Finite Group Databasebeta

← #28349252

Subgroups — #28349252

Order 13696.

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

normal subgroups only

This group has 384 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#20293362C211yesmaximal, characteristic, complemented
2#20293292C211yesmaximal, characteristic, complemented
3#20292988C212yesmaximal, complemented
4#20292975C211yesmaximal, characteristic, complemented
5#20292916C212yesmaximal, complemented
6#20292845C212yesmaximal, complemented
7#20292823C211yesmaximal, characteristic
8#20292754C211yesmaximal, characteristic, complemented
9#20292738C212yesmaximal, complemented
10C4×#13133805C211yesmaximal, characteristic, complemented
11#475×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107—21no—
13C2×#13133805C2211yescharacteristic
14C2×D856C2212yes—
15#16812108C2212yes—
16#16812106C2211yescharacteristic
17#16812098C2212yes—
18#16812097—21no—
19#16812094—22no—
20#16812087—22no—
21#16812085—21no—
22C4×Dic214C2211yescharacteristic
23C4×D428—21no—
24C4×D428C2211yescharacteristic
25#16812071C2212yes—
26#16812071C2212yes—
27#16812040C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812033C2211yescharacteristic
30#16812031C2212yes—
31#16812029C2212yes—
32#16812029C2211yescharacteristic
33#16812029C2212yes—
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×Q8×C107C2211yescharacteristic
38#106×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—22no—
42C2×Dic214D411yescharacteristic
43Q8×D107—44no—
44C2×D428—21no—
45C2×D428—21no—
46C2×D428C2312yes—
47C2×D428C2311yescharacteristic
48C2×C4×D107—21no—
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51#13133805—22no—
52D856—24no—
53C2×C107⋊C8C2311yescharacteristic
54C2×C107⋊C8C2311yescharacteristic
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57Dic107⋊C4—22no—
58Dic107⋊C4D412yes—
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107C2312yes—
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×D107—22no—
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—21no—
86C2×Dic107C2×D412yes—
87C2×Dic107C2×D412yes—
88D428—22no—
89D428—21no—
90D428—22no—
91D428—42no—
92D428—81no—
93D428#4112yes—
94C4×D107—44no—
95C4×D107—42no—
96C4×D107—82no—
97Dic214—22no—
98Dic214—44no—
99C8×C107—22no—
100C107⋊C8—21no—
101C107⋊C8—21no—
102C22×C107C22×D411yescharacteristic
103C2×D107—22no—
104C2×D107—41no—
105C2×D107—41no—
106C2×D107—42no—
107C2×D107—42no—
108C2×D107—42no—
109C4×C107—21no—
110C4×C107—41no—
111C4×C107—42no—
112C4×C107—41no—
113C4×C107AGL(1,ℤ/8)12yes—
114C4×C107C2×#4111yescharacteristic
115C4×C1072−1+411yescharacteristic
116Dic107—22no—
117Dic107—22no—
118Dic107—41no—
119C2×C107#54811yescharacteristic
120C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C107#57711yescharacteristic
122D107—42no—
123D107—82no—
124#3373—1071nomaximal, Sylow
125C107#337311yescharacteristic, complemented, Sylow
126#559—1071no—
127D4×Q8—1071no—
128#489—1072no—
129#486—1071no—
130#476—1072no—
131#475D10711yescharacteristic, complemented
132#463—1072no—
133#459—1071no—
134#446—1071no—
135#442—1072no—
136C4×SD16—1071no—
137C22×Q8—2141no—
138C2×SD16—1071no—
139C2×D8—1072no—
140C4⋊Q8—1072no—
141C4⋊Q8C2×D10711yescharacteristic
142#127—1071no—
143#124—1072no—
144#124—2141no—
145#123—2142no—
146C22⋊Q8—2142no—
147C22⋊Q8—2141no—
148C4×Q8—1071no—
149C4×Q8C2×D10711yescharacteristic
150C4×D4—1072no—
151C4×D4—1072no—
152C4×D4—1071no—
153C4×D4—2141no—
154#106—1071no—
155#106C2×D10712yes—
156C4⋊C8—1072no—
157C4⋊C8C2×D10711yescharacteristic
158#103—1071no—
159#103C2×D10712yes—
160#102—1072no—
161#102—1071no—
162#102—1072no—
163#102—1072no—
164#97—1072no—
165C4×C8—1071no—
166C2×Q8—21no—
167C2×Q8—1071no—
168C2×Q8—4284no—
169C2×Q8—2142no—
170C2×Q8C22×D10711yescharacteristic
171C2×D4—1071no—
172C2×D4—1072no—
173C2×D4—2141no—
174C2×D4—2141no—
175C22×C4—2142no—
176C22×C4—2142no—
177C22×C4—2141no—
178SD16—2142no—
179D8—2144no—
180C2×C8—1071no—
181C2×C8—1071no—
182C2×C8C22×D10712yes—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—1071no—
186C4⋊C4—1072no—
187C4⋊C4—1071no—
188C4⋊C4—2142no—
189C4⋊C4C22×D10712yes—
190C4⋊C4C22×D10711yescharacteristic
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C22⋊C4—2142no—
194C22⋊C4—2142no—
195C22⋊C4—2141no—
196C42—21no—
197C42—1072no—
198C42C22×D10711yescharacteristic
199C23—2142no—
200C23—2141no—
201Q8—21no—
202Q8—21no—
203Q8—42no—
204Q8—4284no—
205Q8—2142no—
206D4—8561no—
207D4—1072no—
208D4—4282no—
209D4—2142no—
210D4—2142no—
211D4—2141no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—8562no—
216C2×C4—1072no—
217C2×C4—1072no—
218C2×C4—4284no—
219C2×C4—4282no—
220C2×C4—2141no—
221C2×C4D4×D10711yescharacteristic
222C2×C4D4×D10711yescharacteristic
223C2×C4C23×D10711yescharacteristic
224C8—22no—
225C8—2141no—
226C8—2141no—
227C22—4282no—
228C22—4282no—
229C22—4282no—
230C22—4281no—
231C22—4281no—
232C22—2142no—
233C22C2×D4×D10711yescharacteristic
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—4281no—
239C4—2142no—
240C4—2142no—
241C4#1681213612yes—
242C4#41×D10711yescharacteristic
243C4#1681215711yescharacteristic
244C2—8562no—
245C2—4282no—
246C2#2029328711yescharacteristic
247C2C2×#1681213611yescharacteristic
248C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#475×C107C2
socleC22×C107C22×D4
prime radical#20292823C2
hypercenter#475D107
hypocenterC107#3373
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. #20293362C2semidirect
2. #20293292C2semidirect
3. #20292988C2semidirect
4. #20292975C2semidirect
5. #20292916C2semidirect
6. #20292845C2semidirect
8. #20292754C2semidirect
9. #20292738C2semidirect
10. C4×#13133805C2semidirect
11. #475×C107C2semidirect
125. C107#3373semidirect
131. #475D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202932922
#202929882
#202929752
#202929162
#202928452
#202928232
#202927542
#202927382
C4×#131338052
#475×C1072
#3373107