Finite Group Databasebeta

← #28347064

Subgroups — #28347064

Order 13696.

364 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 119 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#20293156C211yesmaximal, characteristic
3#20293036C211yesmaximal, characteristic, complemented
4#20292995C211yesmaximal, characteristic, complemented
5#20292986C211yesmaximal, characteristic, complemented
6#20292882C212yesmaximal, complemented
7#20292870C212yesmaximal
8Q16×Dic107C212yesmaximal, complemented
9SD16×Dic107C212yesmaximal, complemented
10#20292579C211yesmaximal, characteristic, complemented
11#497×C107C211yesmaximal, characteristic, complemented
12C2×#13133801C2212yes—
13#16812109C2211yescharacteristic
14#16812102—21no—
15#16812101C2211yescharacteristic
16#16812100—21no—
17#16812091—21no—
18#16812089—21no—
19#16812085C2212yes—
20#16812085C2211yescharacteristic
21#16812080C2211yescharacteristic
22Q8×Dic107C2211yescharacteristic
23Q8×Dic107C2212yes—
24#16812074—21no—
25D4×Dic107C2211yescharacteristic
26(C22⋊D107)⋊C4—21no—
27#16812067—21no—
28#16812065—21no—
29C42×D107C2211yescharacteristic
30#16812043C2212yes—
31#16812041C2212yes—
32#16812033C2212yes—
33#16812033C2212yes—
34#16812033C2212yes—
35#16812030C2212yes—
36#16812020C2211yescharacteristic
37C8×Dic107C2212yes—
38C2×Q16×C107C2212yes—
39C2×SD16×C107C2212yes—
40C4⋊Q8×C107C2211yescharacteristic
41#124×C107C2211yescharacteristic
42#97×C107C2211yescharacteristic
43C2×Dic214—21no—
44C2×C22⋊D107—21no—
45C22×Dic107—21no—
46C2×C4×D107D411yescharacteristic
47C2×C4×D107D411yescharacteristic
48C2×C4×D107C2311yescharacteristic
49#13133801—24no—
50C2×C107⋊C8C2312yes—
51C4⋊Dic107—21no—
52C4⋊Dic107—22no—
53C4⋊Dic107—21no—
54C4⋊Dic107C2311yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2312yes—
57Dic107⋊C4—21no—
58Dic107⋊C4—21no—
59Dic107⋊C4—21no—
60Dic107⋊C4—22no—
61C22⋊Dic107—21no—
62C22⋊Dic107—21no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—21no—
65C4×Dic107—21no—
66C4×Dic107—22no—
67C4×Dic107D411yescharacteristic
68C4×Dic107D411yescharacteristic
69C4×Dic107C2311yescharacteristic
70C2×Q8×C107C2311yescharacteristic
71C2×Q8×C107C2312yes—
72C2×D4×C107C2311yescharacteristic
73Q16×C107—24no—
74SD16×C107—24no—
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C22⋊C4×C107—21no—
79C22⋊C4×C107—21no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85Q8×C107—22no—
86D4×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—42no—
93C2×Dic107—22no—
94C2×Dic107—21no—
95C2×Dic107—21no—
96C2×Dic107—42no—
97C2×Dic107C2×D411yescharacteristic
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100C4×D107—22no—
101C4×D107—22no—
102C4×D107—41no—
103Dic214—42no—
104C8×C107#4114yes—
105C107⋊C8—22no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107—41no—
109C22×C107C22×D411yescharacteristic
110C2×D107—21no—
111C2×D107—21no—
112C2×D107—21no—
113C4×C107—21no—
114C4×C107—21no—
115C4×C107—41no—
116C4×C107—42no—
117C4×C107C2×#4112yes—
118Dic107—21no—
119Dic107—21no—
120Dic107—41no—
121Dic107—42no—
122Dic107—41no—
123Dic107#13712yes—
124C2×C107—81no—
125C2×C107#53211yescharacteristic
126C2×C107C2×#13711yescharacteristic
127C2×C107#57511yescharacteristic
128D107—41no—
129#3172—1071nomaximal, Sylow
130C107#317211yescharacteristic, complemented, Sylow
131#533—1071no—
132#529—1071no—
133#501—1071no—
134#497D10711yescharacteristic, complemented
135#491—1071no—
136#489—1071no—
137#470—1072no—
138#468—1072no—
139C4×Q16—1072no—
140C4×SD16—1072no—
141C4×M16—1071no—
142C2×Q16C2×D10712yes—
143C2×SD16C2×D10712yes—
144C2×M16—1072no—
145C4⋊Q8—1071no—
146C4⋊Q8C2×D10711yescharacteristic
147#126—2141no—
148#125—1071no—
149#125—2141no—
150#124C2×D10711yescharacteristic
151#123—2141no—
152C22⋊Q8—1071no—
153C22⋊Q8—1072no—
154C22⋊Q8—2141no—
155#121—1071no—
156C4×Q8—1071no—
157C4×Q8—1072no—
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#103—1072no—
168#103—1072no—
169#102—1072no—
170#97—1071no—
171#97C2×D10711yescharacteristic
172C4×C8—1072no—
173C2×Q8—2141no—
174C2×Q8C22×D10711yescharacteristic
175C2×Q8C22×D10712yes—
176C2×D4—2141no—
177C2×D4C22×D10711yescharacteristic
178C22×C4—1071no—
179C22×C4—1071no—
180C22×C4—1071no—
181C22×C4—2141no—
182Q16—24no—
183SD16—24no—
184M16—2144no—
185C2×C8—1072no—
186C2×C8C22×D10712yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—1072no—
190C4⋊C4—1071no—
191C4⋊C4—1071no—
192C4⋊C4—2141no—
193C4⋊C4—2142no—
194C4⋊C4—2141no—
195C4⋊C4—2142no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4—2141no—
199C22⋊C4—21no—
200C22⋊C4—21no—
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—21no—
212C23—1071no—
213Q8—22no—
214Q8—22no—
215Q8—22no—
216Q8—4282no—
217D4—22no—
218D4—4282no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—1071no—
222C2×C4—1071no—
223C2×C4—1071no—
224C2×C4—4281no—
225C2×C4—4282no—
226C2×C4—2142no—
227C2×C4—2141no—
228C2×C4—2142no—
229C2×C4—2141no—
230C2×C4—2142no—
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4C23×D10711yescharacteristic
234C8—2142no—
235C8#1313381714yes—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—2141no—
240C22—2141no—
241C22—2141no—
242C22C2×D4×D10711yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—42no—
247C4—1072no—
248C4—4281no—
249C4—4281no—
250C4—4282no—
251C4—2141no—
252C4—2141no—
253C4C2×#1313381712yes—
254C2—81no—
255C2—4281no—
256C2#2029317511yescharacteristic
257C2#137×D10711yescharacteristic
258C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#497×C107C2
socleC22×C107C22×D4
prime radical#16812080C22
hypercenter#497D107
hypocenterC107#3172
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #20293182C2semidirect
3. #20293036C2semidirect
4. #20292995C2semidirect
5. #20292986C2semidirect
6. #20292882C2semidirect
8. Q16×Dic107C2semidirect
9. SD16×Dic107C2semidirect
10. #20292579C2semidirect
11. #497×C107C2semidirect
130. C107#3172semidirect
134. #497D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
#202931562
#202930362
#202929952
#202929862
#202928822
#202928702
Q16×Dic1072
SD16×Dic1072
#202925792
#497×C1072
#3172107