Finite Group Databasebeta

← #28346242

Subgroups — #28346242

Order 13696.

388 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 131 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
2C4×#13133817C211yesmaximal, characteristic
3#20292933C212yesmaximal, complemented
4#20292925C212yesmaximal, complemented
5#20292923C211yesmaximal, characteristic, complemented
6#20292917C211yesmaximal, characteristic, complemented
7#20292909C211yesmaximal, characteristic, complemented
8#20292664C212yesmaximal, complemented
9C4⋊C8×D107C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal
11#474×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2211yescharacteristic
13C2×C8×D107C2212yes—
14#16812109C2211yescharacteristic
15#16812100C2212yes—
16#16812089—21no—
17#16812085—21no—
18Q8×Dic107—21no—
19C4×Dic214C2211yescharacteristic
20#16812074—21no—
21#16812074C2211yescharacteristic
22D4×Dic107C2211yescharacteristic
23C4×C22⋊D107—21no—
24(C22⋊D107)⋊C4—21no—
25#16812067C2212yes—
26#16812067C2211yescharacteristic
27#16812065—21no—
28C2×C4×Dic107—21no—
29C42×D107C2211yescharacteristic
30#16812043C2212yes—
31#16812040C2212yes—
32#16812040C2212yes—
33C4⋊(C107⋊C8)C2211yescharacteristic
34Dic107⋊C8C2212yes—
35#16812034C2212yes—
36#16812032C2212yes—
37#16812030C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic
40#107×C107C2212yes—
41C4⋊C8×C107C2211yescharacteristic
42#102×C107C2212yes—
43#13133817—22no—
44#13133817Q814yescomplemented
45C2×Dic214—21no—
46C2×Dic214C2311yescharacteristic
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C22×Dic107—21no—
50C2×C4×D107D411yescharacteristic
51C2×C4×D107D411yescharacteristic
52C2×C4×D107C2311yescharacteristic
53C2×C107⋊C8C2312yes—
54C8×D107—24no—
55C4⋊Dic107—21no—
56C4⋊Dic107—22no—
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—21no—
60Dic107⋊C4—21no—
61Dic107⋊C4—22no—
62C22⋊Dic107—21no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—21no—
66C4×Dic107—21no—
67C4×Dic107—21no—
68C4×Dic107—22no—
69C4×Dic107D411yescharacteristic
70C4×Dic107D411yescharacteristic
71C4×Dic107C2311yescharacteristic
72C2×Q8×C107—21no—
73C2×D4×C107C2311yescharacteristic
74C22×C4×C107—21no—
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107C2312yes—
78C4⋊C4×C107C2311yescharacteristic
79C22⋊C4×C107—21no—
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C22×D107C2×D411yescharacteristic
83Q8×C107—42no—
84D4×C107—21no—
85D4×C107C2×Q812yes—
86C2×C4×C107—21no—
87C2×C4×C107—22no—
88C2×C4×C107—42no—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—24no—
93C2×Dic107—21no—
94C2×Dic107—24no—
95C2×Dic107—21no—
96C2×Dic107—22no—
97C2×Dic107—42no—
98C2×Dic107C2×D411yescharacteristic
99C2×Dic107C2×D411yescharacteristic
100C2×Dic107C2×D411yescharacteristic
101C4×D107—22no—
102C4×D107—41no—
103C4×D107C2×Q812yes—
104C4×D107#4112yes—
105Dic214—21no—
106Dic214—42no—
107Dic214C2×Q812yes—
108C8×C107—22no—
109C107⋊C8—22no—
110C22×C107—22no—
111C22×C107—41no—
112C22×C107—41no—
113C22×C107C22×D411yescharacteristic
114C2×D107—21no—
115C2×D107—21no—
116C2×D107C22⋊Q812yes—
117C4×C107—21no—
118C4×C107—41no—
119C4×C107—42no—
120C4×C107#13512yes—
121C4×C107C22×Q811yescharacteristic
122C4×C107C2×#4111yescharacteristic
123Dic107—22no—
124Dic107—21no—
125Dic107—21no—
126Dic107—42no—
127Dic107—41no—
128Dic107C22⋊Q812yes—
129C2×C107—42no—
