Finite Group Databasebeta

← #28349048

Subgroups — #28349048

Order 13696.

374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 115 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#20293350C211yesmaximal, characteristic, complemented
2#20293280C211yesmaximal, characteristic
3#20293030C211yesmaximal, characteristic, complemented
4#20292961C212yesmaximal, complemented
5#20292953C212yesmaximal, complemented
6#20292908C211yesmaximal, characteristic, complemented
7#20292876C212yesmaximal, complemented
8C4×#13133805C211yesmaximal, characteristic, complemented
9#20292704C211yesmaximal, characteristic, complemented
10#20292666C212yesmaximal, complemented
11#475×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13C2×#13133805C2211yescharacteristic
14C2×#13133801C2212yes—
15#16812109C2211yescharacteristic
16#16812104—22no—
17#16812097—21no—
18#16812094—22no—
19#16812085—21no—
20#16812085C2212yes—
21#16812084C2212yes—
22C4×Dic214—21no—
23C4×D428C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812067C2212yes—
26C4⋊C4×D107C2212yes—
27#16812043C2212yes—
28#16812042C2212yes—
29#16812042C2212yes—
30#16812040C2211yescharacteristic
31#16812033C2211yescharacteristic
32#16812031C2212yes—
33#16812029C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×Q8×C107C2211yescharacteristic
38#106×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133819—44no—
42C2×Dic214—21no—
43C2×D428—22no—
44C2×D428D411yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C4×D107—22no—
47C2×C4×D107—21no—
48C2×C4×D107C2312yes—
49#13133805—22no—
50#13133801—24no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C4⋊Dic107—21no—
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2312yes—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4D412yes—
64C4×Dic107—22no—
65C2×Q8×C107—21no—
66C2×Q8×C107C2311yescharacteristic
67C2×C8×C107C2312yes—
68C4⋊C4×C107—21no—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2311yescharacteristic
71C4⋊C4×C107C2312yes—
72C42×C107—21no—
73C42×C107C2311yescharacteristic
74C22×D107—21no—
75C22×D107C2×D412yes—
76Q8×C107—21no—
77Q8×C107—21no—
78Q8×C107—42no—
79C2×C4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107C2×D412yes—
88D428—22no—
89D428—21no—
90D428—21no—
91D428—44no—
92C4×D107—22no—
93C4×D107—22no—
94C4×D107—44no—
95C4×D107—42no—
96Dic214—42no—
97C8×C107—22no—
98C107⋊C8—21no—
99C107⋊C8#4112yescomplemented
100C22×C107C22×D411yescharacteristic
101C2×D107—22no—
102C2×D107—22no—
103C2×D107—22no—
104C2×D107—41no—
105C2×D107—41no—
106C2×D107—41no—
107C4×C107—21no—
108C4×C107—41no—
109C4×C107—42no—
110C4×C107—41no—
111C4×C107AGL(1,ℤ/8)12yes—
112C4×C107C2×#4111yescharacteristic
113C4×C1072−1+411yescharacteristic
114Dic107—22no—
115Dic107—42no—
116Dic107—41no—
117C2×C107#54711yescharacteristic
118C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
119C2×C107#57811yescharacteristic
120D107—42no—
121D107—81no—
122#3353—1071nomaximal, Sylow
123C107#335311yescharacteristic, complemented, Sylow
124#556—1071no—
125#547—1071no—
126#500—1071no—
127#484—1072no—
128#482—1072no—
129#475D10711yescharacteristic, complemented
130#474—1071no—
131#468—1072no—
132C4×SD16—1071no—
133#436—1071no—
134#428—1072no—
135C2×#41—2141no—
136C2×SD16—1071no—
137C2×M16—1072no—
138C4⋊Q8—1071no—
139C4⋊Q8C2×D10711yescharacteristic
140#126—2142no—
141#124—2141no—
142#123—2142no—
143C22⋊Q8—1072no—
144C22⋊Q8—2141no—
145#121—1072no—
146C4×Q8—2141no—
147C4×Q8C2×D10711yescharacteristic
148C4×D4—1071no—
149C4×D4—1071no—
150#117—1072no—
151C2×C4⋊C4—1072no—
152#107—1072no—
153#107—1072no—
154#107—1072no—
155#106—1071no—
156#106C2×D10712yes—
157C4⋊C8C2×D10711yescharacteristic
158#103—1071no—
159#103C2×D10712yes—
160#102—1072no—
161#102—1071no—
162C22⋊C8—1072no—
163C4×C8—1071no—
164#41—4284no—
165C2×Q8—21no—
166C2×Q8—2141no—
167C2×Q8C22×D10711yescharacteristic
168C2×D4—1071no—
169C2×D4—1071no—
170C2×D4—2142no—
171C22×C4—1072no—
172C22×C4—2142no—
173C22×C4—2141no—
174SD16—2142no—
175M16—2144no—
176C2×C8—1071no—
177C2×C8—1071no—
178C2×C8C22×D10712yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1071no—
182C4⋊C4—1072no—
183C4⋊C4—1071no—
184C4⋊C4—2142no—
185C4⋊C4—2142no—
186C4⋊C4—2141no—
187C4⋊C4C22×D10711yescharacteristic
188C4⋊C4C22×D10712yes—
189C22⋊C4—1072no—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C22⋊C4—2141no—
194C42—21no—
195C42—2142no—
196C42C22×D10711yescharacteristic
197C23—1072no—
198C23—2141no—
199Q8—21no—
200Q8—21no—
201Q8—42no—
202Q8—4282no—
203D4—4284no—
204D4—2142no—
205D4—2141no—
206D4—2141no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—1072no—
211C2×C4—4284no—
212C2×C4—4282no—
213C2×C4—2142no—
214C2×C4—2141no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4D4×D10711yescharacteristic
218C2×C4D4×D10711yescharacteristic
219C2×C4C23×D10711yescharacteristic
220C8—22no—
221C8—1072no—
222C8—2141no—
223C22—4281no—
224C22—4281no—
225C22—4281no—
226C22—2142no—
227C22—2142no—
228C22—2142no—
229C22C2×D4×D10711yescharacteristic
230C4—21no—
231C4—42no—
232C4—41no—
233C4—41no—
234C4—4281no—
235C4—4282no—
236C4—2142no—
237C4#1681213612yes—
238C4#41×D10711yescharacteristic
239C4#1681215711yescharacteristic
240C2—8561no—
241C2—4282no—
242C2#2029327911yescharacteristic
243C2C2×#1681213611yescharacteristic
244C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#475×C107C2
socleC22×C107C22×D4
prime radical#20293280C2
hypercenter#475D107
hypocenterC107#3353
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #20293350C2semidirect
3. #20293030C2semidirect
4. #20292961C2semidirect
5. #20292953C2semidirect
6. #20292908C2semidirect
7. #20292876C2semidirect
8. C4×#13133805C2semidirect
9. #20292704C2semidirect
10. #20292666C2semidirect
11. #475×C107C2semidirect
99. C107⋊C8#41semidirect
123. C107#3353semidirect
129. #475D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202932802
#202930302
#202929612
#202929532
#202929082
#202928762
C4×#131338052
#202927042
#202926662
#475×C1072
#3353107