Finite Group Databasebeta

← #28348179

Subgroups — #28348179

Order 13696.

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

normal subgroups only

This group has 484 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×#16812129C212yesmaximal, complemented
2#20293282C211yesmaximal, characteristic, complemented
3#20293006C211yesmaximal, characteristic, complemented
4#20292866C212yesmaximal, complemented
5#20292832C211yesmaximal, characteristic, complemented
6#20292829C211yesmaximal, characteristic, complemented
7#20292789C212yesmaximal, complemented
8#20292787C212yesmaximal, complemented
9C4×#13133803C211yesmaximal, characteristic, complemented
10D4×C107⋊C8C211yesmaximal, characteristic, complemented
11#547×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13#16812129—28no—
14C2×#13133808C2212yes—
15C2×#13133808C2212yes—
16C2×#13133806C2212yes—
17C2×#13133805C2212yes—
18C2×#13133803C2211yescharacteristic
19C22×C107⋊C8C2212yes—
20#16812109C2211yescharacteristic
21#16812091—22no—
22#16812088C2212yes—
23#16812080—21no—
24C4×D428C2211yescharacteristic
25C4⋊C4×D107—21no—
26#16812042C2211yescharacteristic
27C4⋊(C107⋊C8)C2211yescharacteristic
28#16812033C2212yes—
29#16812032C2212yes—
30#16812030C2211yescharacteristic
31#16812029C2211yescharacteristic
32C22⋊(C107⋊C8)C2212yes—
33C4×C107⋊C8C2211yescharacteristic
34C2×#41×C107C2212yes—
35C4⋊Q8×C107C2211yescharacteristic
36#123×C107—22no—
37C22⋊Q8×C107C2212yes—
38#121×C107—21no—
39C4×D4×C107C2211yescharacteristic
40C4×D4×C107C2211yescharacteristic
41C2×C4⋊C4×C107—21no—
42#13133818—24no—
43#13133818—42no—
44C2×Dic214C2312yes—
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—22no—
47C2×C4×D107—21no—
48C2×C4×D107—22no—
49#13133808—24no—
50#13133808—42no—
51#13133806—24no—
52#13133805—24no—
53#13133803D414yescomplemented
54C2×C107⋊C8—24no—
55C2×C107⋊C8C2312yes—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C4⋊Dic107—21no—
59C4⋊Dic107—21no—
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—22no—
62C22⋊Dic107—22no—
63(C2×D107)⋊C4—21no—
64#41×C107—24no—
65#41×C107—42no—
66C2×Q8×C107C2312yes—
67C2×D4×C107—22no—
68C2×D4×C107D411yescharacteristic
69C2×D4×C107C2311yescharacteristic
70C22×C4×C107—21no—
71C22×C4×C107—22no—
72C22×C4×C107C2312yes—
73C4⋊C4×C107—21no—
74C4⋊C4×C107—22no—
75C4⋊C4×C107—21no—
76C4⋊C4×C107D411yescharacteristic
77C4⋊C4×C107C2311yescharacteristic
78C22⋊C4×C107—22no—
79C22⋊C4×C107—21no—
80C22⋊C4×C107D412yes—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C23×C107C2×D412yes—
84C22×D107—21no—
85Q8×C107—22no—
86Q8×C107—22no—
87D4×C107—22no—
88D4×C107—21no—
89D4×C107—44no—
90D4×C107C2×D412yes—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—44no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—44no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104D428—21no—
105D428C2×D412yes—
106C4×D107—44no—
107C4×D107—42no—
108Dic214—22no—
109Dic214—22no—
110C107⋊C8—22no—
111C107⋊C8—21no—
112C107⋊C8C2×D412yes—
113C22×C107—22no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—41no—
118C22×C107—41no—
119C22×C107C22×D411yescharacteristic
120C2×D107—22no—
121C2×D107—41no—
122C2×D107—41no—
123C4×C107—22no—
124C4×C107—21no—
125C4×C107—42no—
126C4×C107—41no—
127C4×C107#13512yescomplemented
128C4×C107C22×D411yescharacteristic
129C4×C107H(5,𝔽2)11yescharacteristic
130Dic107—42no—
131Dic107—41no—
132C2×C107—42no—
133C2×C107—42no—
