Finite Group Databasebeta

← #28346833

Subgroups — #28346833

Order 13696.

484 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293180C211yesmaximal, characteristic, complemented
2C2×#16812105C211yesmaximal, characteristic
3#20292990C214yesmaximal, complemented
4#20292972C214yesmaximal, complemented
5#20292633C211yesmaximal, characteristic, complemented
6#20292632C211yesmaximal, characteristic, complemented
7#20292631C211yesmaximal, characteristic, complemented
8M16×Dic107C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C22×C22⋊D107—22no—
12C2×#13133802C2211yescharacteristic
13#16812108C2212yescomplemented
14#16812105—22no—
15#16812105C2212yes—
16#16812105C2212yes—
17#16812100C2212yescomplemented
18#16812086—22no—
19C4×Dic214—22no—
20#16812068C2211yescharacteristic
21C2×C4×Dic107C2211yescharacteristic
22#16812031C2214yescomplemented
23#16812030C2214yescomplemented
24#16812029C2214yescomplemented
25#16812019C2212yescomplemented
26C8×Dic107C2212yescomplemented
27C22×D4×C107C2211yescharacteristic
28C2×M16×C107C2211yescharacteristic
29#117×C107C2211yescharacteristic
30#102×C107C2214yescomplemented
31C23×D107—21no—
32C2×Dic214—22no—
33C2×D428—22no—
34C2×D428—22no—
35C2×D428—21no—
36C2×D428C2312yes—
37C2×C22⋊D107—24no—
38C2×C22⋊D107—24no—
39C2×C22⋊D107—48no—
40C22×Dic107D412yes—
41#13133802—22no—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107—22no—
44C4⋊Dic107C2312yes—
45Dic107⋊C4—22no—
46Dic107⋊C4—22no—
47C22⋊Dic107—21no—
48C4×Dic107—21no—
49C4×Dic107D412yes—
50C4×Dic107C2311yescharacteristic
51C4×Dic107C2311yescharacteristic
52C24×C107—21no—
53C2×D4×C107—21no—
54C2×D4×C107—22no—
55C2×D4×C107—22no—
56C2×D4×C107C2312yes—
57C22×C4×C107C2311yescharacteristic
58M16×C107—22no—
59C2×C8×C107C2312yes—
60C4⋊C4×C107C2312yes—
61C22⋊C4×C107—21no—
62C42×C107—21no—
63C23×C107—22no—
64C23×C107—44no—
65C23×C107—42no—
66C23×C107C2×D411yescharacteristic
67C22×D107—22no—
68C22×D107—44no—
69C22×D107—42no—
70D4×C107—22no—
71D4×C107—22no—
72D4×C107—42no—
73C2×C4×C107—22no—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107#4112yes—
76C2×C4×C107#4112yes—
77C2×C4×C107C2411yescharacteristic
78C22⋊D107—48no—
79C22⋊D107—48no—
80C2×Dic107—24no—
81C2×Dic107—22no—
82C2×Dic107C2×D412yes—
83C2×Dic107C2×D412yes—
84D428—22no—
85D428—22no—
86D428—42no—
87Dic214—44no—
88C8×C107—22no—
89C107⋊C8—22no—
90C22×C107—22no—
91C22×C107—42no—
92C22×C107—44no—
93C22×C107—84no—
94C22×C107#12412yes—
95C22×C107C22×D411yescharacteristic
96C2×D107—42no—
97C2×D107—44no—
98C2×D107—84no—
99C4×C107—42no—
100C4×C107#12412yes—
101C4×C107C2×#4111yescharacteristic
102C4×C107C2×#4111yescharacteristic
103Dic107—22no—
104Dic107—42no—
105Dic107AGL(1,ℤ/8)14yes—
106C2×C107—22no—
107C2×C107—82no—
108C2×C107C2×#12411yescharacteristic
109C2×C107C2×AGL(1,ℤ/8)12yes—
110D107—82no—
111#3138—1071nomaximal, Sylow
112C107#313811yescharacteristic, complemented, Sylow
