Finite Group Databasebeta

← #28346627

Subgroups — #28346627

Order 13696.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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#20293158C211yesmaximal, characteristic
2C2×Q8×Dic107C211yesmaximal, characteristic, complemented
3#20292963C211yesmaximal, characteristic, complemented
4#20292943C211yesmaximal, characteristic
5#20292936C212yesmaximal, complemented
6#20292847C212yesmaximal, complemented
7#20292783C211yesmaximal, characteristic, complemented
8#20292731C212yesmaximal, complemented
9#20292728C212yesmaximal, complemented
10#20292681C211yesmaximal, characteristic, complemented
11#448×C107C211yesmaximal, characteristic
12C2×#13133808C2212yes—
13#16812102—21no—
14#16812100C2211yescharacteristic
15#16812096C2211yescharacteristic
16#16812090—21no—
17#16812088C2211yescharacteristic
18Q8×Dic107—21no—
19Q8×Dic107—21no—
20Q8×Dic107C2212yes—
21Q8×Dic107C2212yes—
22#16812074C2211yescharacteristic
23D4×Dic107—21no—
24D4×Dic107C2211yescharacteristic
25#16812065—21no—
26C2×C4⋊Dic107—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C2×C4×Dic107—21no—
29C2×C4×Dic107C2211yescharacteristic
30#16812043C2212yes—
31#16812040C2212yes—
32Dic107⋊C8C2212yes—
33#16812034C2212yes—
34#16812033C2212yes—
35#16812030C2212yes—
36C22⋊(C107⋊C8)C2211yescharacteristic
37#16812019C2212yes—
38C22×Q8×C107C2211yescharacteristic
39C2×SD16×C107C2212yes—
40#121×C107C2211yescharacteristic
41#103×C107C2212yes—
42C22⋊C8×C107C2211yescharacteristic
43C2×Dic214C2311yescharacteristic
44C22×Dic107—21no—
45C22×Dic107—21no—
46C22×Dic107D411yescharacteristic
47C22×Dic107D411yescharacteristic
48#13133808—24no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—21no—
51C4⋊Dic107—22no—
52C4⋊Dic107—22no—
53C4⋊Dic107—21no—
54C4⋊Dic107C2311yescharacteristic
55C4⋊Dic107C2312yes—
56Dic107⋊C4—21no—
57Dic107⋊C4—21no—
58Dic107⋊C4—21no—
59C22⋊Dic107—21no—
60C22⋊Dic107—21no—
61C22⋊Dic107—21no—
62C22⋊Dic107—21no—
63C4×Dic107—22no—
64C4×Dic107—22no—
65C4×Dic107—21no—
66C4×Dic107D411yescharacteristic
67C4×Dic107D411yescharacteristic
68C4×Dic107C2311yescharacteristic
69C4×Dic107C2311yescharacteristic
70C2×Q8×C107—24no—
71C2×Q8×C107—21no—
72C2×Q8×C107C2312yes—
73C2×D4×C107—21no—
74C2×D4×C107C2311yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77SD16×C107—24no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C23×C107—21no—
82C23×C107C2×D411yescharacteristic
83Q8×C107—24no—
84Q8×C107—22no—
85Q8×C107#4114yes—
86D4×C107—22no—
87D4×C107—42no—
88C2×C4×C107—24no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—21no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—41no—
99C2×Dic107—42no—
100C2×Dic107—42no—
101C2×Dic107C2×D411yescharacteristic
102C2×Dic107C2×D411yescharacteristic
103C2×Dic107C2×D411yescharacteristic
104C2×Dic107C2×D411yescharacteristic
105Dic214—22no—
106C8×C107—22no—
107C107⋊C8—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107—41no—
111C22×C107—41no—
112C22×C107—41no—
113C22×C107#13712yes—
114C22×C107C22×D411yescharacteristic
115C4×C107—24no—
116C4×C107—21no—
117C4×C107—41no—
118C4×C107C2×#4112yes—
119Dic107—21no—
120Dic107—21no—
121Dic107—21no—
122Dic107—21no—
123Dic107—41no—
124Dic107—42no—