134C2×C107D4211yescharacteristic
135C2×C107C2×#13511yescharacteristic
136C2×C107#57811yescharacteristic
137D107—42no—
138#3283—1071nomaximal, Sylow
139C107#328311yescharacteristic, complemented, Sylow
140C2×#135—1072no—
141#547—1071no—
142#547D10711yescharacteristic, complemented
143#494—1071no—
144#467—1072no—
145#461—1071no—
146#461—1071no—
147#452—1072no—
148#452—1072no—
149C4×D8—1071no—
150C8×D4—1071no—
151C2×#41—1072no—
152C2×#41C2×D10712yes—
153#135—2148no—
154C2×Q16—1072no—
155C2×Q16—1072no—
156C2×SD16—1072no—
157C2×SD16—1072no—
158C2×D8—1071no—
159C22×C8—1072no—
160C4⋊Q8—1071no—
161C4⋊Q8C2×D10711yescharacteristic
162#123—22no—
163#123—2142no—
164C22⋊Q8—1072no—
165C22⋊Q8C2×D10712yes—
166#121—21no—
167#121—2141no—
168C4×D4—1071no—
169C4×D4C2×D10711yescharacteristic
170C4×D4C2×D10711yescharacteristic
171C2×C4⋊C4—21no—
172C2×C4⋊C4—2141no—
173#107—1071no—
174C4⋊C8—1071no—
175#103—1072no—
176#103—1072no—
177#102—1071no—
178#102—1071no—
179C22⋊C8—1072no—
180C4×C8—1071no—
181#41—24no—
182#41—42no—
183#41—4282no—
184#41—2144no—
185C2×Q8—1072no—
186C2×Q8C22×D10712yes—
187C2×D4—22no—
188C2×D4—1071no—
189C2×D4—2142no—
190C2×D4C22⋊D10711yescharacteristic
191C2×D4C22×D10711yescharacteristic
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—2141no—
195C22×C4—2142no—
196C22×C4C22×D10712yes—
197Q16—4282no—
198Q16—2144no—
199SD16—2144no—
200SD16—2144no—
201D8—1074no—
202C2×C8—1071no—
203C2×C8—1071no—
204C2×C8—1072no—
205C2×C8—2144no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—1071no—
210C4⋊C4—2141no—
211C4⋊C4—2142no—
212C4⋊C4—2141no—
213C4⋊C4C22⋊D10711yescharacteristic
214C4⋊C4C22×D10711yescharacteristic
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—2142no—
218C22⋊C4—2141no—
219C22⋊C4C22⋊D10712yes—
220C42C22×D10711yescharacteristic
221C23—21no—
222C23—2141no—
223C23C2×C22⋊D10712yes—
224Q8—22no—
225Q8—22no—
226Q8—2142no—
227Q8—2142no—
228D4—22no—
229D4—21no—
230D4—44no—
231D4—1072no—
232D4—4284no—
233D4—2141no—
234D4D4×D10712yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—4284no—
242C2×C4—4282no—
243C2×C4—2141no—
244C2×C4—2142no—
245C2×C4C2×C22⋊D10711yescharacteristic
246C2×C4C2×C22⋊D10712yes—
247C2×C4C2×C22⋊D10711yescharacteristic
248C2×C4C23×D10711yescharacteristic
249C8—1072no—
250C8—2142no—
251C8—2141no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—4281no—
259C22—4281no—
260C22—2142no—
261C22C22×C22⋊D10711yescharacteristic
262C4—21no—
263C4—22no—
264C4—41no—
265C4—42no—
266C4—4281no—
267C4—4282no—
268C4#1681212912yescomplemented
269C4C2×D4×D10711yescharacteristic
270C4#1681215411yescharacteristic
271C2—42no—
272C2—42no—
273C2—4282no—
274C2D4×C22⋊D10711yescharacteristic
275C2C2×#1681212911yescharacteristic
276C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3283
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16812129C2semidirect
2. #20293282C2semidirect
3. #20293006C2semidirect
4. #20292866C2semidirect
5. #20292832C2semidirect
6. #20292829C2semidirect
7. #20292789C2semidirect
8. #20292787C2semidirect
9. C4×#13133803C2semidirect
10. D4×C107⋊C8C2semidirect
11. #547×C107C2semidirect
53. #13133803D4semidirect
127. C4×C107#135semidirect
139. C107#3283semidirect
142. #547D107semidirect
268. C4#16812129semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
#202932822
#202930062
#202928662
#202928322
#202928292
#202927892
#202927872
C4×#131338032
D4×C107⋊C82
#547×C1072
#3283107