Finite Group Databasebeta

← #28348401

Subgroups — #28348401

Order 13696.

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

normal subgroups only

This group has 396 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#20293298C211yesmaximal, characteristic
3#20292974C211yesmaximal, characteristic, complemented
4#20292952C212yesmaximal, complemented
5#20292913C211yesmaximal, characteristic, complemented
6#20292856C212yesmaximal, complemented
7#20292828C211yesmaximal, characteristic, complemented
8#20292791C212yesmaximal, complemented
9C4×Dic428C211yesmaximal, characteristic, complemented
10#20292624C212yesmaximal, complemented
11C8×D4×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13C2×Dic428C2211yescharacteristic
14C2×#13133807C2212yes—
15#16812109C2211yescharacteristic
16#16812106C2211yescharacteristic
17#16812105—21no—
18#16812102—22no—
19#16812096—21no—
20#16812088C2212yes—
21#16812083C2212yes—
22#16812082—22no—
23C4×Dic214C2211yescharacteristic
24C4×Dic214C2211yescharacteristic
25#16812074—21no—
26#16812071—21no—
27#16812068C2212yes—
28#16812043C2211yescharacteristic
29#16812041C2212yes—
30#16812034C2211yescharacteristic
31#16812034C2212yes—
32#16812034C2211yescharacteristic
33#16812031C2212yes—
34#16812031C2212yes—
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×Dic214C2311yescharacteristic
45C2×Dic214C2311yescharacteristic
46C2×D428—21no—
47C2×D428—21no—
48C2×D428C2312yes—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C4×D107—22no—
52Dic428—22no—
53#13133807—42no—
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—22no—
59Dic107⋊C4—21no—
60Dic107⋊C4—21no—
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107—22no—
65C4×Dic107—21no—
66C4×Dic107—21no—
67C2×D4×C107D411yescharacteristic
68C22×C4×C107C2312yes—
69C2×C8×C107—42no—
70C2×C8×C107C2311yescharacteristic
71C2×C8×C107C2312yes—
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107D411yescharacteristic
74C22⋊C4×C107D412yes—
75C42×C107C2311yescharacteristic
76C23×C107C2×D412yes—
77C22×D107—22no—
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—
86C22⋊D107—82no—
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—22no—
92D428—22no—
93D428—22no—
94D428—42no—
95D428—81no—
96C4×D107—44no—
97Dic214—21no—
98Dic214—21no—
99Dic214—21no—
100Dic214—42no—
101Dic214#4112yes—
102C8×C107—21no—
103C8×C107—21no—
104C8×C107—22no—
105C22×C107—22no—
106C22×C107—22no—
107C22×C107—22no—
108C22×C107C22×D411yescharacteristic
109C2×D107—42no—
110C2×D107—42no—
111C2×D107—42no—
112C4×C107—22no—
113C4×C107—21no—
114C4×C107#13512yes—
115C4×C107C2×#4111yescharacteristic
116C4×C107H(5,𝔽2)11yescharacteristic
117Dic107—22no—
118Dic107—41no—
119Dic107—42no—
120Dic107—41no—
121Dic107—41no—
122C2×C107—42no—
123C2×C107#54611yescharacteristic
124C2×C107C2×#13511yescharacteristic
125C2×C107#57711yescharacteristic
126D107—82no—
127#3304—1071nomaximal, Sylow
128C107#330411yescharacteristic, complemented, Sylow
129#556—1071no—
130#550—1071no—
131#486—1071no—
132#482—1072no—
133#475—1071no—
134#465—1072no—
135#460—1071no—
136#452—1072no—
137C4×Q16—1071no—
138C8×D4D10711yescharacteristic, complemented
139#416—1072no—
140C2×#41—1072no—
141C2×Q16—1071no—
142C2×SD16—1072no—
143C22×C8C2×D10712yes—
144C4⋊Q8—1071no—
145#127—1071no—
146#127—2141no—
147#126—2142no—
148#124—2141no—
149C22⋊Q8—1072no—
150#121—1072no—
151#121—2142no—
152C4×Q8—1071no—
153C4×Q8—1071no—
154C4×Q8—2141no—
155C4×D4—2141no—
156C4×D4C2×D10711yescharacteristic
157#117—1072no—
158#107—1071no—
159#106—1072no—
160C4⋊C8C2×D10711yescharacteristic
161#103—1071no—
162#103—1072no—
163#103—1071no—
164#102—1072no—
165#102—1072no—
166#102—1072no—
167C22⋊C8C2×D10712yes—
168C4×C8C2×D10711yescharacteristic
169#41—4282no—
170#41—2144no—
171C2×Q8—1071no—
172C2×Q8—1071no—
173C2×Q8—2141no—
174C2×D4—1072no—
175C2×D4—2142no—
176C2×D4—2142no—
177C2×D4—2141no—
178C2×D4—2141no—
179C2×D4D42811yescharacteristic
180C22×C4—2142no—
181C22×C4C22×D10712yes—
182Q16—2142no—
183SD16—4282no—
184C2×C8—42no—
185C2×C8C22×D10711yescharacteristic
186C2×C8C22×D10711yescharacteristic
187C2×C8C22×D10712yes—
188C4⋊C4—1072no—
189C4⋊C4—1071no—
190C4⋊C4—1071no—
191C4⋊C4—2141no—
192C4⋊C4—2141no—
193C4⋊C4—2142no—
194C4⋊C4—2141no—
195C4⋊C4D42811yescharacteristic
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4D42812yes—
200C42—2142no—
201C42—2141no—
202C42—2141no—
203C42C22×D10711yescharacteristic
204C23—2142no—
205C23C2×D42812yes—
206Q8—1072no—
207Q8—4282no—
208Q8—2141no—
209Q8—2141no—
210Q8—2141no—
211D4—22no—
212D4—8561no—
213D4—8562no—
214D4—4284no—
215D4—4282no—
216D4—2142no—
217D4—2142no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—4284no—
221C2×C4—2141no—
222C2×C4—2142no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2141no—
226C2×C4C2×D42812yes—
227C2×C4C2×D42811yescharacteristic
228C2×C4C2×D42811yescharacteristic
229C2×C4C23×D10711yescharacteristic
230C8—21no—
231C8—21no—
232C8—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—4282no—
237C22—4282no—
238C22—4282no—
239C22C22×D42811yescharacteristic
240C4—21no—
241C4—22no—
242C4—4282no—
243C4—4281no—
244C4—4281no—
245C4—4281no—
246C4—2142no—
247C4#1681213112yes—
248C4#41×D10711yescharacteristic
249C4#1681215411yescharacteristic
250C2—42no—
251C2—8562no—
252C2#2029327211yescharacteristic
253C2C2×#1681213111yescharacteristic
254C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radical#20293298C2
hypercenterC8×D4D107
hypocenterC107#3304
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. #20292974C2semidirect
4. #20292952C2semidirect
5. #20292913C2semidirect
6. #20292856C2semidirect
7. #20292828C2semidirect
8. #20292791C2semidirect
9. C4×Dic428C2semidirect
10. #20292624C2semidirect
11. C8×D4×C107C2semidirect
128. C107#3304semidirect
138. C8×D4D107semidirect

Maximal subgroups

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

subgroupindex
#202933472
#202932982
#202929742
#202929522
#202929132
#202928562
#202928282
#202927912
C4×Dic4282
#202926242
C8×D4×C1072
#3304107