Finite Group Databasebeta

← #28349183

Subgroups — #28349183

Order 13696.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#20293351C211yesmaximal, characteristic, complemented
2D4×Dic214C211yesmaximal, characteristic, complemented
3#20292988C212yesmaximal, complemented
4#20292977C211yesmaximal, characteristic, complemented
5#20292920C212yesmaximal, complemented
6#20292908C211yesmaximal, characteristic, complemented
7#20292844C212yesmaximal
8#20292754C211yesmaximal, characteristic, complemented
9#20292728C212yesmaximal, complemented
10C4×#13133803C211yesmaximal, characteristic
11#461×C107C211yesmaximal, characteristic, complemented
12C22×Dic214—21no—
13C2×#13133803C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812108C2212yes—
16#16812104—22no—
17#16812098C2212yes—
18#16812089—21no—
19#16812088—21no—
20#16812086—22no—
21Q8×Dic107C2212yes—
22C4×Dic214C2211yescharacteristic
23C4×D428C2211yescharacteristic
24D4×Dic107C2212yes—
25#16812067—21no—
26#16812043C2212yes—
27#16812042C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812033C2212yes—
30#16812033C2212yes—
31#16812031C2212yes—
32#16812030C2211yescharacteristic
33#16812029C2211yescharacteristic
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×SD16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214—44no—
44C2×Dic214D411yescharacteristic
45C2×D428C2311yescharacteristic
46C22×Dic107—22no—
47C2×C4×D107—21no—
48#13133803—22no—
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2311yescharacteristic
51C4⋊Dic107—21no—
52C4⋊Dic107—22no—
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107C2312yes—
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—22no—
57Dic107⋊C4D412yes—
58C22⋊Dic107—22no—
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62C4×Dic107—22no—
63C4×Dic107C2312yes—
64C2×Q8×C107C2312yes—
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—21no—
67SD16×C107—24no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107—21no—
71C4⋊C4×C107C2311yescharacteristic
72C22⋊C4×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—21no—
75C22×D107—21no—
76Q8×C107—22no—
77Q8×C107—22no—
78D4×C107—21no—
79D4×C107#4112yes—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—42no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C2×Dic107—21no—
87C2×Dic107—22no—
88C2×Dic107—44no—
89C2×Dic107C2×D412yes—
90C2×Dic107C2×D412yes—
91D428—21no—
92D428—21no—
93C4×D107—42no—
94Dic214—22no—
95Dic214—44no—
96Dic214—42no—
97C8×C107—22no—
98C107⋊C8—21no—
99C107⋊C8—21no—
100C22×C107—22no—
101C22×C107—41no—
102C22×C107—41no—
103C22×C107C22×D411yescharacteristic
104C2×D107—41no—
105C2×D107—41no—
106C2×D107—41no—
107C4×C107—21no—
108C4×C107—42no—
109C4×C107—41no—
110C4×C107AGL(1,ℤ/8)12yes—
111C4×C107C2×#4111yescharacteristic
112C4×C1072−1+411yescharacteristic
113Dic107—22no—
114Dic107—22no—
115Dic107—42no—
116Dic107—41no—
117C2×C107—42no—
118C2×C107#54811yescharacteristic
119C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C107#57811yescharacteristic
121D107—81no—
122#3367—1071nomaximal, Sylow
123C107#336711yescharacteristic, complemented, Sylow
124#556—1071no—
125D4×Q8—1071no—
126#489—1072no—
127#487—1071no—
128#477—1072no—
129#474—1071no—
130#463—1072no—
131#461D10711yescharacteristic, complemented
132#446—1071no—
133#440—1072no—
134C4×D8—1071no—
135C22×Q8—2141no—
136C2×SD16C2×D10712yes—
137C2×D8—1071no—
138C4⋊Q8—1072no—
139C4⋊Q8—1071no—
140C4⋊Q8C2×D10711yescharacteristic
141#126—2142no—
142#124—1072no—
143C22⋊Q8—2142no—
144C22⋊Q8—2141no—
145C22⋊Q8—2141no—
146C4×Q8—1071no—
147C4×Q8—1072no—
148C4×D4—1072no—
149C4×D4—1071no—
150C4×D4C2×D10711yescharacteristic
151#117—2141no—
152#107—1072no—
153#107—1071no—
154C4⋊C8—1072no—
155C4⋊C8C2×D10711yescharacteristic
156#103—1072no—
157#103—1072no—
158#103C2×D10712yes—
159#102—1071no—
160#102—1072no—
161#102—1071no—
162#97—1072no—
163C4×C8—1071no—
164C2×Q8—1071no—
165C2×Q8—4284no—
166C2×Q8—2142no—
167C2×Q8—2141no—
168C2×Q8C22×D10712yes—
169C2×D4—1071no—
170C2×D4C22×D10711yescharacteristic
171C22×C4—21no—
172C22×C4—2142no—
173C22×C4—2141no—
174SD16—24no—
175D8—2142no—
176C2×C8—1071no—
177C2×C8—1071no—
178C2×C8C22×D10712yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1071no—
182C4⋊C4—1072no—
183C4⋊C4—1072no—
184C4⋊C4—1071no—
185C4⋊C4—2142no—
186C4⋊C4—2142no—
187C4⋊C4—2141no—
188C4⋊C4C22×D10711yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C22⋊C4—2141no—
194C42—1072no—
195C42—2142no—
196C42C22×D10711yescharacteristic
197C23—21no—
198C23—2141no—
199Q8—22no—
200Q8—22no—
201Q8—4284no—
202Q8—4282no—
203Q8—2142no—
204D4—21no—
205D4—2141no—
206D4—2141no—
207D4#1313381712yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—1072no—
212C2×C4—1072no—
213C2×C4—4284no—
214C2×C4—4282no—
215C2×C4—2141no—
216C2×C4—2142no—
217C2×C4D4×D10711yescharacteristic
218C2×C4D4×D10711yescharacteristic
219C2×C4C23×D10711yescharacteristic
220C8—22no—
221C8—2141no—
222C8—2141no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—4281no—
227C22—4281no—
228C22—4281no—
229C22C2×D4×D10711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—4282no—
234C4—4281no—
235C4—2142no—
236C4—2142no—
237C4#1681213612yes—
238C4C2×#1313381711yescharacteristic
239C4#1681215611yescharacteristic
240C2—42no—
241C2—8561no—
242C2#2029328311yescharacteristic
243C2C2×#1681213611yescharacteristic
244C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#461D107
hypocenterC107#3367
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #20293351C2semidirect
2. D4×Dic214C2semidirect
3. #20292988C2semidirect
4. #20292977C2semidirect
5. #20292920C2semidirect
6. #20292908C2semidirect
8. #20292754C2semidirect
9. #20292728C2semidirect
11. #461×C107C2semidirect
123. C107#3367semidirect
131. #461D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
D4×Dic2142
#202929882
#202929772
#202929202
#202929082
#202928442
#202927542
#202927282
C4×#131338032
#461×C1072
#3367107