113#533—1071no—
114C2×#127—1071no—
115#490—1074no—
116#486—1074no—
117#418—1071no—
118#418—1071no—
119#418—1071no—
120#418D10711yescharacteristic, complemented
121C4×M16—1071no—
122C22×D4—1071no—
123C22×D4—2142no—
124C22×D4C2×D10711yescharacteristic
125C2×M16—1071no—
126C2×M16C2×D10711yescharacteristic
127C4⋊Q8—1072no—
128#127—1072no—
129#127—1072no—
130#127—2142no—
131#125—1072no—
132C22⋊Q8—2142no—
133C4×Q8—2142no—
134#117—1071no—
135#117C2×D10711yescharacteristic
136C2×C42—1071no—
137#102—1074no—
138#102—1074no—
139#102—1074no—
140#102C2×D10714yescomplemented
141#97—1072no—
142C4×C8—1072no—
143C24—21no—
144C24—2141no—
145C2×Q8—2142no—
146C2×D4—21no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—1072no—
150C2×D4—4288no—
151C2×D4—2144no—
152C2×D4—2144no—
153C2×D4—2141no—
154C2×D4—2142no—
155C2×D4—2142no—
156C2×D4C22×D10712yes—
157C22×C4—1072no—
158C22×C4C22×D10711yescharacteristic
159M16—22no—
160M16—2142no—
161C2×C8—1072no—
162C2×C8C22×D10712yes—
163C4⋊C4—1072no—
164C4⋊C4—2142no—
165C4⋊C4—2142no—
166C4⋊C4—2142no—
167C4⋊C4C22×D10712yes—
168C22⋊C4—21no—
169C22⋊C4—2141no—
170C42—21no—
171C42—1072no—
172C42—1071no—
173C42—1071no—
174C42—2141no—
175C23—22no—
176C23—44no—
177C23—42no—
178C23—4284no—
179C23—4282no—
180C23—2142no—
181C23D4×D10711yescharacteristic
182Q8—4284no—
183D4—22no—
184D4—22no—
185D4—42no—
186D4—4288no—
187D4—4288no—
188D4—4282no—
189D4—2142no—
190D4—2142no—
191C2×C4—22no—
192C2×C4—1072no—
193C2×C4—1072no—
194C2×C4—2144no—
195C2×C4—2142no—
196C2×C4D4×D10711yescharacteristic
197C2×C4#1313381712yes—
198C2×C4#1313381812yes—
199C2×C4C23×D10711yescharacteristic
200C8—22no—
201C8—2142no—
202C22—22no—
203C22—42no—
204C22—44no—
205C22—84no—
206C22—8564no—
207C22—4282no—
208C22—4284no—
209C22#1681209512yes—
210C22C2×D4×D10711yescharacteristic
211C4—42no—
212C4—1074no—
213C4—4282no—
214C4—2142no—
215C4#1681209512yes—
216C4C2×#1313381711yescharacteristic
217C4C2×#1313381811yescharacteristic
218C2—22no—
219C2—82no—
220C2—8562no—
221C2C2×#1681209511yescharacteristic
222C2AGL(1,ℤ/8)×D10712yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812105C2
hypercenter#418D107
hypocenterC107#3138
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. #20293180C2semidirect
3. #20292990C2semidirect
4. #20292972C2semidirect
5. #20292633C2semidirect
6. #20292632C2semidirect
7. #20292631C2semidirect
8. M16×Dic107C2semidirect
9. #418×C107C2semidirect
13. #16812108C22semidirect
17. #16812100C22semidirect
22. #16812031C22semidirect
23. #16812030C22semidirect
24. #16812029C22semidirect
25. #16812019C22semidirect
26. C8×Dic107C22semidirect
30. #102×C107C22semidirect
112. C107#3138semidirect
120. #418D107semidirect
140. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931802
C2×#168121052
#202929902
#202929722
#202926332
#202926322
#202926312
M16×Dic1072
#418×C1072
#3138107