Finite Group Databasebeta

← #28348202

Subgroups — #28348202

Order 13696.

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

normal subgroups only

This group has 482 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
2#20293288C212yesmaximal, complemented
3#20293012C211yesmaximal, characteristic, complemented
4#20292865C211yesmaximal, characteristic, complemented
5#20292863C211yesmaximal, characteristic, complemented
6#20292851C212yesmaximal, complemented
7#20292788C212yesmaximal, complemented
8#20292766C212yesmaximal, complemented
9Q16×Dic107C211yesmaximal, characteristic, complemented
10C8×C22⋊D107C211yesmaximal, characteristic, complemented
11C22×Q16×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13#16812132—24no—
14C2×#13133805C2212yes—
15C2×#13133805C2212yes—
16C2×D856C2211yescharacteristic
17C2×C8×D107C2211yescharacteristic
18#16812098—22no—
19#16812098C2212yes—
20#16812085—22no—
21#16812085C2212yes—
22#16812083—22no—
23#16812083C2212yes—
24Q8×Dic107C2212yes—
25C4×C22⋊D107—22no—
26C4×C22⋊D107C2211yescharacteristic
27#16812043C2211yescharacteristic
28Dic107⋊C8C2211yescharacteristic
29#16812033C2212yes—
30#16812033C2212yes—
31#16812031C2212yes—
32C22⋊(C107⋊C8)C2211yescharacteristic
33(C2×D107)⋊C8C2211yescharacteristic
34C8×Dic107C2211yescharacteristic
35C22×Q8×C107C2212yes—
36C2×Q16×C107—24no—
37C2×Q16×C107C2211yescharacteristic
38C2×Q16×C107C2212yes—
39C2×Q16×C107C2211yescharacteristic
40C22×C8×C107C2211yescharacteristic
41#13133819—24no—
42#13133819—42no—
43C2×D428—22no—
44C2×D428C2312yes—
45C2×C22⋊D107—22no—
46C2×C22⋊D107D411yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107C2311yescharacteristic
49#13133805—24no—
50#13133805—42no—
51D856—22no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107—22no—
55C4⋊Dic107—22no—
56C4⋊Dic107C2312yes—
57Dic107⋊C4—22no—
58Dic107⋊C4D411yescharacteristic
59C22⋊Dic107D411yescharacteristic
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4D411yescharacteristic
65C4×Dic107—22no—
66C4×Dic107C2311yescharacteristic
67C2×Q8×C107—22no—
68C2×Q8×C107—24no—
69C2×Q8×C107—42no—
70C2×Q8×C107C2312yes—
71C2×Q8×C107C2312yes—
72C22×C4×C107—22no—
73C22×C4×C107C2311yescharacteristic
74Q16×C107—24no—
75Q16×C107—22no—
76Q16×C107D414yescomplemented
77C2×C8×C107—22no—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C23×C107C2×D411yescharacteristic
81C22×D107—22no—
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85Q8×C107—22no—
86Q8×C107—44no—
87Q8×C107C2×D414yes—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—44no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—22no—
96C22⋊D107—44no—
97C2×Dic107—22no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101D428—22no—
102D428—22no—
103D428—44no—
104C4×D107—21no—
105C4×D107—21no—
106C4×D107—44no—
107C8×C107—21no—
108C8×C107C2×D412yes—
109C107⋊C8—21no—
110C107⋊C8—21no—
111C22×C107—21no—
112C22×C107—21no—
113C22×C107#13512yescomplemented
114C22×C107C22×D411yescharacteristic
115C2×D107—21no—
116C2×D107—21no—
117C2×D107—21no—
118C2×D107—42no—
119C2×D107—42no—
120C2×D107—42no—
121C4×C107—24no—
122C4×C107—21no—
123C4×C107—42no—
124C4×C107C22×D411yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—21no—
127Dic107—21no—
128Dic107—21no—
129Dic107—42no—
130C2×C107—22no—
131C2×C107D4211yescharacteristic
132C2×C107C2×#13511yescharacteristic
133C2×C107#57811yescharacteristic
134D107—41no—
135D107—82no—
136#3285—1071nomaximal, Sylow
137C107#328511yescharacteristic, complemented, Sylow
138C2×#135—1071no—
139C22×Q16D10711yescharacteristic, complemented
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×#41—1072no—
150C22×Q8C2×D10712yes—
151#135—2144no—
152C2×Q16—24no—
153C2×Q16C2×D10712yes—
154C2×Q16C2×D10711yescharacteristic
155C2×Q16C2×D10711yescharacteristic
156C2×SD16—1072no—
157C2×SD16—1072no—
158C2×D8—1071no—
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—4282no—
179#41—2144no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D10712yes—
184C2×Q8C22×D10712yes—
185C2×D4—1071no—
186C2×D4—1072no—
187C2×D4—2142no—
188C2×D4—2142no—
189C22×C4—22no—
190C22×C4—1071no—
191C22×C4—2142no—
192C22×C4C22×D10711yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D10714yescomplemented
196SD16—4282no—
197SD16—2144no—
198D8—2142no—
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—1071no—
218C23—2142no—
219C23D4×D10711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D10714yes—
225D4—4284no—
226D4—4284no—
227D4—2142no—
228D4—2142no—
229D4—2142no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—1071no—
236C2×C4—1071no—
237C2×C4—1071no—
238C2×C4—4284no—
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—4282no—
251C22—4282no—
252C22—4282no—
253C22—2141no—
254C22—2141no—
255C22—2141no—
256C22#1681213212yescomplemented
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—4282no—
262C4—2141no—
263C4—2141no—
264C4—2141no—
265C4C22×C22⋊D10711yescharacteristic
266C4#1681215411yescharacteristic
267C2—22no—
268C2—8562no—
269C2—4281no—
270C2D4×C22⋊D10711yescharacteristic
271C2C2×#1681213211yescharacteristic
272C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×Q16×C107C2
socleC22×C107C22×D4
hypercenterC22×Q16D107
hypocenterC107#3285
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. #20293288C2semidirect
3. #20293012C2semidirect
4. #20292865C2semidirect
5. #20292863C2semidirect
6. #20292851C2semidirect
7. #20292788C2semidirect
8. #20292766C2semidirect
9. Q16×Dic107C2semidirect
10. C8×C22⋊D107C2semidirect
11. C22×Q16×C107C2semidirect
76. Q16×C107D4semidirect
113. C22×C107#135semidirect
137. C107#3285semidirect
139. C22×Q16D107semidirect
195. Q16C22⋊D107semidirect
256. C22#16812132semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
#202932882
#202930122
#202928652
#202928632
#202928512
#202927882
#202927662
Q16×Dic1072
C8×C22⋊D1072
C22×Q16×C1072
#3285107