Finite Group Databasebeta

← #28345940

Subgroups — #28345940

Order 13696.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic, complemented
3C2×#16812105C211yesmaximal, characteristic, complemented
4#20293095C211yesmaximal, characteristic, complemented
5#20292826C214yesmaximal, complemented
6#20292821C214yesmaximal, complemented
7#20292650C211yesmaximal, characteristic, complemented
8#20292631C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C22×C22⋊D107—22no—
13#16812137—24no—
14#16812135—24no—
15C2×#13133807C2212yescomplemented
16C2×#13133806C2212yescomplemented
17C2×D856C2212yescomplemented
18C2×#13133803C2212yescomplemented
19C2×#13133802C2211yescharacteristic
20#16812105—21no—
21#16812105—21no—
22#16812105C2214yes—
23#16812074C2212yescomplemented
24#16812071C2212yescomplemented
25(C22⋊D107)⋊C4—22no—
26#16812066—21no—
27C2×C4×Dic107C2211yescharacteristic
28C42×D107—21no—
29Dic107⋊C8C2214yescomplemented
30#16812029C2214yescomplemented
31C22×D4×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#102×C107C2214yescomplemented
35C23×D107—21no—
36#13133818—22no—
37#13133818D414yescomplemented
38C2×Dic214C2311yescharacteristic
39C2×D428—22no—
40C2×D428—22no—
41C2×D428—21no—
42C2×D428C2312yes—
43C2×D428C2311yescharacteristic
44C2×C22⋊D107—21no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—48no—
48C22×Dic107D412yes—
49C2×C4×D107—21no—
50C2×C4×D107—22no—
51#13133807—24no—
52#13133806—24no—
53D856—24no—
54#13133803—24no—
55#13133802—22no—
56C2×C107⋊C8C2312yes—
57Dic107⋊C4—22no—
58(C2×D107)⋊C4—22no—
59C4×Dic107—22no—
60C4×Dic107D411yescharacteristic
61C4×Dic107D411yescharacteristic
62C4×Dic107C2312yes—
63C24×C107—21no—
64C2×D4×C107—22no—
65C2×D4×C107—21no—
66C2×D4×C107—22no—
67C2×D4×C107C2312yes—
68C22×C4×C107C2311yescharacteristic
69M16×C107—22no—
70C2×C8×C107C2312yes—
71C4⋊C4×C107C2312yes—
72C22⋊C4×C107—21no—
73C42×C107—21no—
74C23×C107—22no—
75C23×C107—44no—
76C23×C107—42no—
77C23×C107C2×D411yescharacteristic
78C22×D107—21no—
79C22×D107—22no—
80C22×D107—44no—
81C22×D107—42no—
82D4×C107—22no—
83D4×C107—22no—
84D4×C107—42no—
85C2×C4×C107—22no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D412yes—
88C2×C4×C107#4112yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—24no—
91C22⋊D107—48no—
92C22⋊D107—48no—
93C2×Dic107—21no—
94C2×Dic107—24no—
95C2×Dic107C2×D412yes—
96C2×Dic107C2×D412yes—
97D428—22no—
98D428—21no—
99D428—22no—
100D428—42no—
101D428C2×D412yescomplemented
102C4×D107—24no—
103C4×D107—44no—
104Dic214—21no—
105Dic214C2×D412yescomplemented
106C8×C107—22no—
107C107⋊C8—22no—
108C22×C107—22no—
109C22×C107—42no—
110C22×C107—44no—
111C22×C107—84no—
112C22×C107#12112yes—
113C22×C107C22×D411yescharacteristic
114C2×D107—22no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—44no—
118C2×D107—84no—
119C4×C107—42no—
120C4×C107#12112yes—
121C4×C107C22×D411yescharacteristic
122C4×C107C2×#4111yescharacteristic
123Dic107—22no—
124Dic107—22no—
125Dic107AGL(1,ℤ/8)14yescomplemented
126C2×C107—22no—
127C2×C107—82no—
128C2×C107C2×#12111yescharacteristic
129C2×C107C2×AGL(1,ℤ/8)12yes—
130D107—42no—
131D107—82no—
132#3038—1071nomaximal, Sylow
133C107#303811yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1071no—
135C2×AGL(1,ℤ/8)—1071no—
136C2×#127—1071no—
