Finite Group Databasebeta

← #28346044

Subgroups — #28346044

Order 13696.

520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812132C211yesmaximal, characteristic, complemented
2C22×C8×D107C211yesmaximal, characteristic, complemented
3#20293140C212yesmaximal, complemented
4#20292865C211yesmaximal, characteristic, complemented
5#20292864C211yesmaximal, characteristic, complemented
6#20292863C211yesmaximal, characteristic, complemented
7#20292859C212yesmaximal, complemented
8#20292858C212yesmaximal, complemented
9#20292664C211yesmaximal, characteristic, complemented
10#20292622C212yesmaximal, complemented
11#467×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C22×C4×D107C2211yescharacteristic
14#16812132—24no—
15C2×#13133805C2212yescomplemented
16C2×D856C2211yescharacteristic, complemented
17C22×C107⋊C8C2211yescharacteristic
18C2×C8×D107—24no—
19C2×C8×D107C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812094—22no—
24#16812085C2212yes—
25#16812084C2212yes—
26#16812083C2212yes—
27#16812079—22no—
28#16812071—22no—
29(C22⋊D107)⋊C4—22no—
30#16812067C2212yes—
31#16812043C2211yescharacteristic, complemented
32#16812040C2212yescomplemented
33#16812033C2212yescomplemented
34#16812031C2212yescomplemented
35#16812029C2212yescomplemented
36C2×Q16×C107C2211yescharacteristic, complemented
37C22×C8×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#107×C107C2211yescharacteristic, complemented
40#103×C107C2212yescomplemented
41C23×D107D411yescharacteristic
42#13133819—24no—
43#13133819—24no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428C2312yes—
47C2×C22⋊D107—22no—
48C22×Dic107D411yescharacteristic
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52C2×C4×D107D411yescharacteristic
53C2×C4×D107D411yescharacteristic
54C2×C4×D107C2311yescharacteristic
55C2×C4×D107C2311yescharacteristic
56#13133805—24no—
57D856—22no—
58C2×C107⋊C8—22no—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—22no—
62C8×D107—22no—
63C8×D107—22no—
64C8×D107D414yescomplemented
65C4⋊Dic107C2312yes—
66Dic107⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—22no—
72C2×Q8×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74Q16×C107—22no—
75C2×C8×C107—22no—
76C2×C8×C107C2311yescharacteristic
77C2×C8×C107C2311yescharacteristic
78C4⋊C4×C107—22no—
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107C2×D411yescharacteristic
87C22×D107C2×D411yescharacteristic
88Q8×C107—22no—
89Q8×C107—22no—
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—44no—
97C2×Dic107—22no—
98C2×Dic107—22no—
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101D428—22no—
102D428—22no—
103D428—44no—
104D428—44no—
105C4×D107—22no—
106C4×D107—21no—
107C4×D107—21no—
108C4×D107—22no—
109C4×D107—44no—
110C4×D107C2×D412yes—
111C4×D107#4112yes—
112C8×C107—21no—
113C8×C107C2×D412yes—
114C107⋊C8—21no—
115C107⋊C8C2×D412yes—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107#13512yescomplemented
119C22×C107C22×D411yescharacteristic
120C2×D107—28no—
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—21no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—42no—
128C2×D107#12112yes—
129C2×D107#13512yescomplemented
130C4×C107—21no—
131C4×C107—42no—
132C4×C107—42no—
133C4×C107C22×D411yescharacteristic
134C4×C107C2×#4111yescharacteristic
135Dic107—21no—
136Dic107—42no—
137Dic107#12112yes—
138C2×C107—22no—
139C2×C107C2×#12111yescharacteristic
140C2×C107C2×#13511yescharacteristic
141C2×C107C2×#13511yescharacteristic
142D107—22no—
143D107—22no—
144D107—82no—
145#3050—1071nomaximal, Sylow
146C107#305011yescharacteristic, complemented, Sylow
147C2×#135—1071no—
148C23×C8—1071no—
149#525—1072no—
150#467—1071no—
151#467D10711yescharacteristic, complemented
