Finite Group Databasebeta

← #28348379

Subgroups — #28348379

Order 13696.

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

normal subgroups only

This group has 386 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#20293347C211yesmaximal, characteristic, complemented
2#20293282C211yesmaximal, characteristic
3#20292978C211yesmaximal, characteristic, complemented
4#20292969C212yesmaximal, complemented
5#20292908C211yesmaximal, characteristic, complemented
6#20292868C212yesmaximal, complemented
7#20292838C211yesmaximal, characteristic, complemented
8#20292791C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10#20292624C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13C2×Dic428C2212yes—
14C2×#13133807C2211yescharacteristic
15#16812109C2211yescharacteristic
16#16812109C2211yescharacteristic
17#16812102—22no—
18#16812096—21no—
19#16812091—22no—
20#16812088C2212yes—
21#16812088C2212yes—
22#16812080—21no—
23C4×Dic214C2211yescharacteristic
24#16812074—21no—
25C4×D428C2211yescharacteristic
26#16812068C2212yes—
27C4⋊C4×D107—21no—
28#16812043C2212yes—
29#16812041C2211yescharacteristic
30#16812034C2211yescharacteristic
31#16812034C2212yes—
32#16812034C2212yes—
33#16812034C2212yes—
34#16812031C2211yescharacteristic
35#16812031C2212yes—
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×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—22no—
48C2×C4×D107—21no—
49C2×C4×D107—22no—
50Dic428—42no—
51#13133807—22no—
52C4⋊Dic107—21no—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—22no—
59Dic107⋊C4—22no—
60Dic107⋊C4—21no—
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—21no—
65C4×Dic107—22no—
66C4×Dic107—21no—
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—
78D4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D412yes—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2411yescharacteristic
85C22⋊D107—44no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—22no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91D428—21no—
92D428#4112yes—
93C4×D107—44no—
94C4×D107—42no—
95Dic214—21no—
96Dic214—22no—
97Dic214—21no—
98Dic214—22no—
99Dic214—42no—
100C8×C107—21no—
101C8×C107—21no—
102C8×C107—22no—
103C22×C107—22no—
104C22×C107—22no—
105C22×C107—22no—
106C22×C107C22×D411yescharacteristic
107C2×D107—22no—
108C2×D107—41no—
109C2×D107—41no—
110C4×C107—22no—
111C4×C107—21no—
112C4×C107#13512yes—
113C4×C107C2×#4111yescharacteristic
114C4×C107H(5,𝔽2)11yescharacteristic
115Dic107—42no—
116Dic107—41no—
117Dic107—42no—
118Dic107—41no—
119Dic107—41no—
120C2×C107—42no—
121C2×C107#54611yescharacteristic
122C2×C107C2×#13511yescharacteristic
123C2×C107#57811yescharacteristic
124D107—42no—
125#3302—1071nomaximal, Sylow
126C107#330211yescharacteristic, complemented, Sylow
127#556—1071no—
128#547—1071no—
129#487—1071no—
130#485—1072no—
131#474—1071no—
132#467—1072no—
133#462—1071no—
134#452—1072no—
135C4×SD16—1071no—
136C8×D4D10711yescharacteristic, complemented
137#416—1072no—
138C2×#41—1072no—
139C2×Q16—1072no—
140C2×SD16—1071no—
141C22×C8C2×D10712yes—
142C4⋊Q8—1071no—
143C4⋊Q8—1071no—
144#126—2142no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1072no—
148C22⋊Q8—1072no—
149#121—2141no—
150C4×Q8—1071no—
151C4×Q8—2141no—
152C4×D4—1071no—
153C4×D4C2×D10711yescharacteristic
154#117—1072no—
155C2×C4⋊C4—2141no—
156#107—1072no—
157#106—1071no—
158C4⋊C8C2×D10711yescharacteristic
159#103—1072no—
160#103—1072no—
161#103—1071no—
162#103—1072no—
163#102—1071no—
164#102—1072no—
165C22⋊C8C2×D10712yes—
166C4×C8C2×D10711yescharacteristic
167#41—4282no—
168#41—2144no—
169C2×Q8—1072no—
170C2×Q8—1071no—
171C2×Q8—2141no—
172C2×D4—1071no—
173C2×D4—2142no—
174C2×D4D42811yescharacteristic
175C22×C4—2141no—
176C22×C4—2142no—
177C22×C4C22×D10712yes—
178Q16—4282no—
179SD16—2142no—
180C2×C8—42no—
181C2×C8C22×D10711yescharacteristic
182C2×C8C22×D10711yescharacteristic
183C2×C8C22×D10712yes—
184C4⋊C4—1071no—
185C4⋊C4—1072no—
186C4⋊C4—1071no—
187C4⋊C4—2141no—
188C4⋊C4—2142no—
189C4⋊C4—2141no—
190C4⋊C4—2142no—
191C4⋊C4—2141no—
192C4⋊C4—2141no—
193C4⋊C4D42811yescharacteristic
194C22⋊C4—2142no—
195C22⋊C4—2142no—
196C22⋊C4—2142no—
197C22⋊C4—2141no—
198C22⋊C4D42812yes—
199C42—2142no—
200C42—2141no—
201C42C22×D10711yescharacteristic
202C23—2141no—
203C23C2×D42812yes—
204Q8—4282no—
205Q8—2141no—
206Q8—2142no—
207Q8—2141no—
208Q8—2142no—
209D4—22no—
210D4—1072no—
211D4—4284no—
212D4—2141no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—4284no—
216C2×C4—4282no—
217C2×C4—2141no—
218C2×C4—2142no—
219C2×C4—2141no—
220C2×C4—2141no—
221C2×C4—2142no—
222C2×C4C2×D42812yes—
223C2×C4C2×D42811yescharacteristic
224C2×C4C2×D42811yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—21no—
227C8—21no—
228C8—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—4281no—
233C22—4281no—
234C22—2142no—
235C22C22×D42811yescharacteristic
236C4—21no—
237C4—22no—
238C4—4281no—
239C4—4282no—
240C4—4281no—
241C4—4282no—
242C4—4281no—
243C4#1681213112yes—
244C4#41×D10711yescharacteristic
245C4#1681215411yescharacteristic
246C2—42no—
247C2—4282no—
248C2#2029327211yescharacteristic
249C2C2×#1681213111yescharacteristic
250C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radical#20293282C2
hypercenterC8×D4D107
hypocenterC107#3302
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. #20293347C2semidirect
3. #20292978C2semidirect
4. #20292969C2semidirect
5. #20292908C2semidirect
6. #20292868C2semidirect
7. #20292838C2semidirect
8. #20292791C2semidirect
9. C4×#13133807C2semidirect
10. #20292624C2semidirect
11. C8×D4×C107C2semidirect
126. C107#3302semidirect
136. C8×D4D107semidirect

Maximal subgroups

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

subgroupindex
#202933472
#202932822
#202929782
#202929692
#202929082
#202928682
#202928382
#202927912
C4×#131338072
#202926242
C8×D4×C1072
#3302107