Finite Group Databasebeta

← #28340097

Subgroups — #28340097

Order 13696.

1348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×D4×D107C211yesmaximal, characteristic, complemented
2C2×#16812106C211yesmaximal, characteristic, complemented
3C2×#16812105C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic, complemented
5C2×#16812079C214yesmaximal, complemented
6#20292536C214yesmaximal, complemented
7#20292528C211yesmaximal, characteristic, complemented
8C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
9#390×C107C211yesmaximal, characteristic, complemented
10C24×D107C2212yes—
11C22×D428C2212yes—
12C22×D428C2211yescharacteristic
13C22×D428C2211yescharacteristic, complemented
14C22×D428C2214yescomplemented
15C22×C22⋊D107C2212yes—
16C22×C22⋊D107C2214yescomplemented
17C2×D4×D107—24no—
18C2×D4×D107—24no—
19C2×D4×D107—24no—
20C22×C4×D107C2211yescharacteristic
21#16812106—24no—
22#16812105—24no—
23#16812084—24no—
24#16812079—216no—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yescomplemented
27C2×(C2×D107)⋊C4C2214yescomplemented
28C2×C4×Dic107C2211yescharacteristic, complemented
29C107⋊#95C2212yescomplemented
30C22×D4×C107C2211yescharacteristic
31C2×C4⋊C4×C107C2211yescharacteristic, complemented
32C2×C22⋊C4×C107C2214yescomplemented
33C2×C42×C107C2211yescharacteristic, complemented
34C23×D107—24no—
35C23×D107—24no—
36C23×D107—24no—
37C23×D107—28no—
38C23×D107—44no—
39C23×D107C2312yes—
40C23×D107C2312yes—
41C23×D107C2311yescharacteristic
42C2×D428—22no—
43C2×D428—22no—
44C2×D428—28no—
45C2×D428—24no—
46C2×D428—28no—
47C2×D428—22no—
48C2×D428—22no—
49C2×D428—22no—
50C2×D428—28no—
51C2×D428—44no—
52C2×D428D414yescomplemented
53C2×C22⋊D107—24no—
54C2×C22⋊D107—24no—
55C2×C22⋊D107—28no—
56C2×C22⋊D107—24no—
57C2×C22⋊D107—28no—
58C2×C22⋊D107—44no—
59D4×D107—232no—
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2312yes—
62C2×C4×D107—22no—
63C2×C4×D107—22no—
64C2×C4×D107D414yescomplemented
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—28no—
67(C2×D107)⋊C4D418yescomplemented
68C4×Dic107—22no—
69C24×C107C2312yes—
70C2×D4×C107—22no—
71C2×D4×C107—22no—
72C2×D4×C107D414yescomplemented
73C22×C4×C107C2312yes—
74C22×C4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76C4⋊C4×C107—22no—
77C22⋊C4×C107—28no—
78C42×C107—22no—
79C23×C107—24no—
80C23×C107—22no—
81C23×C107—22no—
82C23×C107—24no—
83C23×C107C2×D414yes—
84C23×C107C2411yescharacteristic
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—22no—
88C22×D107—216no—
89C22×D107—24no—
90C22×D107—22no—
91C22×D107—22no—
92C22×D107—24no—
93C22×D107—22no—
94C22×D107—22no—
95C22×D107—24no—
96C22×D107—22no—
97C22×D107—24no—
98C22×D107—416no—
99C22×D107—416no—
100C22×D107—48no—
101C22×D107—48no—
102C22×D107C2×D414yes—
103C22×D107C2×D414yes—
104C22×D107C2×D412yes—
105D4×C107—28no—
106C2×C4×C107—22no—
107C2×C4×C107—22no—
