Finite Group Databasebeta

← #28348381

Subgroups — #28348381

Order 13696.

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

normal subgroups only

This group has 386 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#20292977C211yesmaximal, characteristic, complemented
4#20292967C212yesmaximal, complemented
5#20292908C211yesmaximal, characteristic, complemented
6#20292865C212yesmaximal, complemented
7#20292788C212yesmaximal, complemented
8C4×#13133805C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#20292622C212yesmaximal, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133805C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812104—22no—
17#16812097—21no—
18#16812094—22no—
19#16812085C2212yes—
20#16812085C2212yes—
21#16812084—21no—
22C4×Dic214—21no—
23C4×D428C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812067C2212yes—
26C4⋊C4×D107—21no—
27#16812043C2212yes—
28#16812040C2211yescharacteristic
29#16812033C2211yescharacteristic
30#16812033C2212yes—
31#16812033C2212yes—
32#16812031C2212yes—
33#16812029C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×Q16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133819—24no—
42#13133819—42no—
43C2×Dic214—21no—
44C2×D428—22no—
45C2×D428D411yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C4×D107—21no—
48C2×C4×D107—22no—
49C2×C4×D107C2312yes—
50#13133805—22no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C8×D107—42no—
54C4⋊Dic107—21no—
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—22no—
59Dic107⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4D412yes—
65C4×Dic107—22no—
66C2×Q8×C107C2312yes—
67C2×Q8×C107C2311yescharacteristic
68Q16×C107—42no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107—21no—
71C4⋊C4×C107—21no—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2311yescharacteristic
74C42×C107—21no—
75C42×C107C2311yescharacteristic
76C22×D107—21no—
77C22×D107C2×D412yes—
78Q8×C107—21no—
79Q8×C107—22no—
80Q8×C107—21no—
81Q8×C107—22no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2411yescharacteristic
88C2×Dic107—22no—
89C2×Dic107—21no—
90C2×Dic107C2×D412yes—
91D428—22no—
92D428—21no—
93D428—44no—
94D428#4112yes—
95C4×D107—22no—
96C4×D107—22no—
97C4×D107—44no—
98C4×D107—42no—
99Dic214—42no—
100C8×C107—22no—
101C107⋊C8—21no—
102C107⋊C8—21no—
103C22×C107C22×D411yescharacteristic
104C2×D107—22no—
105C2×D107—22no—
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—41no—
109C2×D107—41no—
110C4×C107—21no—
111C4×C107—42no—
112C4×C107—41no—
113C4×C107—41no—
114C4×C107#13512yes—
115C4×C107C2×#4111yescharacteristic
116C4×C107H(5,𝔽2)11yescharacteristic
117Dic107—22no—
118Dic107—42no—
119Dic107—41no—
120C2×C107#54611yescharacteristic
121C2×C107C2×#13511yescharacteristic
122C2×C107#57811yescharacteristic
123D107—42no—
124D107—42no—
125#3302—1071nomaximal, Sylow
126C107#330211yescharacteristic, complemented, Sylow
127#556—1071no—
128#547—1071no—
129#487—1071no—
130#485—1072no—
131#474—1071no—
132#467—1072no—
133#462D10711yescharacteristic, complemented
134#452—1072no—
135C4×SD16—1071no—
136C8×D4—1071no—
137#416—1072no—
138C2×#41—1072no—
139C2×Q16C2×D10712yes—
140C2×SD16—1071no—
141C22×C8—1072no—
142C4⋊Q8—1071no—
143C4⋊Q8C2×D10711yescharacteristic
144#126—2142no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1072no—
148C22⋊Q8—1072no—
149#121—2141no—
150C4×Q8—2141no—
151C4×Q8C2×D10711yescharacteristic
152C4×D4—1071no—
153C4×D4—1071no—
154#117—1072no—
155C2×C4⋊C4—2141no—
156#107—1072no—
157#106—1071no—
158C4⋊C8C2×D10711yescharacteristic
159#103—1072no—
160#103—1071no—
161#103—1072no—
162#103C2×D10712yes—
163#102—1072no—
164#102—1071no—
165C22⋊C8—1072no—
166C4×C8—1071no—
167#41—4282no—
168#41—2144no—
169C2×Q8—2141no—
170C2×Q8C22×D10712yes—
171C2×Q8C22×D10711yescharacteristic
172C2×D4—1071no—
173C2×D4—1071no—
174C2×D4—2142no—
175C22×C4—1072no—
176C22×C4—2142no—
177C22×C4—2141no—
178Q16—42no—
179SD16—2142no—
180C2×C8—1071no—
181C2×C8—1071no—
182C2×C8—4282no—
183C2×C8C22×D10712yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—1071no—
188C4⋊C4—1072no—
189C4⋊C4—1071no—
190C4⋊C4—2142no—
191C4⋊C4—2142no—
192C4⋊C4—2141no—
193C4⋊C4C22×D10711yescharacteristic
194C22⋊C4—1072no—
195C22⋊C4—2142no—
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2141no—
199C42—21no—
200C42—2142no—
201C42C22×D10711yescharacteristic
202C23—1072no—
203C23—2141no—
204Q8—21no—
205Q8—22no—
206Q8—21no—
207Q8—22no—
208Q8—4282no—
209D4—1072no—
210D4—4284no—
211D4—2142no—
212D4—2141no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—1072no—
217C2×C4—4284no—
218C2×C4—4282no—
219C2×C4—2142no—
220C2×C4—2141no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4D4×D10711yescharacteristic
224C2×C4D4×D10711yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—22no—
227C8—2141no—
228C8—2141no—
229C22—4281no—
230C22—4281no—
231C22—2142no—
232C22—2142no—
233C22—2142no—
234C22—2142no—
235C22C2×D4×D10711yescharacteristic
236C4—21no—
237C4—41no—
238C4—42no—
239C4—41no—
240C4—4281no—
241C4—4282no—
242C4—2142no—
243C4#1681213212yes—
244C4#41×D10711yescharacteristic
245C4#1681215411yescharacteristic
246C2—4282no—
247C2—4282no—
248C2#2029327411yescharacteristic
249C2C2×#1681213211yescharacteristic
250C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radical#20293280C2
hypercenter#462D107
hypocenterC107#3302
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #20293350C2semidirect
3. #20292977C2semidirect
4. #20292967C2semidirect
5. #20292908C2semidirect
6. #20292865C2semidirect
7. #20292788C2semidirect
8. C4×#13133805C2semidirect
9. #20292692C2semidirect
10. #20292622C2semidirect
11. #462×C107C2semidirect
126. C107#3302semidirect
133. #462D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202932802
#202929772
#202929672
#202929082
#202928652
#202927882
C4×#131338052
#202926922
#202926222
#462×C1072
#3302107