Finite Group Databasebeta

← #38741631

Subgroups — #38741631

Order 25216.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161637C211yesmaximal, characteristic, complemented
2#27161215C211yesmaximal, characteristic, complemented
3#27161212C211yesmaximal, characteristic, complemented
4#27160989C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8C22×D788—21no—
9C22×D788C2211yescharacteristic, complemented
10C2×D4×D197—21no—
11C2×D1576—21no—
12C2×#15691343—21no—
13#20263311—21no—
14#20263308—21no—
15#20263300C2211yescharacteristic
16#20263298—21no—
17#20263296—21no—
18#20263294—21no—
19C2×#15691335—21no—
20C2×#15691335C2211yescharacteristic
21#20263248—21no—
22#20263246—21no—
23C22⋊#13096802C2211yescharacteristic, complemented
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic, complemented
26#95×C197C2211yescharacteristic, complemented
27C23×D197—21no—
28C23×D197—21no—
29C23×D197D411yescharacteristic
30C2×D788—21no—
31C2×D788—21no—
32C2×D788—21no—
33C2×D788—42no—
34C2×D788—42no—
35C2×D788—42no—
36C2×D788—42no—
37C2×D788D411yescharacteristic, complemented
38C2×C22⋊D197—21no—
39C2×C22⋊D197—21no—
40C2×C22⋊D197—21no—
41D4×D197—42no—
42D4×D197—42no—
43C22×Dic197D411yescharacteristic
44C2×C4×D197—21no—
45D1576—42no—
46#15691343—42no—
47C2×#13096802—21no—
48C4⋊Dic197D411yescharacteristic, complemented
49#15691337—21no—
50C22⋊Dic197—21no—
51#15691335—21no—
52#15691335—21no—
53#15691335—21no—
54#15691335—21no—
55#15691335—42no—
56C2×D4×C197—21no—
57C2×D4×C197D411yescharacteristic, complemented
58C22×C4×C197—21no—
59C22×C4×C197C2311yescharacteristic
60C2×C8×C197—21no—
61C4⋊C4×C197D411yescharacteristic, complemented
62C22⋊C4×C197—21no—
63C22×D197—21no—
64C22×D197—21no—
65C22×D197—22no—
66C22×D197—21no—
67C22×D197—42no—
68C22×D197—41no—
69C22×D197—41no—
70C22×D197—41no—
71C22×D197—41no—
72C22×D197—41no—
73C22×D197—41no—
74C22×D197—41no—
75C22×D197—41no—
76C22×D197—81no—
77C22×D197—81no—
78C22×D197D812yescomplemented
79D788—22no—
80D788—22no—
81D788—41no—
82D788—41no—
83D788—41no—
84D788—42no—
85D788—48no—
86C22⋊D197—42no—
87C22⋊D197—41no—
88C22⋊D197—41no—
89C22⋊D197—81no—
90C2×Dic197—21no—
91C2×Dic197—21no—
92C2×Dic197—41no—
93C4×D197—42no—
94#13096802—41no—
95C23×C197—21no—
96C23×C197C2×D411yescharacteristic
97D4×C197—41no—
98D4×C197—41no—
99D4×C197—41no—
100C2×C4×C197—22no—
101C2×C4×C197—21no—
102C2×C4×C197—41no—
103C2×C4×C197—41no—
104C2×C4×C197—41no—
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197C2×D411yescharacteristic
107C8×C197—41no—
108C22×C197—21no—
109C22×C197—21no—
110C22×C197—41no—
111C22×C197—41no—
112C22×C197—41no—
113C22×C197#12011yescharacteristic
114C22×C197C2×D811yescharacteristic
115C22×C197AGL(1,ℤ/8)11yescharacteristic
116C2×D197—22no—
117C2×D197—41no—
118C2×D197—41no—
119C2×D197—42no—
120C2×D197—42no—
121C2×D197—42no—
122C2×D197—41no—
123C2×D197—41no—
124C2×D197—42no—
125C2×D197—41no—
126C2×D197—81no—
127C2×D197—84no—
128C2×D197—81no—
129C2×D197—81no—
130C2×D197—81no—
131C2×D197—81no—
132C4×C197—21no—
133C4×C197—21no—
134C4×C197—42no—
135C4×C197—41no—
136Dic197—41no—
137Dic197—41no—
138C2×C197—21no—
139C2×C197—21no—
140C2×C197—81no—
141C2×C197#44711yescharacteristic
142C2×C197#45311yescharacteristic
143C2×C197B(4,𝔽2)11yescharacteristic
144D197—42no—
145D197—82no—
146D197—81no—
147C197#159511yescharacteristic, complemented, Sylow
148#1595—1971nomaximal, Sylow
149#534—1971no—
150#447—1971no—