108C2×C4×C107—24no—
109C2×C4×C107—22no—
110C2×C4×C107C2×D412yes—
111C2×C4×C107C2×D412yes—
112C2×C4×C107C2×D412yes—
113C2×C4×C107C2×D414yescomplemented
114C22⋊D107—216no—
115C22⋊D107—416no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—22no—
120C2×Dic107C2×D412yes—
121C2×Dic107C2×D414yescomplemented
122D428—28no—
123D428—416no—
124D428—48no—
125D428—44no—
126C4×D107—28no—
127C22×C107—24no—
128C22×C107—28no—
129C22×C107—42no—
130C22×C107—42no—
131C22×C107—44no—
132C22×C107C22×D411yescharacteristic
133C22×C107C22×D411yescharacteristic
134C22×C107C22×D412yes—
135C22×C107C22×D411yescharacteristic
136C22×C107H(5,𝔽2)12yes—
137C2×D107—22no—
138C2×D107—22no—
139C2×D107—24no—
140C2×D107—24no—
141C2×D107—24no—
142C2×D107—28no—
143C2×D107—432no—
144C2×D107—42no—
145C2×D107—42no—
146C2×D107—44no—
147C2×D107—42no—
148C2×D107—42no—
149C2×D107—42no—
150C2×D107—44no—
151C2×D107—44no—
152C2×D107#12714yescomplemented
153C4×C107—24no—
154C4×C107—42no—
155C4×C107#12014yescomplemented
156Dic107—22no—
157Dic107—24no—
158C2×C107—44no—
159C2×C107C2×#12011yescharacteristic
160C2×C107C2×#12711yescharacteristic
161C2×C107#53511yescharacteristic
162C2×C107D4212yes—
163C2×C107D4212yes—
164D107—24no—
165D107—44no—
166D107—82no—
167#2433—1071nomaximal, Sylow
168C107#243311yescharacteristic, complemented, Sylow
169C23×D4—1071no—
170C2×#127—1071no—
171C2×#127—1071no—
172C2×#121—1071no—
173C2×#120—1074no—
174#392—1074no—
175#390—1071no—
176#390D10711yescharacteristic, complemented
177C4×C22⋊C4—1071no—
178C25—1072no—
179C22×D4—1074no—
180C22×D4—1072no—
181C22×D4—1071no—
182C22×D4—1074no—
183C22×D4—1072no—
184C22×D4—1071no—
185C22×D4—2144no—
186C22×D4—2144no—
187C22×D4—2144no—
188C22×D4C2×D10711yescharacteristic
189C23×C4—1071no—
190#127—2144no—
191#127—2144no—
192#121—2144no—
193#120—21416no—
194C2×C4⋊C4C2×D10711yescharacteristic, complemented
195C2×C22⋊C4—1072no—
196C2×C22⋊C4—1074no—
197C2×C22⋊C4—1072no—
198C2×C22⋊C4C2×D10714yescomplemented
199C2×C42—1071no—
200C2×C42C2×D10711yescharacteristic, complemented
201#95—1072no—
202C24—1071no—
203C24—1072no—
204C24—1072no—
205C24—4284no—
206C24—2148no—
207C24—2144no—
208C24—2144no—
209C24—2144no—
210C24C22×D10712yes—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—1074no—
214C2×D4—4284no—
215C2×D4—4284no—
216C2×D4—2148no—
217C2×D4—2144no—
218C2×D4—2148no—
219C2×D4—2148no—
220C2×D4—2144no—
221C2×D4—21432no—
222C2×D4—2148no—
223C2×D4—2144no—
224C2×D4—2144no—
225C2×D4—2142no—
226C2×D4—2142no—
227C2×D4—2142no—
228C2×D4—2148no—
229C2×D4—2142no—
230C2×D4—2142no—
231C2×D4D42814yescomplemented
232C22×C4—1071no—
233C22×C4—1072no—
234C22×C4—1074no—
235C22×C4—2142no—
236C22×C4—2142no—
237C22×C4C22×D10712yes—
238C22×C4C22×D10712yes—
239C22×C4C22×D10711yescharacteristic
240C4⋊C4—22no—
241C22⋊C4—28no—
242C22⋊C4—1078no—
243C22⋊C4—2148no—
244C22⋊C4—2144no—
