Finite Group Databasebeta

← #28348196

Subgroups — #28348196

Order 13696.

482 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 131 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×Q16×D107C211yesmaximal, characteristic, complemented
2#20293284C212yesmaximal, complemented
3#20293010C211yesmaximal, characteristic, complemented
4#20292868C211yesmaximal, characteristic, complemented
5#20292860C211yesmaximal, characteristic, complemented
6#20292847C212yesmaximal, complemented
7#20292789C212yesmaximal, complemented
8#20292764C212yesmaximal, complemented
9Q16×Dic107C211yesmaximal, characteristic, complemented
10C8×C22⋊D107C211yesmaximal, characteristic, complemented
11C2×#135×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2212yes—
13C2×Dic428C2211yescharacteristic
14C2×#13133808C2212yes—
15Q16×D107—24no—
16C2×#13133806C2212yes—
17C2×C8×D107C2211yescharacteristic
18#16812096—22no—
19#16812096C2212yes—
20#16812088—22no—
21#16812088C2212yes—
22#16812080—22no—
23#16812080C2212yes—
24Q8×Dic107C2212yes—
25C4×C22⋊D107—22no—
26C4×C22⋊D107C2211yescharacteristic
27#16812043C2211yescharacteristic
28Dic107⋊C8C2211yescharacteristic
29#16812034C2212yes—
30#16812033C2212yes—
31#16812030C2212yes—
32C22⋊(C107⋊C8)C2211yescharacteristic
33(C2×D107)⋊C8C2211yescharacteristic
34C8×Dic107C2211yescharacteristic
35C2×#41×C107C2212yes—
36#135×C107—24no—
37C2×Q16×C107C2211yescharacteristic
38C2×SD16×C107C2212yes—
39C2×D8×C107C2211yescharacteristic
40C22×C8×C107C2211yescharacteristic
41C2×Dic214—22no—
42C2×Dic214C2312yes—
43Q8×D107—24no—
44Q8×D107—42no—
45C2×C22⋊D107—22no—
46C2×C22⋊D107D411yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107C2311yescharacteristic
49Dic428—22no—
50#13133808—24no—
51#13133806—42no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107—22no—
55C4⋊Dic107—22no—
56C4⋊Dic107C2312yes—
57Dic107⋊C4—22no—
58Dic107⋊C4D411yescharacteristic
59C22⋊Dic107—22no—
60C22⋊Dic107—22no—
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63C22⋊Dic107D411yescharacteristic
64(C2×D107)⋊C4D411yescharacteristic
65C4×Dic107—22no—
66C4×Dic107C2311yescharacteristic
67#41×C107—24no—
68#41×C107—42no—
69C2×Q8×C107C2312yes—
70C2×D4×C107—22no—
71C2×D4×C107C2312yes—
72C22×C4×C107—22no—
73C22×C4×C107C2311yescharacteristic
74Q16×C107D414yescomplemented
75SD16×C107—24no—
76D8×C107—22no—
77C2×C8×C107—22no—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C23×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107C2×D414yes—
85D4×C107—22no—
86D4×C107—22no—
87D4×C107—44no—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—44no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—22no—
95C22⋊D107—44no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101C4×D107—21no—
102C4×D107—21no—
103C4×D107—44no—
104Dic214—22no—
105Dic214—22no—
106Dic214—44no—
107C8×C107—21no—
108C8×C107C2×D412yes—
109C107⋊C8—21no—
110C107⋊C8—21no—
111C22×C107—21no—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107—42no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C2×D107—21no—
119C2×D107—21no—
120C2×D107#13512yescomplemented
121C4×C107—24no—
122C4×C107—21no—
123C4×C107C22×D411yescharacteristic
124C4×C107H(5,𝔽2)11yescharacteristic
125Dic107—21no—
126Dic107—21no—
127Dic107—21no—
128Dic107—42no—
129Dic107—42no—
130C2×C107—41no—
131C2×C107—82no—
132C2×C107D4211yescharacteristic
