Finite Group Databasebeta

← #28347066

Subgroups — #28347066

Order 13696.

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

normal subgroups only

This group has 364 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#20293182C211yesmaximal, characteristic, complemented
2#20293162C211yesmaximal, characteristic
3#20293037C211yesmaximal, characteristic, complemented
4#20293019C211yesmaximal, characteristic, complemented
5#20292987C211yesmaximal, characteristic, complemented
6#20292884C212yesmaximal, complemented
7#20292876C212yesmaximal
8#20292719C212yesmaximal, complemented
9#20292712C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#491×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2212yes—
13C2×#13133805C2212yes—
14C2×#13133801C2212yes—
15#16812109C2211yescharacteristic
16#16812104—21no—
17#16812100—21no—
18#16812097C2211yescharacteristic
19#16812094—21no—
20#16812089—21no—
21#16812089C2211yescharacteristic
22#16812085C2212yes—
23#16812084C2211yescharacteristic
24Q8×Dic107C2212yes—
25#16812074—21no—
26#16812074C2211yescharacteristic
27#16812071—21no—
28#16812071C2211yescharacteristic
29#16812067—21no—
30#16812067—21no—
31C42×D107C2211yescharacteristic
32#16812042C2212yes—
33#16812040C2212yes—
34#16812034C2212yes—
35#16812031C2212yes—
36#16812020C2211yescharacteristic
37C8×Dic107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39#125×C107C2211yescharacteristic
40#103×C107C2212yes—
41#103×C107C2212yes—
42#97×C107C2211yescharacteristic
43C2×Dic214—21no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428C2311yescharacteristic
47C2×C4×D107—21no—
48C2×C4×D107D411yescharacteristic
49C2×C4×D107D411yescharacteristic
50C2×C4×D107C2311yescharacteristic
51#13133808—24no—
52#13133805—24no—
53#13133801—24no—
54C2×C107⋊C8C2312yes—
55C4⋊Dic107—21no—
56C4⋊Dic107—21no—
57C4⋊Dic107—22no—
58C4⋊Dic107C2312yes—
59Dic107⋊C4—22no—
60Dic107⋊C4—21no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—21no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—21no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67C4×Dic107—21no—
68C4×Dic107—22no—
69C4×Dic107D411yescharacteristic
70C4×Dic107D411yescharacteristic
71C4×Dic107C2311yescharacteristic
72C2×Q8×C107C2312yes—
73C2×C8×C107C2312yes—
74C4⋊C4×C107—21no—
75C4⋊C4×C107—21no—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2311yescharacteristic
79C4⋊C4×C107C2311yescharacteristic
80C42×C107C2311yescharacteristic
81C22×D107—21no—
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85C2×C4×C107—21no—
86C2×C4×C107—21no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2411yescharacteristic
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107C2×D411yescharacteristic
94C2×Dic107C2×D411yescharacteristic
95C2×Dic107C2×D411yescharacteristic
96D428—22no—
97D428—42no—
98C4×D107—22no—
99C4×D107—22no—
100C4×D107—41no—
101C4×D107—42no—
102Dic214—22no—
103Dic214—42no—
104C8×C107—22no—
105C107⋊C8#4114yes—
106C22×C107C22×D411yescharacteristic
107C2×D107—21no—
108C2×D107—21no—
109C2×D107—21no—
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—41no—
113C4×C107—21no—
114C4×C107—21no—
115C4×C107—42no—
116C4×C107—41no—
117C4×C107—41no—
118C4×C107C2×#4112yes—
119Dic107—21no—
120Dic107—21no—
121Dic107—41no—
122Dic107—42no—
123Dic107#13712yes—
124C2×C107#53211yescharacteristic
125C2×C107C2×#13711yescharacteristic
126C2×C107#57511yescharacteristic
127D107—41no—
128D107—81no—
129#3172—1071nomaximal, Sylow
130C107#317211yescharacteristic, complemented, Sylow
131#533—1071no—
132#529—1071no—
133#501—1071no—
134#497—1071no—
135#491D10711yescharacteristic, complemented
136#489—1071no—
137#470—1072no—
138#468—1072no—
139C4×Q16—1072no—
140C4×SD16—1072no—
141C4×M16—1071no—
142C2×Q16—1072no—
143C2×SD16—1072no—
144C2×M16—1072no—
145C4⋊Q8—1071no—
146C4⋊Q8C2×D10711yescharacteristic
147#126—2141no—
148#125—2141no—
149#125C2×D10711yescharacteristic
150#124—1071no—
151#123—2141no—
152C22⋊Q8—1072no—
153C22⋊Q8—1071no—
154C22⋊Q8—2141no—
155#121—1071no—
156C4×Q8—1072no—
157C4×Q8—1071no—
158C4×Q8—2141no—
159C4×D4—1071no—
160C4×D4—2141no—
161#117—2141no—
162#117—2141no—
163C2×C42—1071no—
164#107—1072no—
165#106—1072no—
166#103—1072no—
167#103C2×D10712yes—
168#103C2×D10712yes—
169#102—1072no—
170#97—1071no—
171#97C2×D10711yescharacteristic
172C4×C8—1072no—
173C2×Q8—1071no—
174C2×Q8—2141no—
175C2×Q8C22×D10712yes—
176C2×D4—1071no—
177C2×D4—2141no—
178C22×C4—1071no—
179C22×C4—1071no—
180C22×C4—1071no—
181C22×C4—2141no—
182Q16—2144no—
183SD16—2144no—
184M16—2144no—
185C2×C8—1072no—
186C2×C8C22×D10712yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—1072no—
192C4⋊C4—2141no—
193C4⋊C4—2142no—
194C4⋊C4—2141no—
195C4⋊C4—2142no—
196C4⋊C4—2141no—
197C4⋊C4C22×D10711yescharacteristic
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—2141no—
200C22⋊C4—2141no—
201C22⋊C4—2142no—
202C22⋊C4—2141no—
203C22⋊C4—2141no—
204C22⋊C4—2141no—
205C42—1071no—
206C42—1071no—
207C42—1071no—
208C42—2141no—
209C42—2142no—
210C42C22×D10711yescharacteristic
211C23—1071no—
212C23—2141no—
213Q8—22no—
214Q8—22no—
215Q8—4282no—
216Q8—2142no—
217D4—4282no—
218D4—2142no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—1071no—
223C2×C4—1071no—
224C2×C4—1071no—
225C2×C4—4281no—
226C2×C4—4282no—
227C2×C4—2142no—
228C2×C4—2141no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4C23×D10711yescharacteristic
234C8—22no—
235C8—1074no—
236C22—4281no—
237C22—4281no—
238C22—4281no—
239C22—2141no—
240C22—2141no—
241C22—2141no—
242C22C2×D4×D10711yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—1072no—
249C4—4282no—
250C4—4281no—
251C4—2141no—
252C4—2141no—
253C4#41×D10712yes—
254C2—8561no—
255C2—4281no—
256C2#2029317611yescharacteristic
257C2#137×D10711yescharacteristic
258C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#491×C107C2
socleC22×C107C22×D4
prime radical#16812084C22
hypercenter#491D107
hypocenterC107#3172
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #20293182C2semidirect
3. #20293037C2semidirect
4. #20293019C2semidirect
5. #20292987C2semidirect
6. #20292884C2semidirect
8. #20292719C2semidirect
9. #20292712C2semidirect
10. #20292579C2semidirect
11. #491×C107C2semidirect
130. C107#3172semidirect
135. #491D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
#202931622
#202930372
#202930192
#202929872
#202928842
#202928762
#202927192
#202927122
#202925792
#491×C1072
#3172107