Finite Group Databasebeta

← C2×#20292759 #28345635

Subgroups — C2×#20292759 #28345635

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×#13133803C211yesmaximal, characteristic, complemented
3C2×#16812080C211yesmaximal, characteristic, complemented
4#20292759C218yesmaximal, direct product factor
5C2×#16812031C211yesmaximal, characteristic, complemented
6C2×#16812030C211yesmaximal, characteristic, complemented
7C2×(C2×D107)⋊C8C211yesmaximal, characteristic, complemented
8C22×D8×C107C211yesmaximal, characteristic, complemented
9C24×D107—21no—
10C22×D428C2211yescharacteristic
11C22×C22⋊D107—21no—
12C22×C22⋊D107—21no—
13C2×D4×D107—24no—
14C2×D4×D107—22no—
15C2×D4×D107—24no—
16C2×D4×D107C2214yescomplemented
17C22×C4×D107C2211yescharacteristic
18C2×#13133803—24no—
19C2×#13133803C2214yescomplemented
20C22×C107⋊C8C2211yescharacteristic, complemented
21#16812080—22no—
22#16812080C2214yescomplemented
23C2×C4⋊Dic107C2211yescharacteristic, complemented
24C2×C22⋊Dic107—21no—
25#16812031C2214yescomplemented
26#16812030C2214yescomplemented
27(C2×D107)⋊C8C2214yescomplemented
28C22×D4×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic, complemented
30C2×D8×C107—24no—
31C2×D8×C107C2214yescomplemented
32C22×C8×C107C2211yescharacteristic, complemented
33C23×D107—28no—
34C23×D107—21no—
35C23×D107—24no—
36C23×D107—44no—
37C23×D107—44no—
38C23×D107D411yescharacteristic
39C2×D428—21no—
40C2×D428—22no—
41C2×D428D414yescomplemented
42C2×D428C2312yes—
43C2×C22⋊D107—28no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—44no—
47D4×D107—216no—
48D4×D107—28no—
49D4×D107—28no—
50C22×Dic107—21no—
51C22×Dic107D411yescharacteristic
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107D412yes—
55C2×C4×D107C2312yes—
56#13133803—28no—
57C2×C107⋊C8—22no—
58C2×C107⋊C8C2312yescomplemented
59C4⋊Dic107—21no—
60C4⋊Dic107C2312yescomplemented
61C22⋊Dic107—24no—
62C24×C107—21no—
63C24×C107—21no—
64C2×D4×C107—21no—
65C2×D4×C107—22no—
66C2×D4×C107—22no—
67C2×D4×C107—22no—
68C2×D4×C107—21no—
69C2×D4×C107D414yescomplemented
70C2×D4×C107C2312yes—
71C2×D4×C107C2312yescomplemented
72C22×C4×C107C2311yescharacteristic
73D8×C107—28no—
74C2×C8×C107—22no—
75C2×C8×C107C2312yescomplemented
76C23×C107—24no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—42no—
80C23×C107—42no—
81C23×C107—42no—
82C23×C107—42no—
83C23×C107—42no—
84C23×C107C2×D411yescharacteristic
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—216no—
88C22×D107—24no—
89C22×D107—22no—
90C22×D107—48no—
91C22×D107—432no—
92C22×D107—42no—
93C22×D107—42no—
94C22×D107—48no—
95C22×D107D814yescomplemented
96C22×D107C2×D412yes—
97D4×C107—24no—
98D4×C107—22no—
99D4×C107—22no—
100D4×C107—22no—
101D4×C107—42no—
102D4×C107C2×D414yescomplemented
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D412yes—
106C2×C4×C107C2411yescharacteristic
107C22⋊D107—216no—
108C22⋊D107—48no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—42no—
112C2×Dic107C2×D412yes—
113D428—24no—
114D428—22no—
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—42no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—84no—
129C22×C107#12012yes—
130C22×C107C2×D812yes—
131C22×C107AGL(1,ℤ/8)12yes—
132C22×C107C22×D411yescharacteristic
133C2×D107—22no—
134C2×D107—22no—
135C2×D107—28no—
136C2×D107—24no—
137C2×D107—432no—
138C2×D107—42no—
139C2×D107—42no—
140C2×D107—48no—
141C2×D107C2×D814yescomplemented
142C4×C107#12012yes—
143C4×C107C22×D411yescharacteristic
144C4×C107C22×D411yescharacteristic
145Dic107—22no—
146Dic107—42no—
147C2×C107—44no—
148C2×C107—82no—
149C2×C107#44714yescomplemented
150C2×C107C2×#12011yescharacteristic
151C2×C107C22×D811yescharacteristic
152C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
153D107—24no—
154D107—44no—
155C2×#447—1071nomaximal, Sylow
156C107C2×#44711yescharacteristic, complemented, Sylow
157C23×D4—1071no—
158C22×D8—1071no—
159C22×D8D10711yescharacteristic, complemented
160C2×#121—1071no—
161#447—1078no—
162C2×#102—1071no—
163C2×#102—1071no—
164C2×C22⋊C8—1071no—
165C25—2141no—
166C22×D4—1074no—
167C22×D4—1071no—