133C2×C107C2×#13511yescharacteristic
134C2×C107#57811yescharacteristic
135D107—22no—
136#3285—1071nomaximal, Sylow
137C107#328511yescharacteristic, complemented, Sylow
138C2×#135D10711yescharacteristic, complemented
139C22×Q16—1071no—
140#548—1072no—
141#495—1071no—
142#467—1071no—
143#466—1071no—
144#464—1072no—
145#452—1072no—
146#448—1072no—
147C4×Q16—1071no—
148C8×D4—1071no—
149C2×#41C2×D10712yes—
150C22×Q8—1072no—
151#135—24no—
152C2×Q16—1072no—
153C2×Q16—1071no—
154C2×Q16—2144no—
155C2×Q16C2×D10711yescharacteristic
156C2×SD16—1072no—
157C2×SD16C2×D10712yes—
158C2×D8C2×D10711yescharacteristic
159C22×C8—1071no—
160C22×C8C2×D10711yescharacteristic
161#124—1072no—
162#124—2142no—
163C22⋊Q8—1072no—
164C22⋊Q8—2142no—
165#121—1072no—
166#121—2142no—
167C4×Q8—1072no—
168C4×D4—1071no—
169C4×D4—2142no—
170#107—1071no—
171C4⋊C8—1071no—
172#103—1072no—
173#103—1072no—
174#102—1072no—
175C22⋊C8—1071no—
176C22⋊C8—1071no—
177C4×C8—1071no—
178#41—24no—
179#41—42no—
180C2×Q8—1072no—
181C2×Q8—4282no—
182C2×Q8—2142no—
183C2×Q8—2144no—
184C2×Q8C22×D10712yes—
185C2×D4—22no—
186C2×D4—1071no—
187C2×D4—2142no—
188C2×D4C22×D10712yes—
189C22×C4—22no—
190C22×C4—1071no—
191C22×C4—2142no—
192C22×C4C22×D10711yescharacteristic
193Q16—2144no—
194Q16—2142no—
195Q16C22⋊D10714yescomplemented
196SD16—24no—
197SD16—4282no—
198D8—22no—
199C2×C8—22no—
200C2×C8—1071no—
201C2×C8—1071no—
202C2×C8—2142no—
203C2×C8C22×D10711yescharacteristic
204C2×C8C22×D10711yescharacteristic
205C4⋊C4—1071no—
206C4⋊C4—1072no—
207C4⋊C4—2142no—
208C4⋊C4—2142no—
209C22⋊C4—1071no—
210C22⋊C4—1071no—
211C22⋊C4—2142no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C42—1071no—
216C42—2142no—
217C23—22no—
218C23—1071no—
219C23D4×D10711yescharacteristic
220Q8—22no—
221Q8—4284no—
222Q8—2142no—
223Q8—2142no—
224Q8C2×C22⋊D10714yes—
225D4—22no—
226D4—22no—
227D4—44no—
228D4—4284no—
229D4—2142no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—44no—
234C2×C4—1071no—
235C2×C4—1071no—
236C2×C4—1071no—
237C2×C4—4284no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2141no—
241C2×C4—2141no—
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—2141no—
246C8—2141no—
247C8C2×C22⋊D10712yes—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—1072no—
255C22—2141no—
256C22—2141no—
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—24no—
260C4—4282no—
261C4—4282no—
262C4—2141no—
263C4—2141no—
264C4—2141no—
265C4C22×C22⋊D10711yescharacteristic
266C4#1681215411yescharacteristic
267C2—41no—
268C2—82no—
269C2—2142no—
270C2D4×C22⋊D10711yescharacteristic
271C2#135×D10711yescharacteristic
272C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#135×C107C2
socleC22×C107C22×D4
hypercenterC2×#135D107
hypocenterC107#3285
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D107C2semidirect
2. #20293284C2semidirect
3. #20293010C2semidirect
4. #20292868C2semidirect
5. #20292860C2semidirect
6. #20292847C2semidirect
7. #20292789C2semidirect
8. #20292764C2semidirect
9. Q16×Dic107C2semidirect
10. C8×C22⋊D107C2semidirect
11. C2×#135×C107C2semidirect
74. Q16×C107D4semidirect
120. C2×D107#135semidirect
137. C107#3285semidirect
138. C2×#135D107semidirect
195. Q16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
#202932842
#202930102
#202928682
#202928602
#202928472
#202927892
#202927642
Q16×Dic1072
C8×C22⋊D1072
C2×#135×C1072
#3285107