Finite Group Databasebeta

← C2×#20292762 #28345638

Subgroups — C2×#20292762 #28345638

Order 13696.

1048 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1048 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
1C22×D4×D107C211yesmaximal, characteristic, complemented
2C22×D856C211yesmaximal, characteristic, complemented
3C22×#13133803C211yesmaximal, characteristic, complemented
4C2×#16812084C211yesmaximal, characteristic, complemented
5#20292762C218yesmaximal, direct product factor
6C2×#16812029C211yesmaximal, characteristic, complemented
7C2×(C2×D107)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C107C211yesmaximal, characteristic, complemented
9C24×D107—21no—
10C22×D428—21no—
11C22×D428C2211yescharacteristic
12C22×D428C2211yescharacteristic, complemented
13C22×C22⋊D107—21no—
14C2×D4×D107—24no—
15C2×D4×D107—22no—
16C2×D4×D107—24no—
17C2×D4×D107C2214yescomplemented
18C22×C4×D107C2211yescharacteristic
19C2×D856—24no—
20C2×D856C2214yescomplemented
21C2×#13133803—24no—
22C2×#13133803C2214yescomplemented
23C22×C107⋊C8C2211yescharacteristic, complemented
24#16812084—22no—
25#16812084C2214yescomplemented
26C2×(C2×D107)⋊C4—21no—
27#16812029C2214yescomplemented
28(C2×D107)⋊C8C2214yescomplemented
29C22×D4×C107C2211yescharacteristic
30C22×C8×C107C2211yescharacteristic, complemented
31C2×C4⋊C4×C107C2211yescharacteristic, complemented
32#102×C107C2214yescomplemented
33C23×D107—28no—
34C23×D107—21no—
35C23×D107—24no—
36C23×D107—21no—
37C23×D107—44no—
38C23×D107—44no—
39C23×D107D411yescharacteristic
40C2×D428—21no—
41C2×D428—22no—
42C2×D428—22no—
43C2×D428—22no—
44C2×D428—24no—
45C2×D428—21no—
46C2×D428—44no—
47C2×D428D414yescomplemented
48C2×D428C2312yes—
49C2×D428C2312yescomplemented
50C2×C22⋊D107—28no—
51C2×C22⋊D107—24no—
52D4×D107—216no—
53D4×D107—28no—
54D4×D107—28no—
55C22×Dic107D411yescharacteristic
56C2×C4×D107—22no—
57C2×C4×D107—22no—
58C2×C4×D107D412yes—
59C2×C4×D107C2312yes—
60D856—28no—
61#13133803—28no—
62C2×C107⋊C8—22no—
63C2×C107⋊C8C2312yescomplemented
64(C2×D107)⋊C4—24no—
65C24×C107—21no—
66C2×D4×C107—21no—
67C2×D4×C107—22no—
68C2×D4×C107D414yescomplemented
69C2×D4×C107C2312yes—
70C22×C4×C107—21no—
71C22×C4×C107C2311yescharacteristic
72C2×C8×C107—22no—
73C2×C8×C107C2312yescomplemented
74C4⋊C4×C107—21no—
75C4⋊C4×C107C2312yescomplemented
76C23×C107—24no—
77C23×C107—22no—
78C23×C107—42no—
79C23×C107—42no—
80C23×C107C2×D411yescharacteristic
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—216no—
84C22×D107—24no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—48no—
88C22×D107—432no—
89C22×D107—42no—
90C22×D107—42no—
91C22×D107—48no—
92C22×D107—42no—
93C22×D107—42no—
94C22×D107—42no—
95C22×D107D814yescomplemented
96C22×D107C2×D412yes—
97D4×C107—24no—
98D4×C107—22no—
99D4×C107C2×D414yescomplemented
100C2×C4×C107—22no—
101C2×C4×C107—42no—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D412yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—216no—
107C2×Dic107—22no—
108C2×Dic107C2×D412yes—
109D428—24no—
110D428—22no—
111D428—22no—
112D428—22no—
113D428—48no—
114D428—42no—
115D428C2×D414yescomplemented
116C4×D107—24no—
117C4×D107—22no—
118C4×D107—22no—
119C8×C107—22no—
120C107⋊C8—22no—
121C22×C107—24no—
122C22×C107—42no—
123C22×C107—42no—
124C22×C107—48no—
125C22×C107#12012yes—
126C22×C107C2×D812yes—
127C22×C107AGL(1,ℤ/8)12yes—
128C22×C107C22×D411yescharacteristic
129C2×D107—22no—
130C2×D107—22no—
131C2×D107—28no—
132C2×D107—24no—
133C2×D107—432no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107—48no—
137C2×D107—42no—
138C2×D107—42no—
139C2×D107—42no—
140C2×D107—84no—
141C2×D107C2×D814yescomplemented
142C4×C107—42no—
143C4×C107#12012yes—
144C4×C107C22×D411yescharacteristic
145C4×C107C22×D411yescharacteristic
146Dic107—22no—
147C2×C107—44no—
148C2×C107#44714yescomplemented
149C2×C107C2×#12011yescharacteristic
150C2×C107C22×D811yescharacteristic
151C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
152D107—24no—
153D107—44no—
154D107—82no—
155C2×#447—1071nomaximal, Sylow
156C107C2×#44711yescharacteristic, complemented, Sylow
157C23×D4—1071no—
158C22×D8—1071no—
159C22×D8—1071no—
160C2×#121—1071no—
161#447—1078no—
162C2×#102—1071no—
163C2×#102D10711yescharacteristic, complemented
