Finite Group Databasebeta

← #47986304

Subgroups — #47986304

Order 35456.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#22579848C211yesmaximal, characteristic
2#31799902C212yesmaximal, complemented
3#31799893C212yesmaximal, complemented
4#31799892C212yesmaximal, complemented
5#31799890C212yesmaximal, complemented
6#31799882C211yesmaximal, characteristic, complemented
7#31799879C212yesmaximal, complemented
8#31799857C212yesmaximal, complemented
9#390×C277C211yesmaximal, characteristic, complemented
10C24×D277C2211yescharacteristic
11C22×C22⋊D277C2212yes—
12C23×Dic277C2211yescharacteristic, complemented
13#22579848C418yescomplemented
14C2×#17984542C2212yescomplemented
15C2×C22⋊Dic277C2212yes—
16C2×C22⋊Dic277C2211yescharacteristic
17C2×C22⋊Dic277C2211yescharacteristic, complemented
18C2×#17984540C2212yes—
19C2×#17984540C2212yes—
20C2×#17984540C2212yes—
21C2×#17984540C2212yes—
22C2×#17984540C2212yes—
23C2×C4×Dic277C2212yescomplemented
24#22579787C2212yescomplemented
25#22579787C2212yescomplemented
26#22579787C2212yescomplemented
27C22×D4×C277C2211yescharacteristic
28C2×C4⋊C4×C277C2211yescharacteristic, complemented
29C2×C22⋊C4×C277C2212yes—
30C2×C22⋊C4×C277C2212yescomplemented
31C2×C42×C277C2211yescharacteristic, complemented
32C23×D277—42no—
33C23×D277—41no—
34C23×D277—41no—
35C23×D277—42no—
36C23×D277C2×C414yescomplemented
37C23×D277C2312yes—
38C2×C22⋊D277—42no—
39C2×C22⋊D277—42no—
40C2×C22⋊D277C2×C418yes—
41C22×Dic277—41no—
42C22×Dic277—42no—
43C22×Dic277C2312yes—
44C22×Dic277C2312yes—
45C22×Dic277C2312yes—
46#17984542—42no—
47C22⋊Dic277—42no—
48C22⋊Dic277—41no—
49C22⋊Dic277C2×C414yescomplemented
50#17984540—42no—
51#17984540—42no—
52#17984540—42no—
53#17984540—42no—
54#17984540C2×C418yescomplemented
55C4×Dic277—42no—
56C24×C277C2311yescharacteristic
57C24×C277C2311yescharacteristic
58C2×D4×C277—41no—
59C2×D4×C277—41no—
60C2×D4×C277C2×C414yes—
61C22×C4×C277C2312yes—
62C22×C4×C277C2312yes—
63C22×C4×C277C2311yescharacteristic
64C4⋊C4×C277—41no—
65C22⋊C4×C277—42no—
66C22⋊C4×C277—42no—
67C42×C277—41no—
68C22×D277—22no—
69C22×D277—22no—
70C22×D277—22no—
71C22×D277—22no—
72C22×D277—22no—
73C22×D277—22no—
74C22×D277—48no—
75C22×D277—44no—
76C22×D277—44no—
77C22×D277—48no—
78C22×D277—44no—
79C22×D277C22×C414yes—
80C22⋊D277—48no—
81C2×Dic277—22no—
82C2×Dic277—22no—
83C2×Dic277—22no—
84C2×Dic277—22no—
85C2×Dic277—22no—
86C2×Dic277—22no—
87C2×Dic277—22no—
88C2×Dic277—22no—
89C2×Dic277—44no—
90C2×Dic277C22×C414yes—
91C23×C277—22no—
92C23×C277—22no—
93C23×C277—21no—
94C23×C277—21no—
95C23×C277—21no—
96C23×C277—21no—
97C23×C277—22no—
98C23×C277—22no—
99C23×C277C22×C412yes—
100C23×C277C22×C412yes—
101C23×C277C2411yescharacteristic
102D4×C277—44no—
103C2×C4×C277—21no—
104C2×C4×C277—21no—
105C2×C4×C277—22no—
106C2×C4×C277—22no—
107C2×C4×C277—22no—
108C2×C4×C277—22no—
109C2×C4×C277—22no—
110C2×C4×C277—22no—
111C2×C4×C277C22×C412yes—
112C22×C277—22no—
113C22×C277—22no—
114C22×C277—24no—
115C22×C277—24no—
116C22×C277—41no—
117C22×C277—42no—
118C22×C277—41no—
119C22×C277—42no—
120C22×C277—41no—
121C22×C277—41no—
122C22×C277C23×C411yescharacteristic
123C22×C277H(5,𝔽2)12yes—
124C22×C277H(5,𝔽2)11yescharacteristic
125C22×C277H(5,𝔽2)11yescharacteristic
126C22×C277H(5,𝔽2)12yes—
127C2×D277—24no—
128C2×D277—24no—
129C2×D277—24no—
130C2×D277—416no—
131C2×D277—42no—
132C2×D277—42no—
133C2×D277—42no—
134C2×D277—42no—
135C4×C277—22no—
136C4×C277—42no—
137C4×C277—42no—
138Dic277—24no—
139Dic277—42no—
140Dic277—42no—
141C2×C277—42no—
142C2×C277—42no—
143C2×C277#51812yes—
144C2×C277#51811yescharacteristic
145C2×C277#56011yescharacteristic
146C2×C277#56012yes—
147C2×C277H(3,𝔽4)11yescharacteristic
148D277—44no—
149C277#237511yescharacteristic, complemented, Sylow
150#2375—2771nomaximal, Sylow
