Finite Group Databasebeta

← #28348390

Subgroups — #28348390

Order 13696.

398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293282C211yesmaximal, characteristic, complemented
3#20292978C211yesmaximal, characteristic, complemented
4#20292969C212yesmaximal, complemented
5#20292918C211yesmaximal, characteristic, complemented
6#20292856C212yesmaximal, complemented
7#20292833C211yesmaximal, characteristic, complemented
8#20292774C212yesmaximal, complemented
9C4×D856C211yesmaximal, characteristic, complemented
10#20292624C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13C2×#13133807C2212yes—
14C2×D856C2211yescharacteristic
15#16812109C2211yescharacteristic
16#16812109C2211yescharacteristic
17#16812095—22no—
18#16812091—22no—
19#16812088C2212yes—
20#16812083C2212yes—
21#16812080—21no—
22#16812080—21no—
23C4×D428C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812068C2212yes—
26#16812067—21no—
27C4⋊C4×D107—21no—
28#16812043C2211yescharacteristic
29#16812041C2212yes—
30#16812034C2212yes—
31#16812034C2212yes—
32#16812034C2212yes—
33#16812031C2211yescharacteristic
34#16812031C2212yes—
35#16812031C2211yescharacteristic
36C22×C8×C107C2212yes—
37C4×D4×C107C2211yescharacteristic
38C4⋊C8×C107C2211yescharacteristic
39C22⋊C8×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133818—24no—
42#13133818—42no—
43C2×Dic214—21no—
44C2×Dic214C2312yes—
45C2×D428C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C4×D107—21no—
50C2×C4×D107—22no—
51C2×C4×D107—21no—
52#13133807—42no—
53D856—22no—
54C4⋊Dic107—21no—
55C4⋊Dic107—21no—
56C4⋊Dic107—21no—
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—22no—
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—21no—
65(C2×D107)⋊C4—21no—
66C4×Dic107—22no—
67C2×D4×C107D411yescharacteristic
68C22×C4×C107C2312yes—
69C2×C8×C107—42no—
70C2×C8×C107C2311yescharacteristic
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2312yes—
73C4⋊C4×C107D411yescharacteristic
74C22⋊C4×C107D412yes—
75C42×C107C2311yescharacteristic
76C23×C107C2×D412yes—
77C22×D107—21no—
78C22×D107—21no—
79D4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D412yes—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—44no—
87C22⋊D107—82no—
88C2×Dic107—21no—
89C2×Dic107—22no—
90C2×Dic107—21no—
91C2×Dic107—22no—
92D428—21no—
93D428—21no—
94D428—21no—
95D428#4112yes—
96C4×D107—44no—
97C4×D107—42no—
98C4×D107—42no—
99Dic214—22no—
100Dic214—22no—
101Dic214—81no—
102C8×C107—21no—
103C8×C107—21no—
104C8×C107—22no—
105C22×C107—22no—
106C22×C107—22no—
107C22×C107—22no—
108C22×C107C22×D411yescharacteristic
109C2×D107—22no—
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—22no—
116C4×C107—21no—
117C4×C107#13512yes—
118C4×C107C2×#4111yescharacteristic
119C4×C107H(5,𝔽2)11yescharacteristic
120Dic107—42no—
121Dic107—41no—
122Dic107—41no—
123Dic107—42no—
124C2×C107—42no—
125C2×C107#54611yescharacteristic
126C2×C107C2×#13511yescharacteristic
127C2×C107#57711yescharacteristic
128D107—42no—
129D107—81no—
130#3303—1071nomaximal, Sylow
131C107#330311yescharacteristic, complemented, Sylow
