Finite Group Databasebeta

← #28348414

Subgroups — #28348414

Order 13696.

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

normal subgroups only

This group has 408 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#20293342C211yesmaximal, characteristic, complemented
2#20293297C211yesmaximal, characteristic, complemented
3#20292973C211yesmaximal, characteristic, complemented
4#20292949C212yesmaximal, complemented
5#20292923C211yesmaximal, characteristic, complemented
6#20292860C212yesmaximal, complemented
7#20292769C212yesmaximal, complemented
8C4×#13133806C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal, complemented
11#459×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×#13133806C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812105—21no—
17#16812095—22no—
18#16812083—21no—
19#16812081—22no—
20#16812080C2212yes—
21#16812080C2212yes—
22C4×Dic214C2211yescharacteristic
23C4×D428C2211yescharacteristic
24D4×Dic107—21no—
25#16812068—21no—
26#16812067C2212yes—
27#16812043C2212yes—
28#16812040C2211yescharacteristic
29#16812034C2212yes—
30#16812032C2211yescharacteristic
31#16812030C2212yes—
32#16812030C2212yes—
33#16812030C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×D8×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133817—24no—
42#13133817—42no—
43C2×Dic214—21no—
44C2×Dic214C2311yescharacteristic
45C2×D428D411yescharacteristic
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C22×Dic107—22no—
50C2×C4×D107C2312yes—
51#13133806—22no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107—42no—
55C4⋊Dic107—21no—
56C4⋊Dic107D411yescharacteristic
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—21no—
60C22⋊Dic107—22no—
61C22⋊Dic107—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4D412yes—
64C4×Dic107—21no—
65C4×Dic107—22no—
66C2×D4×C107—21no—
67C2×D4×C107—21no—
68C2×D4×C107C2311yescharacteristic
69C2×D4×C107C2312yes—
70C22×C4×C107—21no—
71D8×C107—42no—
72C2×C8×C107C2312yes—
73C4⋊C4×C107C2311yescharacteristic
74C22⋊C4×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—21no—
77C23×C107—22no—
78C22×D107C2×D412yes—
79D4×C107—21no—
80D4×C107—22no—
81D4×C107—21no—
82D4×C107—22no—
83D4×C107—42no—
84D4×C107—81no—
85C2×C4×C107—21no—
86C2×C4×C107—42no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—44no—
91C22⋊D107—82no—
92C22⋊D107—82no—
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107—21no—
96C2×Dic107—44no—
97C2×Dic107C2×D412yes—
98D428—22no—
99C4×D107—22no—
100C4×D107—22no—
101Dic214—21no—
102Dic214—81no—
103Dic214#4112yes—
104C8×C107—22no—
105C107⋊C8—21no—
106C107⋊C8—21no—
107C22×C107—42no—
108C22×C107—41no—
109C22×C107—41no—
110C22×C107—42no—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—22no—
116C2×D107—22no—
117C4×C107—21no—
118C4×C107—41no—
119C4×C107#13512yes—
120C4×C107C2×#4111yescharacteristic
121C4×C107H(5,𝔽2)11yescharacteristic
122Dic107—22no—
123Dic107—22no—
124Dic107—42no—
125Dic107—41no—
126C2×C107—82no—
127C2×C107—81no—
128C2×C107#54611yescharacteristic
129C2×C107C2×#13511yescharacteristic
130C2×C107#57611yescharacteristic
131D107—42no—
132#3305—1071nomaximal, Sylow
133C107#330511yescharacteristic, complemented, Sylow
134#555—1071no—
135#550—1071no—
136#486—1071no—
137#482—1072no—
138#477—1071no—
139#466—1072no—
140#459D10711yescharacteristic, complemented
141#449—1072no—
142C4×SD16—1071no—
143C8×D4—1071no—
144#416—1072no—
145C2×#41—1072no—
146C2×SD16—1071no—
147C2×D8C2×D10712yes—
148C22×C8—1072no—
149C4⋊Q8—1071no—
150#127—2141no—
151#127C2×D10711yescharacteristic
152#124—2142no—
153#121—1072no—
154#121—1072no—
155#121—2142no—
156#121—2141no—
157C4×Q8—1071no—
158C4×D4—1071no—
159C4×D4—2141no—
160C4×D4C2×D10711yescharacteristic
161#117—1072no—
162#117—2141no—
163#107—1072no—
164#106—1071no—
165C4⋊C8C2×D10711yescharacteristic
166#103—1072no—
167#103—1071no—
168#102—1072no—
169#102—1071no—
170#102—1072no—
171#102C2×D10712yes—
172C22⋊C8—1072no—
173C4×C8—1071no—
174#41—4282no—
175#41—2144no—
176C2×Q8—1071no—
177C2×Q8—2141no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—1071no—
181C2×D4—2142no—
182C2×D4—2142no—
183C2×D4—2142no—
184C2×D4C22×D10712yes—
185C2×D4C22×D10711yescharacteristic
186C22×C4—21no—
187C22×C4—1072no—
188C22×C4—2142no—
189SD16—2142no—
190D8—42no—
191C2×C8—1071no—
192C2×C8—1071no—
193C2×C8—4282no—
194C2×C8C22×D10712yes—
195C4⋊C4—1071no—
196C4⋊C4—1072no—
197C4⋊C4—1071no—
198C4⋊C4—2141no—
199C4⋊C4—2141no—
200C4⋊C4C22×D10711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1072no—
203C22⋊C4—2142no—
204C22⋊C4—2142no—
205C22⋊C4—2142no—
206C42—2142no—
207C42—2141no—
208C42C22×D10711yescharacteristic
209C23—21no—
210C23—22no—
211C23—1072no—
212Q8—8561no—
213Q8—1072no—
214Q8—2141no—
215D4—21no—
216D4—22no—
217D4—21no—
218D4—22no—
219D4—42no—
220D4—81no—
221D4—8562no—
222D4—8562no—
223D4—4284no—
224D4—2142no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—1072no—
228C2×C4—4284no—
229C2×C4—2141no—
230C2×C4—2142no—
231C2×C4—2141no—
232C2×C4—2142no—
233C2×C4—2142no—
234C2×C4D4×D10711yescharacteristic
235C2×C4D4×D10711yescharacteristic
236C2×C4C23×D10711yescharacteristic
237C8—22no—
238C8—2141no—
239C8—2141no—
240C22—41no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—2142no—
247C22—2142no—
248C22—2142no—
249C22C2×D4×D10711yescharacteristic
250C4—21no—
251C4—41no—
252C4—4281no—
253C4—4282no—
254C4—2142no—
255C4—2142no—
256C4#1681212812yes—
257C4#41×D10711yescharacteristic
258C4#1681215411yescharacteristic
259C2—81no—
260C2—82no—
261C2—4282no—
262C2#2029327411yescharacteristic
263C2C2×#1681212811yescharacteristic
264C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3305
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293297C2semidirect
3. #20292973C2semidirect
4. #20292949C2semidirect
5. #20292923C2semidirect
6. #20292860C2semidirect
7. #20292769C2semidirect
8. C4×#13133806C2semidirect
9. #20292692C2semidirect
10. #20292620C2semidirect
11. #459×C107C2semidirect
133. C107#3305semidirect
140. #459D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932972
#202929732
#202929492
#202929232
#202928602
#202927692
C4×#131338062
#202926922
#202926202
#459×C1072
#3305107