151#447—1971no—
152#392—1971no—
153#327—1971no—
154#327D19711yescharacteristic, complemented
155#323—1971no—
156C22×D4—3941no—
157C22×D4—3941no—
158C22×D4—1971no—
159C2×D8—3941no—
160C2×D8—3941no—
161#123—3941no—
162#123—3941no—
163#121—3941no—
164#121—1971no—
165#121C2×D19711yescharacteristic
166#120—3941no—
167#120—3941no—
168C2×C22⋊C4—3941no—
169C2×C22⋊C4—1971no—
170#102—3941no—
171#102—3941no—
172C22⋊C8—1971no—
173C22⋊C8C2×D19711yescharacteristic, complemented
174#95C2×D19711yescharacteristic, complemented
175C24—3941no—
176C24—3941no—
177C24—1971no—
178C2×D4—21no—
179C2×D4—7882no—
180C2×D4—7882no—
181C2×D4—7882no—
182C2×D4—7882no—
183C2×D4—7882no—
184C2×D4—7882no—
185C2×D4—3941no—
186C2×D4—3941no—
187C2×D4—3941no—
188C2×D4—3941no—
189C2×D4—3941no—
190C2×D4—3941no—
191C2×D4—1971no—
192C2×D4D78811yescharacteristic, complemented
193C22×C4—21no—
194C22×C4—3941no—
195C22×C4—1971no—
196C22×C4C22×D19711yescharacteristic
197D8—7882no—
198D8—7882no—
199C2×C8—21no—
200C2×C8—3941no—
201C4⋊C4—3941no—
202C4⋊C4—1971no—
203C4⋊C4D78811yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—7882no—
206C22⋊C4—3941no—
207C22⋊C4—3941no—
208C22⋊C4—3941no—
209C22⋊C4—3941no—
210C22⋊C4—3941no—
211C23—21no—
212C23—7881no—
213C23—7881no—
214C23—7881no—
215C23—7881no—
216C23—7881no—
217C23—7882no—
218C23—7881no—
219C23—7881no—
220C23—7881no—
221C23—15761no—
222C23—15761no—
223C23—3942no—
224C23—3941no—
225C23—3941no—
226C23—3941no—
227C23—1972no—
228C23D4×D19711yescharacteristic
229D4—41no—
230D4—41no—
231D4—41no—
232D4—7881no—
233D4—7881no—
234D4—7881no—
235D4—7888no—
236D4—7882no—
237D4—7882no—
238D4—7881no—
239D4—7881no—
240D4—15761no—
241D4—3942no—
242D4—3942no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—7882no—
249C2×C4—7881no—
250C2×C4—3941no—
251C2×C4—3941no—
252C2×C4D4×D19711yescharacteristic
253C2×C4C2×D78811yescharacteristic
254C8—41no—
255C8—7881no—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—7882no—
262C22—7882no—
263C22—7882no—
264C22—7881no—
265C22—7881no—
266C22—7881no—
267C22—7881no—
268C22—7882no—
269C22—7881no—
270C22—15761no—
271C22—15761no—
272C22—15761no—
273C22—15764no—
274C22—15761no—
275C22—15761no—
276C22—3942no—
277C22#2026329611yescharacteristic
278C22D8×D19711yescharacteristic
279C22#2026335311yescharacteristic
280C4—21no—
281C4—21no—
282C4—42no—
283C4—41no—
284C4—7881no—
285C4—7881no—
286C2—21no—
287C2—21no—
288C2—81no—
289C2—7882no—
290C2—15762no—
291C2—15761no—
292C2#2716121411yescharacteristic
293C2#2716124811yescharacteristic
294C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
hypercenter#327D197
hypocenterC197#1595
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #27161637C2semidirect
2. #27161215C2semidirect
3. #27161212C2semidirect
4. #27160989C2semidirect
5. #27160812C2semidirect
6. #27160802C2semidirect
7. #327×C197C2semidirect
9. C22×D788C22semidirect
23. C22⋊#13096802C22semidirect
25. C22⋊C8×C197C22semidirect
26. #95×C197C22semidirect
37. C2×D788D4semidirect
48. C4⋊Dic197D4semidirect
57. C2×D4×C197D4semidirect
61. C4⋊C4×C197D4semidirect
78. C22×D197D8semidirect
147. C197#1595semidirect
154. #327D197semidirect
173. C22⋊C8C2×D197semidirect
174. #95C2×D197semidirect
192. C2×D4D788semidirect
203. C4⋊C4D788semidirect

Maximal subgroups

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

subgroupindex
#271616372
#271612152
#271612122
#271609892
#271608122
#271608022
#327×C1972
#1595197