Finite Group Databasebeta

← #28346397

Subgroups — #28346397

Order 13696.

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

normal subgroups only

This group has 462 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×#16812068C211yesmaximal, characteristic, complemented
2#20292948C214yesmaximal, complemented
3#20292943C212yesmaximal, complemented
4#20292938C212yesmaximal, complemented
5#20292667C212yesmaximal, complemented
6#20292632C212yesmaximal, complemented
7#20292591C211yesmaximal, characteristic, complemented
8C2×#121×C107C211yesmaximal, characteristic
9C2×#13133802C2212yes—
10#16812068—21no—
11#16812068—21no—
12#16812068C2212yes—
13#16812068C2212yescomplemented
14C2×C4⋊Dic107C2211yescharacteristic, complemented
15C2×C4⋊Dic107C2211yescharacteristic, complemented
16C2×C22⋊Dic107—21no—
17C2×C4×Dic107—21no—
18#16812042C2214yescomplemented
19#16812040C2214yescomplemented
20#16812030C2214yescomplemented
21#16812030C2214yescomplemented
22C22⋊(C107⋊C8)C2212yescomplemented
23C22⋊(C107⋊C8)C2212yescomplemented
24C22×D4×C107—21no—
25C22×D4×C107C2211yescharacteristic
26C23×C4×C107C2211yescharacteristic
27#121×C107—21no—
28#121×C107—21no—
29#121×C107C2212yes—
30#121×C107C2212yes—
31C2×C4⋊C4×C107C2211yescharacteristic
32C2×C22⋊C4×C107—21no—
33C22×Dic107—21no—
34C22×Dic107—21no—
35#13133802—24no—
36C2×C107⋊C8C2314yes—
37C4⋊Dic107—21no—
38C4⋊Dic107—21no—
39C4⋊Dic107C2312yes—
40C4⋊Dic107C2312yes—
41C22⋊Dic107—22no—
42C22⋊Dic107—22no—
43C4×Dic107—22no—
44C4×Dic107—22no—
45C24×C107—21no—
46C24×C107D411yescharacteristic
47C2×D4×C107—22no—
48C2×D4×C107—22no—
49C2×D4×C107—21no—
50C2×D4×C107—24no—
51C2×D4×C107—44no—
52C2×D4×C107C2312yes—
53C22×C4×C107—24no—
54C22×C4×C107—21no—
55C22×C4×C107D411yescharacteristic
56C22×C4×C107D411yescharacteristic
57C22×C4×C107D411yescharacteristic
58C22×C4×C107C2311yescharacteristic
59C22×C4×C107C2311yescharacteristic
60C22×C4×C107C2311yescharacteristic
61C4⋊C4×C107—21no—
62C4⋊C4×C107C2312yes—
63C22⋊C4×C107—22no—
64C22⋊C4×C107—22no—
65C23×C107—22no—
66C23×C107—22no—
67C23×C107—22no—
68C23×C107—41no—
69C23×C107—42no—
70C23×C107—42no—
71C23×C107—42no—
72C23×C107C2×D411yescharacteristic
73C23×C107C2×D411yescharacteristic
74C23×C107C2×D411yescharacteristic
75D4×C107—24no—
76D4×C107—44no—
77D4×C107—44no—
78D4×C107—42no—
79C2×C4×C107—22no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—42no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107#4114yes—
90C2×C4×C107C2411yescharacteristic
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—42no—
94C2×Dic107—42no—
95C107⋊C8—24no—
96C22×C107—21no—
97C22×C107—21no—
98C22×C107—21no—
99C22×C107—21no—
100C22×C107—21no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107—44no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107—42no—
107C22×C107—84no—
108C22×C107#12312yes—
109C22×C107AGL(1,ℤ/8)12yes—
110C22×C107C22×D411yescharacteristic
111C4×C107—21no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107#12312yes—
115C4×C107C2×#4112yes—
116Dic107—42no—
117Dic107—42no—
118C2×C107—22no—
119C2×C107—22no—
120C2×C107—41no—
121C2×C107—82no—
122C2×C107C2×#12311yescharacteristic
123C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C107#57511yescharacteristic
125#3094—1071nomaximal, Sylow
126C107#309411yescharacteristic, complemented, Sylow
127C2×#121D10711yescharacteristic
128C2×#117—1071no—
129#482—1074no—
130#481—1072no—
131#480—1072no—
132#428—1072no—
133#418—1072no—
134#407—1071no—
135C22×D4—21no—
136C22×D4C2×D10711yescharacteristic
137C23×C4C2×D10711yescharacteristic
138C2×M16—1072no—
139#121—21no—
140#121—21no—
141#121C2×D10712yes—
142#121C2×D10712yes—
143#117—1072no—
144#117—1072no—
145#117—2141no—
146#117—2141no—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4C2×D10711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2141no—
152C2×C42—2141no—
153#107—1074no—
154#106—1074no—
155#102—1074no—
156#102—1074no—
157C22⋊C8—1072no—
158C22⋊C8—1072no—
159C24—21no—
160C24C22⋊D10711yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—24no—
165C2×D4—44no—
166C2×D4C22×D10712yes—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—2141no—
170C22×C4—2141no—
171C22×C4C22⋊D10711yescharacteristic
172C22×C4C22⋊D10711yescharacteristic
173C22×C4C22⋊D10711yescharacteristic
174C22×C4C22×D10711yescharacteristic
175C22×C4C22×D10711yescharacteristic
176C22×C4C22×D10711yescharacteristic
177M16—2144no—
178C2×C8—1074no—
179C4⋊C4—21no—
180C4⋊C4—1072no—
181C4⋊C4—1072no—
182C4⋊C4—2141no—
183C4⋊C4—2141no—
184C4⋊C4C22×D10712yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2142no—
188C22⋊C4—2142no—
189C42—2142no—
190C42—2142no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C2×C22⋊D10711yescharacteristic
199C23C2×C22⋊D10711yescharacteristic
200C23C2×C22⋊D10711yescharacteristic
201D4—24no—
202D4—44no—
203D4—44no—
204D4—42no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—4282no—
213C2×C4—4282no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4C2×C22⋊D10711yescharacteristic
217C2×C4C2×C22⋊D10711yescharacteristic
218C2×C4C2×C22⋊D10711yescharacteristic
219C2×C4#1313381714yes—
220C2×C4C23×D10711yescharacteristic
221C8—2144no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—84no—
234C22#1681209012yes—
235C22#1681213512yes—
236C22C22×C22⋊D10711yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—4282no—
241C4—4282no—
242C4#1681209012yes—
243C4C2×#1313381712yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—82no—
248C2C2×#1681209011yescharacteristic
249C2C2×#1681213511yescharacteristic
250C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radicalC2×#121×C107C2
socleC22×C107C22×D4
hypercenterC2×#121D107
hypocenterC107#3094
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16812068C2semidirect
2. #20292948C2semidirect
3. #20292943C2semidirect
4. #20292938C2semidirect
5. #20292667C2semidirect
6. #20292632C2semidirect
7. #20292591C2semidirect
13. #16812068C22semidirect
14. C2×C4⋊Dic107C22semidirect
15. C2×C4⋊Dic107C22semidirect
18. #16812042C22semidirect
19. #16812040C22semidirect
20. #16812030C22semidirect
21. #16812030C22semidirect
22. C22⋊(C107⋊C8)C22semidirect
23. C22⋊(C107⋊C8)C22semidirect
126. C107#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#168120682
#202929482
#202929432
#202929382
#202926672
#202926322
#202925912
C2×#121×C1072
#3094107