137C4×#41—1071no—
138#460—1074no—
139#459—1074no—
140#423—1071no—
141#418—1071no—
142#418D10711yescharacteristic, complemented
143C2×#41—1071no—
144C22×D4—1071no—
145C22×D4—2142no—
146C22×D4C2×D10711yescharacteristic
147AGL(1,ℤ/8)—2144no—
148AGL(1,ℤ/8)—2144no—
149C2×SD16—1072no—
150C2×SD16—1072no—
151C2×D8—1072no—
152C2×D8—1072no—
153C2×M16—1071no—
154C2×M16C2×D10711yescharacteristic
155#127—1074no—
156#127—2141no—
157#127—2141no—
158C4×Q8—1072no—
159C4×D4—1072no—
160C4×D4—2142no—
161#117—2141no—
162#117C2×D10711yescharacteristic
163C2×C42—1071no—
164C2×C42—2141no—
165C4⋊C8—1074no—
166#102—1074no—
167#102C2×D10714yescomplemented
168C24—21no—
169C24—2141no—
170#41—1074no—
171#41—2142no—
172C2×Q8—1071no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—1071no—
177C2×D4—1072no—
178C2×D4—4288no—
179C2×D4—2144no—
180C2×D4—2144no—
181C2×D4—2141no—
182C2×D4—2141no—
183C2×D4—2142no—
184C2×D4—2142no—
185C2×D4C22×D10712yes—
186C22×C4—1072no—
187C22×C4—2142no—
188C22×C4—2141no—
189C22×C4C22×D10711yescharacteristic
190SD16—2144no—
191SD16—2144no—
192D8—2144no—
193D8—2144no—
194M16—22no—
195M16—2142no—
196C2×C8—1072no—
197C2×C8C22×D10712yes—
198C4⋊C4—2142no—
199C4⋊C4C22×D10712yes—
200C22⋊C4—21no—
201C22⋊C4—2142no—
202C42—21no—
203C42—1071no—
204C42—1071no—
205C42—1072no—
206C42—2142no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—4284no—
211C23—4282no—
212C23—2142no—
213C23—2141no—
214C23D4×D10711yescharacteristic
215Q8—1072no—
216Q8—2141no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—1072no—
221D4—4288no—
222D4—4288no—
223D4—4282no—
224D4—2144no—
225D4—2142no—
226D4—2142no—
227D4—2141no—
228C2×C4—22no—
229C2×C4—1072no—
230C2×C4—1072no—
231C2×C4—4284no—
232C2×C4—2144no—
233C2×C4—2141no—
234C2×C4—2144no—
235C2×C4D4×D10711yescharacteristic
236C2×C4D4×D10712yes—
237C2×C4#1313381812yes—
238C2×C4C23×D10711yescharacteristic
239C8—22no—
240C8—2142no—
241C22—22no—
242C22—44no—
243C22—42no—
244C22—84no—
245C22—8564no—
246C22—4284no—
247C22—4282no—
248C22—4282no—
249C22—2142no—
250C22#1681208212yes—
251C22C2×D4×D10711yescharacteristic
252C4—42no—
253C4—1074no—
254C4—2142no—
255C4—2142no—
256C4#1681208212yes—
257C4C2×D4×D10711yescharacteristic
258C4C2×#1313381811yescharacteristic
259C2—22no—
260C2—82no—
261C2—8562no—
262C2—4282no—
263C2C2×#1681208211yescharacteristic
264C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
hypercenter#418D107
hypocenterC107#3038
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812135C2semidirect
3. C2×#16812105C2semidirect
4. #20293095C2semidirect
5. #20292826C2semidirect
6. #20292821C2semidirect
7. #20292650C2semidirect
8. #20292631C2semidirect
9. #418×C107C2semidirect
15. C2×#13133807C22semidirect
16. C2×#13133806C22semidirect
17. C2×D856C22semidirect
18. C2×#13133803C22semidirect
23. #16812074C22semidirect
24. #16812071C22semidirect
29. Dic107⋊C8C22semidirect
30. #16812029C22semidirect
34. #102×C107C22semidirect
37. #13133818D4semidirect
101. D428C2×D4semidirect
105. Dic214C2×D4semidirect
125. Dic107AGL(1,ℤ/8)semidirect
133. C107#3038semidirect
142. #418D107semidirect
167. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121352
C2×#168121052
#202930952
#202928262
#202928212
#202926502
#202926312
#418×C1072
#3038107