Finite Group Databasebeta

← #28348391

Subgroups — #28348391

Order 13696.

398 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293281C211yesmaximal, characteristic, complemented
3#20292978C211yesmaximal, characteristic, complemented
4#20292970C212yesmaximal, complemented
5#20292917C211yesmaximal, characteristic, complemented
6#20292855C212yesmaximal, complemented
7#20292832C211yesmaximal, characteristic, complemented
8#20292769C212yesmaximal, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal, complemented
11C4×D8×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×#13133806C2212yes—
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812109C2211yescharacteristic
17#16812095—22no—
18#16812093—22no—
19#16812089C2212yes—
20#16812083—21no—
21#16812083—21no—
22#16812080C2212yes—
23C4×D428C2211yescharacteristic
24#16812068—21no—
25#16812067C2212yes—
26C2×C4⋊Dic107—21no—
27#16812040C2212yes—
28C4⋊(C107⋊C8)C2211yescharacteristic
29#16812034C2212yes—
30#16812034C2212yes—
31#16812032C2212yes—
32#16812030C2212yes—
33(C2×D107)⋊C8C2212yes—
34C2×D8×C107C2211yescharacteristic
35C4×D4×C107C2211yescharacteristic
36C4×D4×C107C2211yescharacteristic
37#107×C107C2211yescharacteristic
38#102×C107C2211yescharacteristic
39#102×C107C2211yescharacteristic
40C4×C8×C107C2211yescharacteristic
41#13133817—24no—
42#13133817—42no—
43C2×Dic214—21no—
44C2×Dic214C2312yes—
45C2×D428D411yescharacteristic
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C22×Dic107—22no—
49C2×C4×D107C2312yes—
50#13133806—42no—
51C2×C107⋊C8C2312yes—
52C8×D107—42no—
53C4⋊Dic107—21no—
54C4⋊Dic107—21no—
55C4⋊Dic107—21no—
56C4⋊Dic107—22no—
57C4⋊Dic107D411yescharacteristic
58C4⋊Dic107C2312yes—
59C22⋊Dic107—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4D412yes—
63C4×Dic107—22no—
64C2×D4×C107C2311yescharacteristic
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—21no—
67C22×C4×C107—21no—
68D8×C107—22no—
69C2×C8×C107C2311yescharacteristic
70C2×C8×C107C2311yescharacteristic
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—21no—
77C23×C107—21no—
78C22×D107C2×D412yes—
79D4×C107—21no—
80D4×C107—21no—
81D4×C107—21no—
82D4×C107#4112yes—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—42no—
86C2×C4×C107—42no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—44no—
91C22⋊D107—82no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—44no—
95C2×Dic107C2×D412yes—
96D428—22no—
97C4×D107—22no—
98C4×D107—22no—
99Dic214—22no—
100Dic214—22no—
101Dic214—81no—
102C8×C107—21no—
103C8×C107—21no—
104C107⋊C8—22no—
105C22×C107—22no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107—41no—
109C22×C107—41no—
110C22×C107—41no—
111C22×C107C22×D411yescharacteristic
112C2×D107—22no—
113C2×D107—22no—
114C2×D107—22no—
115C4×C107—21no—
116C4×C107—41no—
117C4×C107—41no—
118C4×C107#13512yes—
119C4×C107C2×#4111yescharacteristic
120C4×C107H(5,𝔽2)11yescharacteristic
121Dic107—22no—
122Dic107—42no—
123Dic107—42no—
124C2×C107—42no—
125C2×C107—81no—
126C2×C107#54611yescharacteristic
127C2×C107C2×#13511yescharacteristic
128C2×C107#57711yescharacteristic
129D107—42no—
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×D8D10711yescharacteristic, complemented
141C8×D4—1071no—
142#416—1072no—
143C2×#41—1072no—
144C2×SD16—1072no—
145C2×D8C2×D10711yescharacteristic
146C22×C8—1072no—
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×D4C2×D10711yescharacteristic
157C4×D4C2×D10711yescharacteristic
158#117—1072no—
159#117—2141no—
160C2×C4⋊C4—2141no—
161#107C2×D10711yescharacteristic
162#106—1072no—
163C4⋊C8—1071no—
164#103—1072no—
165#103—1072no—
166#103—1072no—
167#102—1072no—
168#102C2×D10711yescharacteristic
169#102C2×D10711yescharacteristic
170C22⋊C8—1072no—
171C4×C8C2×D10711yescharacteristic
172#41—4282no—
173#41—2144no—
174C2×Q8—1072no—
175C2×Q8—2141no—
176C2×D4—1071no—
177C2×D4—2142no—
178C2×D4—2142no—
179C2×D4C22×D10711yescharacteristic
180C2×D4C22×D10711yescharacteristic
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—1072no—
184C22×C4—2142no—
185SD16—4282no—
186D8—22no—
187C2×C8—1072no—
188C2×C8—4282no—
189C2×C8C22×D10711yescharacteristic
190C2×C8C22×D10711yescharacteristic
191C4⋊C4—1071no—
192C4⋊C4—1072no—
193C4⋊C4—2142no—
194C4⋊C4—2141no—
195C4⋊C4—2141no—
196C4⋊C4—2141no—
197C4⋊C4C22×D10711yescharacteristic
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—1072no—
202C22⋊C4—2142no—
203C22⋊C4—2142no—
204C22⋊C4—2142no—
205C42—2142no—
206C42C22×D10711yescharacteristic
207C23—21no—
208C23—21no—
209C23—1072no—
210Q8—8561no—
211Q8—2142no—
212Q8—2142no—
213D4—21no—
214D4—21no—
215D4—21no—
216D4—8562no—
217D4—4284no—
218D4—2142no—
219D4#1313381812yes—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—1072no—
225C2×C4—4284no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4D4×D10711yescharacteristic
231C2×C4D4×D10711yescharacteristic
232C2×C4C23×D10711yescharacteristic
233C8—21no—
234C8—21no—
235C8—2142no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—2142no—
243C22—2142no—
244C22—2142no—
245C22C2×D4×D10711yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—4282no—
250C4—4282no—
251C4—2142no—
252C4#1681212812yes—
253C4C2×#1313381811yescharacteristic
254C4#1681215511yescharacteristic
255C2—42no—
256C2—81no—
257C2—4282no—
258C2#2029327311yescharacteristic
259C2C2×#1681212811yescharacteristic
260C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×D8×C107C2
socleC22×C107C22×D4
hypercenterC4×D8D107
hypocenterC107#3303
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293281C2semidirect
3. #20292978C2semidirect
4. #20292970C2semidirect
5. #20292917C2semidirect
6. #20292855C2semidirect
7. #20292832C2semidirect
8. #20292769C2semidirect
9. C8×D428C2semidirect
10. #20292620C2semidirect
11. C4×D8×C107C2semidirect
131. C107#3303semidirect
140. C4×D8D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932812
#202929782
#202929702
#202929172
#202928552
#202928322
#202927692
C8×D4282
#202926202
C4×D8×C1072
#3303107