152#466—1071no—
153#466—1071no—
154#465—1072no—
155#465—1072no—
156#427—1071no—
157#416—1072no—
158C2×#41—1072no—
159C23×C4—1071no—
160#135—2144no—
161C2×Q16C2×D10711yescharacteristic, complemented
162C2×SD16—1072no—
163C2×D8—1071no—
164C22×C8—1071no—
165C22×C8—1071no—
166C22×C8—1071no—
167C22×C8—1071no—
168C22×C8—1071no—
169C22×C8—2144no—
170C22×C8C2×D10711yescharacteristic
171#123—2142no—
172C22⋊Q8—1072no—
173C22⋊Q8C2×D10712yes—
174#121—1072no—
175#121—1072no—
176#120—2142no—
177C4×D4—2142no—
178C4×D4—2142no—
179#117—1072no—
180#107—1071no—
181#107C2×D10711yescharacteristic, complemented
182#106—1072no—
183#103—1072no—
184#103C2×D10712yescomplemented
185#102—1072no—
186#102—1072no—
187C24—1071no—
188#41—2144no—
189#41—2144no—
190C2×Q8C22×D10712yes—
191C2×D4—1072no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2142no—
195C22×C4—1071no—
196C22×C4—1071no—
197C22×C4—1071no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—2142no—
201C22×C4—2142no—
202C22×C4—2142no—
203C22×C4C22×D10711yescharacteristic
204Q16—22no—
205SD16—2144no—
206D8—2142no—
207C2×C8—22no—
208C2×C8—1071no—
209C2×C8—1074no—
210C2×C8—1071no—
211C2×C8—2142no—
212C2×C8—2142no—
213C2×C8—2142no—
214C2×C8—2142no—
215C2×C8C22×D10711yescharacteristic
216C2×C8C22×D10711yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—1072no—
219C4⋊C4—2142no—
220C4⋊C4C22×D10712yes—
221C22⋊C4—22no—
222C22⋊C4—2142no—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C42—2142no—
227C23—1071no—
228C23—1071no—
229C23—2142no—
230C23—2142no—
231C23—2142no—
232C23—2142no—
233C23D4×D10711yescharacteristic
234Q8—22no—
235Q8—22no—
236D4—4284no—
237D4—4284no—
238D4—4284no—
239D4—2142no—
240D4—2142no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—1071no—
246C2×C4—1072no—
247C2×C4—1072no—
248C2×C4—1071no—
249C2×C4—4284no—
250C2×C4—2142no—
251C2×C4—2142no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2141no—
255C2×C4—2141no—
256C2×C4D4×D10711yescharacteristic
257C2×C4C23×D10711yescharacteristic
258C8—21no—
259C8—1072no—
260C8—2141no—
261C8D4×D10712yes—
262C22—21no—
263C22—21no—
264C22—1072no—
265C22—1072no—
266C22—4282no—
267C22—4282no—
268C22—4282no—
269C22—2148no—
270C22—2141no—
271C22—2141no—
272C22—2141no—
273C22—2141no—
274C22#1681213212yescomplemented
275C22C2×D4×D10711yescharacteristic
276C4—21no—
277C4—42no—
278C4—42no—
279C4—1072no—
280C4—4282no—
281C4—2141no—
282C4C2×D4×D10711yescharacteristic
283C4#41×D10711yescharacteristic
284C2—22no—
285C2—8562no—
286C2—2142no—
287C2—2142no—
288C2#121×D10711yescharacteristic
289C2#135×D10711yescharacteristic
290C2C2×#1681213211yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
hypercenter#467D107
hypocenterC107#3050
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. C22×C8×D107C2semidirect
3. #20293140C2semidirect
4. #20292865C2semidirect
5. #20292864C2semidirect
6. #20292863C2semidirect
7. #20292859C2semidirect
8. #20292858C2semidirect
9. #20292664C2semidirect
10. #20292622C2semidirect
11. #467×C107C2semidirect
15. C2×#13133805C22semidirect
16. C2×D856C22semidirect
31. #16812043C22semidirect
32. #16812040C22semidirect
33. #16812033C22semidirect
34. #16812031C22semidirect
35. #16812029C22semidirect
36. C2×Q16×C107C22semidirect
39. #107×C107C22semidirect
40. #103×C107C22semidirect
64. C8×D107D4semidirect
118. C22×C107#135semidirect
129. C2×D107#135semidirect
146. C107#3050semidirect
151. #467D107semidirect
161. C2×Q16C2×D107semidirect
181. #107C2×D107semidirect
184. #103C2×D107semidirect
274. C22#16812132semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
C22×C8×D1072
#202931402
#202928652
#202928642
#202928632
#202928592
#202928582
#202926642
#202926222
#467×C1072
#3050107