132#555—1071no—
133#547—1071no—
134#487—1071no—
135#485—1072no—
136#476—1071no—
137#465—1072no—
138#461—1071no—
139#449—1072no—
140C4×D8—1071no—
141C8×D4D10711yescharacteristic, complemented
142#416—1072no—
143C2×#41—1072no—
144C2×SD16—1072no—
145C2×D8—1071no—
146C22×C8C2×D10712yes—
147C4⋊Q8—1071no—
148C4⋊Q8—1071no—
149#124—2142no—
150#123—2142no—
151C22⋊Q8—1072no—
152#121—1072no—
153#121—2141no—
154#121—2141no—
155C4×D4—1071no—
156C4×D4—1071no—
157C4×D4C2×D10711yescharacteristic
158#117—1072no—
159#117—2141no—
160C2×C4⋊C4—2141no—
161#107—1071no—
162#106—1072no—
163C4⋊C8C2×D10711yescharacteristic
164#103—1072no—
165#103—1072no—
166#103—1072no—
167#102—1071no—
168#102—1072no—
169#102—1071no—
170C22⋊C8C2×D10712yes—
171C4×C8C2×D10711yescharacteristic
172#41—4282no—
173#41—2144no—
174C2×Q8—1072no—
175C2×Q8—2141no—
176C2×D4—1071no—
177C2×D4—1071no—
178C2×D4—2142no—
179C2×D4—2142no—
180C2×D4D42811yescharacteristic
181C22×C4—2141no—
182C22×C4—2142no—
183C22×C4—2141no—
184C22×C4C22×D10712yes—
185SD16—4282no—
186D8—2142no—
187C2×C8—42no—
188C2×C8C22×D10711yescharacteristic
189C2×C8C22×D10711yescharacteristic
190C2×C8C22×D10712yes—
191C4⋊C4—1072no—
192C4⋊C4—1071no—
193C4⋊C4—1071no—
194C4⋊C4—2141no—
195C4⋊C4—2142no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4D42811yescharacteristic
199C22⋊C4—2142no—
200C22⋊C4—2142no—
201C22⋊C4—2142no—
202C22⋊C4—2141no—
203C22⋊C4—2141no—
204C22⋊C4D42812yes—
205C42—2142no—
206C42C22×D10711yescharacteristic
207C23—2141no—
208C23—2141no—
209C23C2×D42812yes—
210Q8—8561no—
211Q8—2142no—
212Q8—2142no—
213D4—22no—
214D4—8562no—
215D4—1072no—
216D4—4284no—
217D4—2141no—
218D4—2141no—
219D4—2141no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—4284no—
223C2×C4—4282no—
224C2×C4—4282no—
225C2×C4—2141no—
226C2×C4—2142no—
227C2×C4—2141no—
228C2×C4—2142no—
229C2×C4C2×D42812yes—
230C2×C4C2×D42811yescharacteristic
231C2×C4C2×D42811yescharacteristic
232C2×C4C23×D10711yescharacteristic
233C8—21no—
234C8—21no—
235C8—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—4281no—
240C22—4281no—
241C22—4281no—
242C22—4281no—
243C22—4281no—
244C22—2142no—
245C22C22×D42811yescharacteristic
246C4—21no—
247C4—22no—
248C4—4281no—
249C4—4281no—
250C4—4282no—
251C4—4282no—
252C4#1681213112yes—
253C4#41×D10711yescharacteristic
254C4#1681215411yescharacteristic
255C2—42no—
256C2—8561no—
257C2—4282no—
258C2#2029327211yescharacteristic
259C2C2×#1681213111yescharacteristic
260C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
hypercenterC8×D4D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293282C2semidirect
3. #20292978C2semidirect
4. #20292969C2semidirect
5. #20292918C2semidirect
6. #20292856C2semidirect
7. #20292833C2semidirect
8. #20292774C2semidirect
9. C4×D856C2semidirect
10. #20292624C2semidirect
11. C8×D4×C107C2semidirect
131. C107#3303semidirect
141. C8×D4D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932822
#202929782
#202929692
#202929182
#202928562
#202928332
#202927742
C4×D8562
#202926242
C8×D4×C1072
#3303107