Finite Group Databasebeta

← #28349178

Subgroups — #28349178

Order 13696.

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

normal subgroups only

This group has 372 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#20293347C211yesmaximal, characteristic, complemented
2#20293292C211yesmaximal, characteristic, complemented
3#20292985C212yesmaximal, complemented
4#20292979C211yesmaximal, characteristic, complemented
5#20292921C212yesmaximal, complemented
6#20292841C212yesmaximal
7#20292833C211yesmaximal, characteristic, complemented
8#20292754C211yesmaximal, characteristic, complemented
9#20292726C212yesmaximal, complemented
10C4×#13133803C211yesmaximal, characteristic
11#474×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107—21no—
13C2×#13133807C2212yes—
14C2×#13133803C2211yescharacteristic
15#16812109C2211yescharacteristic
16#16812108C2212yes—
17#16812102—22no—
18#16812096C2212yes—
19#16812088—21no—
20#16812087—22no—
21#16812085—21no—
22C4×Dic214C2211yescharacteristic
23#16812074C2212yes—
24C4×D428C2211yescharacteristic
25#16812071C2212yes—
26#16812068—21no—
27#16812042C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812034C2212yes—
30#16812032C2212yes—
31#16812032C2212yes—
32#16812030C2211yescharacteristic
33#16812029C2211yescharacteristic
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×D4×C107C2211yescharacteristic
38#107×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41C2×Dic214—22no—
42C2×Dic214D411yescharacteristic
43C2×Dic214C2312yes—
44Q8×D107—44no—
45C2×D428C2311yescharacteristic
46C2×C4×D107—21no—
47C2×C4×D107—22no—
48#13133807—24no—
49#13133803—22no—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2311yescharacteristic
52C4⋊Dic107—21no—
53C4⋊Dic107—21no—
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—22no—
57Dic107⋊C4—22no—
58Dic107⋊C4D412yes—
59C22⋊Dic107—22no—
60C22⋊Dic107—22no—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—22no—
64C4×Dic107C2312yes—
65C2×Q8×C107—21no—
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2312yes—
71C4⋊C4×C107C2311yescharacteristic
72C22⋊C4×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—21no—
75C22×D107—21no—
76Q8×C107—42no—
77D4×C107—21no—
78D4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107—42no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107C2×D412yes—
88C2×Dic107C2×D412yes—
89D428—21no—
90D428#4112yes—
91C4×D107—44no—
92C4×D107—42no—
93Dic214—22no—
94Dic214—22no—
95Dic214—22no—
96Dic214—44no—
97C8×C107—22no—
98C107⋊C8—21no—
99C107⋊C8—21no—
100C22×C107—41no—
101C22×C107—41no—
102C22×C107—41no—
103C22×C107C22×D411yescharacteristic
104C2×D107—22no—
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—81no—
118C2×C107#54811yescharacteristic
119C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C107#57811yescharacteristic
121D107—42no—
122#3367—1071nomaximal, Sylow
123C107#336711yescharacteristic, complemented, Sylow
124#556—1071no—
125D4×Q8—1071no—
126#489—1072no—
127#487—1071no—
128#477—1072no—
129#474D10711yescharacteristic, complemented
130#463—1072no—
131#461—1071no—
132#446—1071no—
133#440—1072no—
134C4×D8—1071no—
135C22×Q8—2141no—
136C2×SD16—1072no—
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—1071no—
153#107C2×D10712yes—
154C4⋊C8—1072no—
155C4⋊C8C2×D10711yescharacteristic
156#103—1072no—
157#103—1072no—
158#103—1072no—
159#102—1071no—
160#102—1071no—
161#102C2×D10712yes—
162#97—1072no—
163C4×C8—1071no—
164C2×Q8—21no—
165C2×Q8—1071no—
166C2×Q8—1072no—
167C2×Q8—4284no—
168C2×Q8—2142no—
169C2×D4—1071no—
170C2×D4C22×D10711yescharacteristic
171C22×C4—21no—
172C22×C4—2142no—
173C22×C4—2141no—
174SD16—2144no—
175D8—2142no—
176C2×C8—1071no—
177C2×C8—1071no—
178C2×C8C22×D10712yes—
179C4⋊C4—21no—
180C4⋊C4—1071no—
181C4⋊C4—1072no—
182C4⋊C4—1071no—
183C4⋊C4—2142no—
184C4⋊C4—2142no—
185C4⋊C4—2141no—
186C4⋊C4—2141no—
187C4⋊C4C22×D10711yescharacteristic
188C4⋊C4C22×D10712yes—
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—42no—
200Q8—4284no—
201Q8—2142no—
202Q8—2142no—
203Q8—2142no—
204D4—21no—
205D4—21no—
206D4—1072no—
207D4—2141no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—1072no—
212C2×C4—1072no—
213C2×C4—4284no—
214C2×C4—4282no—
215C2×C4—2142no—
216C2×C4—2141no—
217C2×C4D4×D10711yescharacteristic
218C2×C4D4×D10711yescharacteristic
219C2×C4C23×D10711yescharacteristic
220C8—22no—
221C8—2141no—
222C8—2141no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—4281no—
227C22—4281no—
228C22—2142no—
229C22C2×D4×D10711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—4281no—
234C4—4282no—
235C4—2142no—
236C4—2142no—
237C4#1681213312yes—
238C4#41×D10711yescharacteristic
239C4#1681215711yescharacteristic
240C2—81no—
241C2—4282no—
242C2#2029328711yescharacteristic
243C2C2×#1681213311yescharacteristic
244C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#474×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133803C22
hypercenter#474D107
hypocenterC107#3367
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20293347C2semidirect
2. #20293292C2semidirect
3. #20292985C2semidirect
4. #20292979C2semidirect
5. #20292921C2semidirect
7. #20292833C2semidirect
8. #20292754C2semidirect
9. #20292726C2semidirect
11. #474×C107C2semidirect
123. C107#3367semidirect
129. #474D107semidirect

Maximal subgroups

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

subgroupindex
#202933472
#202932922
#202929852
#202929792
#202929212
#202928412
#202928332
#202927542
#202927262
C4×#131338032
#474×C1072
#3367107