125Dic107—41no—
126C2×C107—22no—
127C2×C107—81no—
128C2×C107#52511yescharacteristic
129C2×C107C2×#13711yescharacteristic
130C2×C107#57511yescharacteristic
131#3117—1071nomaximal, Sylow
132C107#311711yescharacteristic, complemented, Sylow
133#529—1071no—
134C2×C4×Q8—1071no—
135#484—1071no—
136#481—1071no—
137#479—1072no—
138#464—1072no—
139#451—1071no—
140#448D10711yescharacteristic
141#441—1072no—
142#440—1072no—
143#432—1071no—
144C22×Q8C2×D10711yescharacteristic
145C2×Q16—1072no—
146C2×SD16C2×D10712yes—
147#126—2141no—
148#125—1071no—
149#124—1071no—
150#123—2141no—
151C22⋊Q8—1071no—
152#121C2×D10711yescharacteristic
153C4×Q8—1072no—
154C4×Q8—1072no—
155C4×Q8—1071no—
156C4×Q8—2141no—
157C4×Q8—2141no—
158C4×D4—1071no—
159C4×D4—2141no—
160#117—2141no—
161C2×C4⋊C4—1071no—
162C2×C4⋊C4—2141no—
163C2×C42—1071no—
164C2×C42—2141no—
165#107—1072no—
166#106—1072no—
167C4⋊C8—1072no—
168#103—1072no—
169#103—1072no—
170#103C2×D10712yes—
171#102—1072no—
172C22⋊C8—1071no—
173C22⋊C8C2×D10711yescharacteristic
174#97—1072no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1071no—
178C2×Q8C22×D10712yes—
179C2×D4—21no—
180C2×D4C22×D10711yescharacteristic
181C22×C4—21no—
182C22×C4—1071no—
183C22×C4—1071no—
184C22×C4—2141no—
185C22×C4—2141no—
186C22×C4C22×D10711yescharacteristic
187Q16—2144no—
188SD16—24no—
189C2×C8—1072no—
190C2×C8C22×D10712yes—
191C4⋊C4—1072no—
192C4⋊C4—1071no—
193C4⋊C4—2142no—
194C4⋊C4—2142no—
195C4⋊C4—2141no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4—2141no—
199C4⋊C4—2141no—
200C4⋊C4C22×D10711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2141no—
203C22⋊C4—2141no—
204C22⋊C4—2141no—
205C22⋊C4—2141no—
206C42—1071no—
207C42—1071no—
208C42—1071no—
209C42—1071no—
210C42—2142no—
211C42—2142no—
212C42—2141no—
213C23—21no—
214C23D4×D10711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—2142no—
218Q8#1313381714yes—
219D4—22no—
220D4—42no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—1071no—
226C2×C4—1071no—
227C2×C4—1071no—
228C2×C4—1071no—
229C2×C4—4282no—
230C2×C4—4282no—
231C2×C4—4281no—
232C2×C4—2142no—
233C2×C4—2141no—
234C2×C4—2141no—
235C2×C4—2142no—
236C2×C4D4×D10711yescharacteristic
237C2×C4C23×D10711yescharacteristic
238C8—22no—
239C8—2142no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1681213812yes—
246C22C2×D4×D10711yescharacteristic
247C4—24no—
248C4—21no—
249C4—41no—
250C4—4281no—
251C4—4282no—
252C4—4281no—
253C4—2141no—
254C4—2141no—
255C4—2141no—
256C4—2141no—
257C4C2×#1313381712yes—
258C2—22no—
259C2—81no—
260C2#2029313311yescharacteristic
261C2C2×#1681213811yescharacteristic
262C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
prime radical#121×C107C22
hypercenter#448D107
hypocenterC107#3117
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic107C2semidirect
3. #20292963C2semidirect
5. #20292936C2semidirect
6. #20292847C2semidirect
7. #20292783C2semidirect
8. #20292731C2semidirect
9. #20292728C2semidirect
10. #20292681C2semidirect
132. C107#3117semidirect

Maximal subgroups

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

subgroupindex
#202931582
C2×Q8×Dic1072
#202929632
#202929432
#202929362
#202928472
#202927832
#202927312
#202927282
#202926812
#448×C1072
#3117107