164C2×C22⋊C8—1071no—
165C25—2141no—
166C22×D4—1074no—
167C22×D4—1071no—
168C22×D4—1071no—
169C22×D4—2141no—
170C22×D4—2142no—
171C22×D4—2141no—
172C22×D4—2144no—
173C22×D4—2144no—
174C22×D4C2×D10711yescharacteristic
175C23×C4—1071no—
176C2×D8—1074no—
177C2×D8—1074no—
178C2×D8—2144no—
179C2×D8—2144no—
180C22×C8—1071no—
181C22×C8C2×D10711yescharacteristic, complemented
182#121—1074no—
183#121—2142no—
184C2×C4⋊C4C2×D10711yescharacteristic, complemented
185C2×C22⋊C4—2141no—
186#102—1074no—
187#102C2×D10714yescomplemented
188C22⋊C8—1074no—
189C24—21no—
190C24—1071no—
191C24—4284no—
192C24—4284no—
193C24—2144no—
194C24—2148no—
195C24—2141no—
196C24—2141no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—1074no—
200C2×D4—1072no—
201C2×D4—1072no—
202C2×D4—4284no—
203C2×D4—2144no—
204C2×D4—2144no—
205C2×D4—2148no—
206C2×D4—21416no—
207C2×D4—2148no—
208C2×D4—2148no—
209C2×D4—2141no—
210C2×D4—2142no—
211C2×D4—2141no—
212C2×D4—2142no—
213C2×D4—2142no—
214C2×D4D42814yescomplemented
215C2×D4C22×D10712yes—
216C22×C4—21no—
217C22×C4—1071no—
218C22×C4—1072no—
219C22×C4—1072no—
220C22×C4—2142no—
221C22×C4—2142no—
222C22×C4C22×D10711yescharacteristic
223D8—2148no—
224D8—2148no—
225C2×C8—22no—
226C2×C8—1072no—
227C2×C8—2142no—
228C2×C8C22×D10712yescomplemented
229C4⋊C4—21no—
230C4⋊C4C22×D10712yescomplemented
231C22⋊C4—2144no—
232C23—24no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—1074no—
237C23—1072no—
238C23—4288no—
239C23—4288no—
240C23—42832no—
241C23—4282no—
242C23—4282no—
243C23—4282no—
244C23—4282no—
245C23—4282no—
246C23—21416no—
247C23—2142no—
248C23—2144no—
249C23—2142no—
250C23—2142no—
251C23—2142no—
252C23D4×D10711yescharacteristic
253D4—24no—
254D4—22no—
255D4—1074no—
256D4—4288no—
257D4—4282no—
258D4—21416no—
259D4—2144no—
260D4—2142no—
261D4—2142no—
262D4—2142no—
263D4C2×D42814yescomplemented
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—1072no—
267C2×C4—2142no—
268C2×C4—2144no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4D4×D10711yescharacteristic
272C2×C4D4×D10712yes—
273C2×C4C2×D42812yes—
274C2×C4C23×D10711yescharacteristic
275C8—22no—
276C8—2142no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—48no—
281C22—8564no—
282C22—1074no—
283C22—42832no—
284C22—4282no—
285C22—4282no—
286C22—4282no—
287C22—4282no—
288C22—4282no—
289C22—4288no—
290C22—2144no—
291C22—2142no—
292C22—2148no—
293C22—2142no—
294C22#1681207912yes—
295C22D8×D10712yes—
296C22#1681213612yes—
297C22C2×D4×D10711yescharacteristic
298C4—42no—
299C4—2142no—
300C4#1681207912yes—
301C4C2×D4×D10711yescharacteristic
302C4C22×D42811yescharacteristic
303C2—44no—
304C2—8562no—
305C2—4284no—
306C2—2144no—
307C2#2029276214yesdirect product factor
308C2C2×#1681207911yescharacteristic
309C2C2×D8×D10711yescharacteristic
310C2C2×#1681213611yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#102×C107C2
socleC23×C107C2×D4
hypercenterC2×#102D107
hypocenterC107C2×#447
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×D4×D107C2semidirect
2. C22×D856C2semidirect
3. C22×#13133803C2semidirect
4. C2×#16812084C2semidirect
5. #20292762C2direct (N × H)
6. C2×#16812029C2semidirect
7. C2×(C2×D107)⋊C8C2semidirect
8. C2×#102×C107C2semidirect
12. C22×D428C22semidirect
17. C2×D4×D107C22semidirect
20. C2×D856C22semidirect
22. C2×#13133803C22semidirect
23. C22×C107⋊C8C22semidirect
25. #16812084C22semidirect
27. #16812029C22semidirect
28. (C2×D107)⋊C8C22semidirect
30. C22×C8×C107C22semidirect
31. C2×C4⋊C4×C107C22semidirect
32. #102×C107C22semidirect
47. C2×D428D4semidirect
49. C2×D428C23semidirect
63. C2×C107⋊C8C23semidirect
68. C2×D4×C107D4semidirect
73. C2×C8×C107C23semidirect
75. C4⋊C4×C107C23semidirect
95. C22×D107D8semidirect
99. D4×C107C2×D4semidirect
115. D428C2×D4semidirect
141. C2×D107C2×D8semidirect
148. C2×C107#447semidirect
156. C107C2×#447semidirect
163. C2×#102D107semidirect
181. C22×C8C2×D107semidirect
184. C2×C4⋊C4C2×D107semidirect
187. #102C2×D107semidirect
214. C2×D4D428semidirect
228. C2×C8C22×D107semidirect
230. C4⋊C4C22×D107semidirect
263. D4C2×D428semidirect
307. C2#20292762direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C22×D8562
C22×#131338032
C2×#168120842
#202927622
C2×#168120292
C2×(C2×D107)⋊C82
C2×#102×C1072
C2×#447107