168C22×D4—2142no—
169C22×D4—2141no—
170C22×D4—2144no—
171C22×D4—2141no—
172C22×D4—2144no—
173C22×D4C2×D10711yescharacteristic
174C22×D4C2×D10711yescharacteristic, complemented
175C23×C4—1071no—
176C2×D8—24no—
177C2×D8—1074no—
178C2×D8—2144no—
179C2×D8C2×D10714yescomplemented
180C22×C8—1071no—
181C22×C8C2×D10711yescharacteristic, complemented
182#121—1074no—
183#121—2142no—
184C2×C4⋊C4—1071no—
185C2×C22⋊C4—2141no—
186#102—1074no—
187#102—1074no—
188C22⋊C8—1074no—
189C24—21no—
190C24—21no—
191C24—1071no—
192C24—4284no—
193C24—4284no—
194C24—2144no—
195C24—2148no—
196C24—2141no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—22no—
202C2×D4—1074no—
203C2×D4—1072no—
204C2×D4—4284no—
205C2×D4—2144no—
206C2×D4—2148no—
207C2×D4—21416no—
208C2×D4—2144no—
209C2×D4—2148no—
210C2×D4—2148no—
211C2×D4—2141no—
212C2×D4—2142no—
213C2×D4C22⋊D10714yescomplemented
214C2×D4C22×D10712yes—
215C2×D4C22×D10712yescomplemented
216C22×C4—1071no—
217C22×C4—1072no—
218C22×C4—1072no—
219C22×C4—2141no—
220C22×C4—2142no—
221C22×C4—2142no—
222C22×C4C22×D10711yescharacteristic
223D8—28no—
224D8—2148no—
225C2×C8—22no—
226C2×C8—1072no—
227C2×C8—2142no—
228C2×C8C22×D10712yescomplemented
229C4⋊C4—1072no—
230C4⋊C4—2141no—
231C22⋊C4—2144no—
232C23—24no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—1074no—
241C23—1072no—
242C23—4288no—
243C23—4288no—
244C23—42832no—
245C23—4282no—
246C23—4282no—
247C23—21416no—
248C23—2144no—
249C23—2142no—
250C23—2142no—
251C23—2142no—
252C23D4×D10711yescharacteristic
253D4—24no—
254D4—22no—
255D4—22no—
256D4—22no—
257D4—42no—
258D4—1074no—
259D4—4288no—
260D4—21416no—
261D4—2144no—
262D4—2142no—
263D4C2×C22⋊D10714yescomplemented
264C2×C4—1072no—
265C2×C4—4282no—
266C2×C4—2142no—
267C2×C4—2142no—
268C2×C4—2144no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4D4×D10711yescharacteristic
272C2×C4D4×D10712yes—
273C2×C4C2×C22⋊D10712yes—
274C2×C4C23×D10711yescharacteristic
275C8—22no—
276C8—2142no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—48no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—1074no—
286C22—42832no—
287C22—4282no—
288C22—4282no—
289C22—4288no—
290C22—2144no—
291C22—2142no—
292C22—2148no—
293C22—2142no—
294C22#1681207712yes—
295C22D8×D10712yes—
296C22#1681213312yes—
297C22C2×D4×D10711yescharacteristic
298C4—4282no—
299C4—2142no—
300C4#1681207712yes—
301C4C2×D4×D10711yescharacteristic
302C4C22×C22⋊D10711yescharacteristic
303C2—44no—
304C2—82no—
305C2—4284no—
306C2—2144no—
307C2#2029275914yesdirect product factor
308C2C2×#1681207711yescharacteristic
309C2C2×D8×D10711yescharacteristic
310C2C2×#1681213311yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×D8×C107C2
socleC23×C107C2×D4
hypercenterC22×D8D107
hypocenterC107C2×#447
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×D4×D107C2semidirect
2. C22×#13133803C2semidirect
3. C2×#16812080C2semidirect
4. #20292759C2direct (N × H)
5. C2×#16812031C2semidirect
6. C2×#16812030C2semidirect
7. C2×(C2×D107)⋊C8C2semidirect
8. C22×D8×C107C2semidirect
16. C2×D4×D107C22semidirect
19. C2×#13133803C22semidirect
20. C22×C107⋊C8C22semidirect
22. #16812080C22semidirect
23. C2×C4⋊Dic107C22semidirect
25. #16812031C22semidirect
26. #16812030C22semidirect
27. (C2×D107)⋊C8C22semidirect
29. C22×D4×C107C22semidirect
31. C2×D8×C107C22semidirect
32. C22×C8×C107C22semidirect
41. C2×D428D4semidirect
58. C2×C107⋊C8C23semidirect
60. C4⋊Dic107C23semidirect
69. C2×D4×C107D4semidirect
71. C2×D4×C107C23semidirect
75. C2×C8×C107C23semidirect
95. C22×D107D8semidirect
102. D4×C107C2×D4semidirect
115. D428C2×D4semidirect
141. C2×D107C2×D8semidirect
149. C2×C107#447semidirect
156. C107C2×#447semidirect
159. C22×D8D107semidirect
174. C22×D4C2×D107semidirect
179. C2×D8C2×D107semidirect
181. C22×C8C2×D107semidirect
213. C2×D4C22⋊D107semidirect
215. C2×D4C22×D107semidirect
228. C2×C8C22×D107semidirect
263. D4C2×C22⋊D107semidirect
307. C2#20292759direct (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×#131338032
C2×#168120802
#202927592
C2×#168120312
C2×#168120302
C2×(C2×D107)⋊C82
C22×D8×C1072
C2×#447107