—41no—
126C22×C199C2×C4⋊C411yescharacteristic
127C22×C199C4×D411yescharacteristic
128C22×C199C4×D411yescharacteristic
129C22×C199#12011yescharacteristic
130C22×C199C22⋊Q811yescharacteristic
131C22×C199C22⋊Q811yescharacteristic
132C22×C199#12311yescharacteristic
133C2×D199—22no—
134C2×D199—41no—
135C2×D199—41no—
136C2×D199—42no—
137C2×D199—81no—
138C4×C199—42no—
139C4×C199—41no—
140C4×C199—41no—
141C4×C199—81no—
142Dic199—22no—
143Dic199—42no—
144Dic199—41no—
145Dic199—42no—
146Dic199—41no—
147C2×C199—21no—
148C2×C199—21no—
149C2×C199—21no—
150C2×C199—21no—
151C2×C199—21no—
152C2×C199—21no—
153C2×C199#38511yescharacteristic
154C2×C199B(4,𝔽2)11yescharacteristic
155C2×C199#45811yescharacteristic
156D199—42no—
157C199#186711yescharacteristic, complemented, Sylow
158#1867—1991nomaximal, Sylow
159#518—1991no—
160C2×#99—1991no—
161#386—1991no—
162#385—1991no—
163#385—1991no—
164#342—1991no—
165#342D19911yescharacteristic, complemented
166C22×D4—1991no—
167C23×C4—1991no—
168C4×D4—3981no—
169C4×D4—3981no—
170C4×D4—3981no—
171C4×D4—3981no—
172#117—3981no—
173C2×C4⋊C4—3981no—
174C2×C4⋊C4—3981no—
175C2×C22⋊C4—3981no—
176C2×C22⋊C4—3981no—
177C2×C22⋊C4—1991no—
178C2×C22⋊C4—1991no—
179C2×C22⋊C4C2×D19911yescharacteristic
180C2×C22⋊C4C2×D19911yescharacteristic
181C2×C22⋊C4C2×D19911yescharacteristic
182#99—1994no—
183#95—3981no—
184#95—3981no—
185#95—3981no—
186C24—1991no—
187C24C22×D19911yescharacteristic
188C2×D4—3981no—
189C2×D4—3981no—
190C2×D4—3981no—
191C2×D4—3981no—
192C2×D4—1992no—
193C2×D4—1992no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—7962no—
197C22×C4—3981no—
198C22×C4—3981no—
199C22×C4—3981no—
200C22×C4—3981no—
201C22×C4—3981no—
202C22×C4—3981no—
203C22×C4—1991no—
204C22×C4—1991no—
205C22×C4—1991no—
206C22×C4D79611yescharacteristic
207C22×C4D79611yescharacteristic
208C4⋊C4—7962no—
209C4⋊C4—7962no—
210C4⋊C4—3981no—
211C4⋊C4—3981no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—7962no—
218C22⋊C4—3981no—
219C22⋊C4—3981no—
220C22⋊C4—3981no—
221C22⋊C4—3981no—
222C22⋊C4—3981no—
223C22⋊C4—1992no—
224C22⋊C4C4×D19912yescomplemented
225C42—3981no—
226C42—3981no—
227C23—41no—
228C23—41no—
229C23—7961no—
230C23—7961no—
231C23—3981no—
232C23—3981no—
233C23—1992no—
234C23C2×C4×D19911yescharacteristic
235C23D4×D19911yescharacteristic
236C23D4×D19911yescharacteristic
237C23C2×D79611yescharacteristic
238C23Q8×D19911yescharacteristic
239C23#1569188511yescharacteristic
240C23#1569188611yescharacteristic
241D4—3982no—
242D4—3982no—
243D4—3982no—
244D4—3982no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—7961no—
256C2×C4—7961no—
257C2×C4—7961no—
258C2×C4—7961no—
259C2×C4—7961no—
260C2×C4—7961no—
261C2×C4—7961no—
262C2×C4—7961no—
263C2×C4—15921no—
264C2×C4—15921no—
265C2×C4—3982no—
266C2×C4—3982no—
267C2×C4—3982no—
268C2×C4—3982no—
269C2×C4—3981no—
270C2×C4—3981no—
271C2×C4—3981no—
272C2×C4—3981no—
273C2×C4—1992no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—7962no—
285C22—7961no—
286C22—7961no—
287C22—15921no—
288C22—3982no—
289C22C4⋊C4×D19911yescharacteristic
290C22(C22⋊D199)⋊C411yescharacteristic
291C22C4×D79611yescharacteristic
292C22#2026392511yescharacteristic
293C22#2026393311yescharacteristic
294C22#2026393511yescharacteristic
295C22#2026393711yescharacteristic
296C4—42no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—7962no—
301C4—7962no—
302C4—7961no—
303C4—7961no—
304C4—3982no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—7962no—
312C2#2716389211yescharacteristic
313C2#2716420611yescharacteristic
314C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutatorC23×C199C22×C4
FrattiniC24C22×D199
Fitting#342×C199C2
socleC22×C199#120
prime radicalC22×C22⋊D199C22
cosocleC24×C199C23
hypercenter#342D199
hypocenterC199#1867
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #27163893C2semidirect
5. #27163890C2semidirect
6. #27163784C2semidirect
7. #342×C199C2semidirect
21. #20263869C4semidirect
52. C22⋊Dic199C2×C4semidirect
68. C22⋊C4×C199C2×C4semidirect
73. C22×D199C4⋊C4semidirect
94. C2×Dic199C4⋊C4semidirect
157. C199#1867semidirect
165. #342D199semidirect
224. C22⋊C4C4×D199semidirect

Maximal subgroups

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

subgroupindex
#271644912
C2×#202638692
#271638952
#271638932
#271638902
#271637842
#342×C1992
#1867199