245C42—22no—
246C42—2142no—
247C23—24no—
248C23—22no—
249C23—22no—
250C23—24no—
251C23—1074no—
252C23—1072no—
253C23—1074no—
254C23—42816no—
255C23—4288no—
256C23—4288no—
257C23—42816no—
258C23—21416no—
259C23—2142no—
260C23—2144no—
261C23—2142no—
262C23—2142no—
263C23—2144no—
264C23—2144no—
265C23—2142no—
266C23—2142no—
267C23—2144no—
268C23—2141no—
269C23—2141no—
270C23—2142no—
271C23C2×D42814yes—
272C23C23×D10711yescharacteristic
273D4—28no—
274D4—42816no—
275D4—4288no—
276D4—42816no—
277D4—4284no—
278D4—21416no—
279D4—2148no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—1074no—
285C2×C4—1072no—
286C2×C4—2141no—
287C2×C4—2142no—
288C2×C4—2141no—
289C2×C4—2142no—
290C2×C4—2148no—
291C2×C4D4×D10712yes—
292C2×C4D4×D10712yes—
293C2×C4D4×D10714yescomplemented
294C2×C4C2×D42812yes—
295C22—28no—
296C22—24no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—1074no—
301C22—42832no—
302C22—4282no—
303C22—4282no—
304C22—4284no—
305C22—4284no—
306C22—4282no—
307C22—4282no—
308C22—4282no—
309C22—4284no—
310C22—2148no—
311C22—2144no—
312C22—2142no—
313C22—2144no—
314C22—2144no—
315C22—2142no—
316C22C2×D4×D10711yescharacteristic
317C22C2×D4×D10712yes—
318C22C2×D4×D10711yescharacteristic
319C22C22×D42811yescharacteristic
320C22#1681215512yes—
321C4—24no—
322C4—42no—
323C4—2144no—
324C4—2142no—
325C4#1681207914yescomplemented
326C2—44no—
327C2—8562no—
328C2—4284no—
329C2—2144no—
330C2C2×#1681207911yescharacteristic
331C2#127×D10711yescharacteristic
332C2#2029319511yescharacteristic
333C2#2029326112yes—
334C2D4×D42812yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
hypercenter#390D107
hypocenterC107#2433
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C22×D4×D107C2semidirect
2. C2×#16812106C2semidirect
3. C2×#16812105C2semidirect
4. C2×#16812084C2semidirect
5. C2×#16812079C2semidirect
6. #20292536C2semidirect
7. #20292528C2semidirect
8. C4×(C2×D107)⋊C4C2semidirect
9. #390×C107C2semidirect
13. C22×D428C22semidirect
14. C22×D428C22semidirect
16. C22×C22⋊D107C22semidirect
26. C2×(C2×D107)⋊C4C22semidirect
27. C2×(C2×D107)⋊C4C22semidirect
28. C2×C4×Dic107C22semidirect
29. C107⋊#95C22semidirect
31. C2×C4⋊C4×C107C22semidirect
32. C2×C22⋊C4×C107C22semidirect
33. C2×C42×C107C22semidirect
52. C2×D428D4semidirect
64. C2×C4×D107D4semidirect
67. (C2×D107)⋊C4D4semidirect
72. C2×D4×C107D4semidirect
113. C2×C4×C107C2×D4semidirect
121. C2×Dic107C2×D4semidirect
152. C2×D107#127semidirect
155. C4×C107#120semidirect
168. C107#2433semidirect
176. #390D107semidirect
194. C2×C4⋊C4C2×D107semidirect
198. C2×C22⋊C4C2×D107semidirect
200. C2×C42C2×D107semidirect
231. C2×D4D428semidirect
293. C2×C4D4×D107semidirect
325. C4#16812079semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168121062
C2×#168121052
C2×#168120842
C2×#168120792
#202925362
#202925282
C4×(C2×D107)⋊C42
#390×C1072
#2433107