Finite Group Databasebeta

← #28348177

Subgroups — #28348177

Order 13696.

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

normal subgroups only

This group has 484 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×#16812128C211yesmaximal, characteristic, complemented
2#20293276C212yesmaximal, complemented
3#20293005C211yesmaximal, characteristic, complemented
4#20292868C211yesmaximal, characteristic, complemented
5#20292865C211yesmaximal, characteristic, complemented
6#20292830C212yesmaximal, complemented
7#20292789C212yesmaximal, complemented
8#20292786C212yesmaximal, complemented
9D8×Dic107C211yesmaximal, characteristic, complemented
10C8×C22⋊D107C211yesmaximal, characteristic, complemented
11C2×#135×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13#16812128—24no—
14C2×Dic428C2211yescharacteristic
15C2×#13133808C2212yes—
16C2×#13133806C2212yes—
17C2×C8×D107C2211yescharacteristic
18#16812107C2212yes—
19#16812092—22no—
20#16812090—22no—
21#16812088C2212yes—
22#16812085C2212yes—
23#16812081—22no—
24D4×Dic107C2212yes—
25C4×C22⋊D107C2211yescharacteristic
26C2×Dic107⋊C4—22no—
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
41#13133817—24no—
42#13133817—42no—
43C2×Dic214C2312yes—
44C2×C22⋊D107—22no—
45C2×C22⋊D107D411yescharacteristic
46C22×Dic107—22no—
47C22×Dic107—22no—
48C2×C4×D107C2311yescharacteristic
49Dic428—22no—
50#13133808—42no—
51#13133806—24no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107—22no—
55C4⋊Dic107C2312yes—
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59Dic107⋊C4—22no—
60Dic107⋊C4D411yescharacteristic
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63C22⋊Dic107D411yescharacteristic
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4D411yescharacteristic
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×C107—22no—
75SD16×C107—24no—
76D8×C107D414yescomplemented
77C2×C8×C107—22no—
78C2×C8×C107C2311yescharacteristic
79C2×C8×C107C2311yescharacteristic
80C23×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85D4×C107—22no—
86D4×C107—44no—
87D4×C107C2×D414yes—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107—44no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C22⋊D107—22no—
95C22⋊D107—44no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107—44no—
99C2×Dic107—44no—
100C2×Dic107C2×D411yescharacteristic
101C2×Dic107C2×D411yescharacteristic
102C2×Dic107C2×D411yescharacteristic
103C4×D107—21no—
104C4×D107—21no—
105Dic214—22no—
106Dic214—22no—
107C8×C107—21no—
108C8×C107C2×D412yes—
109C107⋊C8—21no—
110C107⋊C8—21no—
111C22×C107—24no—
112C22×C107—21no—
113C22×C107—21no—
114C22×C107—21no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107C22×D411yescharacteristic
118C2×D107—21no—
119C2×D107—21no—
120C2×D107—21no—
121C4×C107—21no—
122C4×C107—42no—
123C4×C107C22×D411yescharacteristic
124C4×C107H(5,𝔽2)11yescharacteristic
125Dic107—21no—
126Dic107—21no—
127Dic107—42no—
128Dic107—42no—
129Dic107#13512yescomplemented
130C2×C107—41no—
131C2×C107—44no—
132C2×C107D4211yescharacteristic
133C2×C107C2×#13511yescharacteristic
134C2×C107#57811yescharacteristic
135D107—41no—
136#3283—1071nomaximal, Sylow
137C107#328311yescharacteristic, complemented, Sylow
138C2×#135—1071no—
139C2×#135D10711yescharacteristic, complemented
140#547—1072no—
141#494—1071no—
142#467—1071no—
143#467—1071no—
144#461—1072no—
145#452—1072no—
146#452—1072no—
147C4×D8—1071no—
148C8×D4—1071no—
149C2×#41—1072no—
150C2×#41C2×D10712yes—
151#135—24no—
152#135—2144no—
153C2×Q16—1072no—
154C2×Q16—1071no—
155C2×Q16C2×D10711yescharacteristic
156C2×SD16—1072no—
157C2×SD16C2×D10712yes—
158C2×D8C2×D10711yescharacteristic
159C22×C8—1071no—
160C22×C8C2×D10711yescharacteristic
161C4⋊Q8—1072no—
162#123—2142no—
163#123—2142no—
164C22⋊Q8—1072no—
165C22⋊Q8—1072no—
166#121—2142no—
167C4×D4—1071no—
168C4×D4—1072no—
169C2×C4⋊C4—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—
180#41—4282no—
181#41—2144no—
182C2×Q8—1072no—
183C2×Q8C22×D10712yes—
184C2×D4—22no—
185C2×D4—1071no—
186C2×D4—2142no—
187C2×D4C22×D10712yes—
188C22×C4—22no—
189C22×C4—1071no—
190C22×C4—2142no—
191C22×C4—2142no—
192C22×C4C22×D10711yescharacteristic
193Q16—22no—
194Q16—4282no—
195Q16—2142no—
196SD16—24no—
197SD16—2144no—
198D8C22⋊D10714yescomplemented
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—
209C4⋊C4—2142no—
210C4⋊C4—2142no—
211C22⋊C4—1071no—
212C22⋊C4—1071no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C22⋊C4—2142no—
216C42—1071no—
217C23—22no—
218C23—1071no—
219C23D4×D10711yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—2142no—
223Q8—2142no—
224D4—22no—
225D4—44no—
226D4—4284no—
227D4—2142no—
228D4C2×C22⋊D10714yes—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—44no—
233C2×C4—1071no—
234C2×C4—1071no—
235C2×C4—1071no—
236C2×C4—4284no—
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—24no—
252C22—42no—
253C22—42no—
254C22—2141no—
255C22—2141no—
256C22—2141no—
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—42no—
260C4—1072no—
261C4—4282no—
262C4—4282no—
263C4—2141no—
264C4—2141no—
265C4C22×C22⋊D10711yescharacteristic
266C4#1681215411yescharacteristic
267C2—41no—
268C2—44no—
269C2—4281no—
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#3283
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×#16812128C2semidirect
2. #20293276C2semidirect
3. #20293005C2semidirect
4. #20292868C2semidirect
5. #20292865C2semidirect
6. #20292830C2semidirect
7. #20292789C2semidirect
8. #20292786C2semidirect
9. D8×Dic107C2semidirect
10. C8×C22⋊D107C2semidirect
11. C2×#135×C107C2semidirect
76. D8×C107D4semidirect
129. Dic107#135semidirect
137. C107#3283semidirect
139. C2×#135D107semidirect
198. D8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121282
#202932762
#202930052
#202928682
#202928652
#202928302
#202927892
#202927862
D8×Dic1072
C8×C22⋊D1072
C2×#135×C1072
#3283107