Finite Group Databasebeta

← #28348103

Subgroups — #28348103

Order 13696.

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

normal subgroups only

This group has 472 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×#16812139C212yesmaximal, complemented
2D4×Dic214C211yesmaximal, characteristic, complemented
3#20293006C211yesmaximal, characteristic, complemented
4#20292883C212yesmaximal, complemented
5#20292834C211yesmaximal, characteristic, complemented
6#20292832C211yesmaximal, characteristic, complemented
7#20292789C212yesmaximal, complemented
8#20292782C212yesmaximal, complemented
9C4×#13133806C211yesmaximal, characteristic, complemented
10#20292705C211yesmaximal, characteristic, complemented
11#547×C107C211yesmaximal, characteristic
12C22×Dic214C2212yes—
13#16812139—28no—
14C2×#13133808C2212yes—
15C2×#13133808C2212yes—
16C2×#13133808C2212yes—
17C2×#13133806C2212yes—
18C2×#13133806C2211yescharacteristic
19C2×#13133802C2212yes—
20#16812109C2211yescharacteristic
21#16812108—21no—
22#16812088C2212yes—
23#16812086—22no—
24C4×Dic214C2211yescharacteristic
25D4×Dic107—21no—
26#16812040C2211yescharacteristic
27C4⋊(C107⋊C8)C2211yescharacteristic
28#16812033C2212yes—
29#16812032C2212yes—
30#16812032C2211yescharacteristic
31#16812030C2211yescharacteristic
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—
42C2×Dic214—24no—
43C2×Dic214—22no—
44C2×Dic214—42no—
45C2×Dic214C2312yes—
46C2×Dic214C2311yescharacteristic
47C22×Dic107—22no—
48#13133808—24no—
49#13133808—24no—
50#13133808—42no—
51#13133806—24no—
52#13133806D414yescomplemented
53#13133802—24no—
54C2×C107⋊C8C2312yes—
55C2×C107⋊C8C2311yescharacteristic
56C2×C107⋊C8C2311yescharacteristic
57C4⋊Dic107—21no—
58C4⋊Dic107—21no—
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—21no—
61Dic107⋊C4—22no—
62C22⋊Dic107—22no—
63C4×Dic107—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—
84Q8×C107—22no—
85Q8×C107—22no—
86D4×C107—22no—
87D4×C107—21no—
88D4×C107—44no—
89D4×C107C2×D412yes—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—44no—
95C2×C4×C107—42no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—44no—
104Dic214—21no—
105Dic214—22no—
106Dic214—22no—
107Dic214—44no—
108Dic214C2×D412yes—
109C107⋊C8—22no—
110C107⋊C8—21no—
111C107⋊C8C2×D412yes—
112C22×C107—22no—
113C22×C107—22no—
114C22×C107—22no—
115C22×C107—22no—
116C22×C107—41no—
117C22×C107—41no—
118C22×C107C22×D411yescharacteristic
119C4×C107—22no—
120C4×C107—21no—
121C4×C107—42no—
122C4×C107—41no—
123C4×C107#13712yescomplemented
124C4×C107C22×D411yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—22no—
127Dic107—42no—
128Dic107—41no—
129C2×C107—42no—
130C2×C107—42no—
131C2×C107D4211yescharacteristic
132C2×C107C2×#13711yescharacteristic
133C2×C107#57811yescharacteristic
134#3276—1071nomaximal, Sylow
135C107#327611yescharacteristic, complemented, Sylow
136C2×#137—1072no—
137D4×Q8—1071no—
138#547D10711yescharacteristic
139#494—1071no—
140#470—1072no—
141#462—1071no—
142#461—1071no—
143#452—1072no—
144#451—1072no—
145C4×SD16—1071no—
146#436—1071no—
147C2×#41C2×D10712yes—
148C22×Q8—1072no—
149#137—2148no—
150C2×Q16—1072no—
151C2×Q16—1072no—
152C2×Q16—1072no—
153C2×SD16—1071no—
154C2×SD16—1072no—
155C2×M16—1072no—
156C4⋊Q8—1071no—
157C4⋊Q8—2141no—
158C4⋊Q8C2×D10711yescharacteristic
159#123—22no—
160C22⋊Q8—1072no—
161C22⋊Q8—2142no—
162C22⋊Q8C2×D10712yes—
163#121—21no—
164C4×Q8—1071no—
165C4×D4—2141no—
166C4×D4C2×D10711yescharacteristic
167C4×D4C2×D10711yescharacteristic
168C2×C4⋊C4—21no—
169#106—1071no—
170C4⋊C8—1071no—
171#103—1072no—
172#103—1071no—
173#103—1072no—
174#102—1071no—
175C22⋊C8—1072no—
176C4×C8—1071no—
177#41—24no—
178#41—42no—
179C2×Q8—1071no—
180C2×Q8—1072no—
181C2×Q8—4282no—
182C2×Q8—2144no—
183C2×Q8—2142no—
184C2×Q8C22×D10712yes—
185C2×D4—22no—
186C2×D4C22⋊D10711yescharacteristic
187C2×D4C22×D10711yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—2142no—
191C22×C4C22×D10712yes—
192Q16—4282no—
193Q16—2144no—
194Q16—2144no—
195SD16—1074no—
196SD16—2144no—
197M16—2144no—
198C2×C8—1071no—
199C2×C8—1071no—
200C2×C8—1072no—
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—1071no—
205C4⋊C4—2141no—
206C4⋊C4—2141no—
207C4⋊C4—2142no—
208C4⋊C4—2141no—
209C4⋊C4C22⋊D10711yescharacteristic
210C4⋊C4C22×D10711yescharacteristic
211C22⋊C4—22no—
212C22⋊C4—21no—
213C22⋊C4—2142no—
214C22⋊C4C22⋊D10712yes—
215C42—2141no—
216C42C22×D10711yescharacteristic
217C23—21no—
218C23C2×C22⋊D10712yes—
219Q8—22no—
220Q8—22no—
221Q8—1072no—
222Q8—4284no—
223Q8—2142no—
224Q8—2142no—
225Q8—2141no—
226D4—22no—
227D4—21no—
228D4—44no—
229D4D4×D10712yes—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—4284no—
237C2×C4—2141no—
238C2×C4—2142no—
239C2×C4—2141no—
240C2×C4C2×C22⋊D10711yescharacteristic
241C2×C4C2×C22⋊D10712yes—
242C2×C4C2×C22⋊D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—1072no—
245C8—2142no—
246C8—2141no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—41no—
252C22—41no—
253C22C22×C22⋊D10711yescharacteristic
254C4—21no—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—4281no—
259C4—4282no—
260C4—2142no—
261C4#1681213912yescomplemented
262C4C2×D4×D10711yescharacteristic
263C4#1681215411yescharacteristic
264C2—42no—
265C2—42no—
266C2D4×C22⋊D10711yescharacteristic
267C2C2×#1681213911yescharacteristic
268C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3276
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×#16812139C2semidirect
2. D4×Dic214C2semidirect
3. #20293006C2semidirect
4. #20292883C2semidirect
5. #20292834C2semidirect
6. #20292832C2semidirect
7. #20292789C2semidirect
8. #20292782C2semidirect
9. C4×#13133806C2semidirect
10. #20292705C2semidirect
52. #13133806D4semidirect
123. C4×C107#137semidirect
135. C107#3276semidirect
261. C4#16812139semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
D4×Dic2142
#202930062
#202928832
#202928342
#202928322
#202927892
#202927822
C4×#131338062
#202927052
#547×C1072
#3276107