130C2×C107C2×C22⋊Q811yescharacteristic
131C2×C107C2×#13511yescharacteristic
132C2×C107#57511yescharacteristic
133D107—22no—
134#3076—1071nomaximal, Sylow
135C107#307611yescharacteristic, complemented, Sylow
136#533—1071no—
137C4×#41—1071no—
138#479—1072no—
139#478—1072no—
140#477—1071no—
141#476—1071no—
142#475—1071no—
143#474D10711yescharacteristic, complemented
144#427—1072no—
145C2×C4⋊C8—1071no—
146#416—1072no—
147C2×#41—1071no—
148C22×C8—1072no—
149C4⋊Q8—1071no—
150C4⋊Q8C2×D10711yescharacteristic
151#125—1072no—
152C22⋊Q8—2141no—
153C22⋊Q8—2141no—
154C4×Q8—1071no—
155C4×Q8—1071no—
156C4×Q8—2141no—
157C4×Q8—2141no—
158C4×D4—1071no—
159C4×D4—2141no—
160C4×D4—2141no—
161C4×D4C2×D10711yescharacteristic
162#117—1071no—
163#117—1072no—
164#117—2141no—
165C2×C42—1071no—
166C2×C42—2141no—
167#107—1072no—
168#107C2×D10712yes—
169#106—1072no—
170#106—1072no—
171C4⋊C8—1072no—
172C4⋊C8—1071no—
173C4⋊C8C2×D10711yescharacteristic
174#103—1072no—
175#103—1072no—
176#102—1072no—
177#102C2×D10712yes—
178#41—1074no—
179#41—2142no—
180C2×Q8—21no—
181C2×Q8—1071no—
182C2×Q8—2141no—
183C2×D4—2141no—
184C2×D4C22×D10711yescharacteristic
185C22×C4—21no—
186C22×C4—1071no—
187C22×C4—1071no—
188C22×C4—1071no—
189C22×C4—2141no—
190C22×C4—2141no—
191C2×C8—1072no—
192C2×C8—2144no—
193C2×C8C22×D10712yes—
194C4⋊C4—21no—
195C4⋊C4—1072no—
196C4⋊C4—1071no—
197C4⋊C4—2141no—
198C4⋊C4—2142no—
199C4⋊C4—2142no—
200C4⋊C4—2141no—
201C4⋊C4—2141no—
202C4⋊C4C22×D10712yes—
203C4⋊C4C22×D10711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2141no—
206C22⋊C4—2142no—
207C22⋊C4—2141no—
208C42—1071no—
209C42—1071no—
210C42—1071no—
211C42—2141no—
212C42—2141no—
213C42—2142no—
214C42—2141no—
215C42C22×D10711yescharacteristic
216C23—21no—
217C23—1071no—
218Q8—42no—
219Q8—1072no—
220Q8—4282no—
221Q8—2141no—
222D4—21no—
223D4—2144no—
224D4Q8×D10712yes—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—1071no—
229C2×C4—1072no—
230C2×C4—1072no—
231C2×C4—1071no—
232C2×C4—1071no—
233C2×C4—4281no—
234C2×C4—4282no—
235C2×C4—2144no—
236C2×C4—2141no—
237C2×C4—2142no—
238C2×C4—2141no—
239C2×C4—2142no—
240C2×C4D4×D10711yescharacteristic
241C2×C4D4×D10711yescharacteristic
242C2×C4C23×D10711yescharacteristic
243C8—22no—
244C8—2142no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—1072no—
249C22—2141no—
250C22—2141no—
251C22C2×D4×D10711yescharacteristic
252C4—21no—
253C4—41no—
254C4—42no—
255C4—1072no—
256C4—4281no—
257C4—4282no—
258C4—2142no—
259C4—2141no—
260C4—2141no—
261C4#1681212812yes—
262C4C2×Q8×D10711yescharacteristic
263C4#41×D10711yescharacteristic
264C2—42no—
265C2—2142no—
266C2C22⋊Q8×D10711yescharacteristic
267C2C2×#1681212811yescharacteristic
268C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#474×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133817C22
hypercenter#474D107
hypocenterC107#3076
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. #20293182C2semidirect
3. #20292933C2semidirect
4. #20292925C2semidirect
5. #20292923C2semidirect
6. #20292917C2semidirect
7. #20292909C2semidirect
8. #20292664C2semidirect
9. C4⋊C8×D107C2semidirect
11. #474×C107C2semidirect
44. #13133817Q8semidirect
135. C107#3076semidirect
143. #474D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
C4×#131338172
#202929332
#202929252
#202929232
#202929172
#202929092
#202926642
C4⋊C8×D1072
#202926202
#474×C1072
#3076107