151C2×#120—2771no—
152#390—2772no—
153#390D27711yescharacteristic, complemented
154#388—2772no—
155#388—2772no—
156#388—2772no—
157#386—2771no—
158#386—2772no—
159#379—2772no—
160C25—2771no—
161C22×D4—2772no—
162C22×D4C2×D27711yescharacteristic
163C23×C4—2771no—
164#120—2778no—
165C2×C4⋊C4—2772no—
166C2×C4⋊C4C2×D27711yescharacteristic, complemented
167C2×C22⋊C4—2771no—
168C2×C22⋊C4—2772no—
169C2×C22⋊C4—2772no—
170C2×C22⋊C4—2771no—
171C2×C22⋊C4—2772no—
172C2×C22⋊C4—2772no—
173C2×C22⋊C4—2772no—
174C2×C22⋊C4—2772no—
175C2×C22⋊C4C2×D27712yes—
176C2×C22⋊C4C2×D27712yescomplemented
177C2×C42—2772no—
178C2×C42C2×D27711yescharacteristic, complemented
179#95—2772no—
180#95—2772no—
181#95—2772no—
182C24—2772no—
183C24—2774no—
184C24—11082no—
185C24—11081no—
186C24—11081no—
187C24—11082no—
188C24C22×D27711yescharacteristic
189C24C22×D27711yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—2778no—
193C2×D4—11082no—
194C2×D4—11082no—
195C2×D4C4×D27714yes—
196C22×C4—2772no—
197C22×C4—2772no—
198C22×C4—2772no—
199C22×C4—11081no—
200C22×C4—11082no—
201C22×C4C22×D27712yes—
202C22×C4C22×D27712yes—
203C22×C4C22×D27711yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—11082no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—2774no—
209C22⋊C4—2778no—
210C22⋊C4—11082no—
211C22⋊C4—11082no—
212C22⋊C4—11082no—
213C22⋊C4—11082no—
214C22⋊C4—11081no—
215C22⋊C4—11082no—
216C42—41no—
217C42—11082no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—2774no—
227C23—5542no—
228C23—5542no—
229C23—5542no—
230C23—5542no—
231C23—5542no—
232C23—5542no—
233C23—11088no—
234C23—11084no—
235C23—11084no—
236C23—11088no—
237C23—11084no—
238C23C2×C4×D27712yes—
239C23C2×C4×D27712yes—
240C23C23×D27711yescharacteristic
241D4—44no—
242D4—11088no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—2774no—
252C2×C4—5542no—
253C2×C4—5542no—
254C2×C4—5542no—
255C2×C4—5542no—
256C2×C4—5542no—
257C2×C4—5542no—
258C2×C4—5542no—
259C2×C4—5542no—
260C2×C4—11084no—
261C2×C4C2×C4×D27712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—5544no—
273C22—5544no—
274C22—5544no—
275C22—110816no—
276C22—11082no—
277C22—11082no—
278C22—11082no—
279C22—11082no—
280C22C22×C4×D27711yescharacteristic
281C22#2257992512yes—
282C22#2257992511yescharacteristic
283C22#2257992511yescharacteristic
284C22#2257992612yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—5544no—
289C4—11082no—
290C4—11082no—
291C2—42no—
292C2—42no—
293C2—11084no—
294C2#3180047412yes—
295C2#3180047511yescharacteristic
296C2#3180073911yescharacteristic
297C2#3180074012yes—
298C2#3180074411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D277
commutatorC22×C277C23×C4
Fitting#390×C277C2
socle, cosocleC23×C277C24
prime radicalC2×#22579848C2
hypercenter#390D277
hypocenterC277#2375
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #31799902C2semidirect
3. #31799893C2semidirect
4. #31799892C2semidirect
5. #31799890C2semidirect
6. #31799882C2semidirect
7. #31799879C2semidirect
8. #31799857C2semidirect
9. #390×C277C2semidirect
12. C23×Dic277C22semidirect
13. #22579848C4semidirect
14. C2×#17984542C22semidirect
17. C2×C22⋊Dic277C22semidirect
23. C2×C4×Dic277C22semidirect
24. #22579787C22semidirect
25. #22579787C22semidirect
26. #22579787C22semidirect
28. C2×C4⋊C4×C277C22semidirect
30. C2×C22⋊C4×C277C22semidirect
31. C2×C42×C277C22semidirect
36. C23×D277C2×C4semidirect
49. C22⋊Dic277C2×C4semidirect
54. #17984540C2×C4semidirect
149. C277#2375semidirect
153. #390D277semidirect
166. C2×C4⋊C4C2×D277semidirect
176. C2×C22⋊C4C2×D277semidirect
178. C2×C42C2×D277semidirect

Maximal subgroups

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

subgroupindex
C2×#225798482
#317999022
#317998932
#317998922
#317998902
#317998822
#317998792
#317998572
#390×C2772
#2375277