Finite Group Databasebeta

← #28347042

Subgroups — #28347042

Order 13696.

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

normal subgroups only

This group has 368 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#20293179C212yesmaximal
2#20293031C211yesmaximal, characteristic, complemented
3#20293005C211yesmaximal, characteristic, complemented
4#20292976C212yesmaximal, complemented
5#20292868C211yesmaximal, characteristic
6#20292866C212yesmaximal
7Q16×Dic107C211yesmaximal, characteristic, complemented
8#20292721C211yesmaximal, characteristic, complemented
9#20292719C212yesmaximal, complemented
10C2×C8×Dic107C211yesmaximal, characteristic
11#467×C107C211yesmaximal, characteristic
12C2×Dic428C2211yescharacteristic
13C2×#13133808C2212yes—
14C22×C107⋊C8C2211yescharacteristic
15#16812108C2212yes—
16#16812107C2212yes—
17#16812099—22no—
18#16812088C2212yes—
19#16812086—22no—
20Q8×Dic107C2212yes—
21#16812074—22no—
22#16812074C2212yes—
23#16812065—22no—
24C2×C4×Dic107C2211yescharacteristic
25#16812043C2211yescharacteristic
26#16812042C2212yes—
27#16812034C2212yes—
28#16812033C2212yes—
29#16812032C2212yes—
30C8×Dic107C2211yescharacteristic
31C8×Dic107C2211yescharacteristic
32C8×Dic107C2211yescharacteristic
33C8×Dic107C2211yescharacteristic
34C2×Q16×C107C2211yescharacteristic
35C22×C8×C107C2211yescharacteristic
36C22⋊Q8×C107C2212yes—
37#107×C107C2211yescharacteristic
38#103×C107C2212yes—
39C2×Dic214—22no—
40C2×Dic214C2312yes—
41C22×Dic107D411yescharacteristic
42C22×Dic107D411yescharacteristic
43Dic428—22no—
44#13133808—24no—
45C2×C107⋊C8—22no—
46C2×C107⋊C8C2311yescharacteristic
47C2×C107⋊C8C2311yescharacteristic
48C4⋊Dic107—22no—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—22no—
51Dic107⋊C4—22no—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55C22⋊Dic107—22no—
56C4×Dic107—22no—
57C4×Dic107—22no—
58C4×Dic107D411yescharacteristic
59C4×Dic107D411yescharacteristic
60C4×Dic107C2311yescharacteristic
61C4×Dic107C2311yescharacteristic
62C2×Q8×C107C2312yes—
63C22×C4×C107C2311yescharacteristic
64Q16×C107—22no—
65C2×C8×C107—22no—
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C4⋊C4×C107C2312yes—
70C22⋊C4×C107—22no—
71C23×C107C2×D411yescharacteristic
72Q8×C107—22no—
73Q8×C107—22no—
74C2×C4×C107—21no—
75C2×C4×C107—21no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2411yescharacteristic
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107Q1614yescomplemented
84C2×Dic107C2×D411yescharacteristic
85C2×Dic107C2×D411yescharacteristic
86C2×Dic107C2×D411yescharacteristic
87C2×Dic107C2×D411yescharacteristic
88Dic214—22no—
89Dic214—22no—
90Dic214—44no—
91C8×C107—21no—
92C8×C107#4112yes—
93C107⋊C8—21no—
94C107⋊C8#4112yes—
95C22×C107—21no—
96C22×C107—21no—
97C22×C107C2×Q1612yes—
98C22×C107C22×D411yescharacteristic
99C4×C107—21no—
100C4×C107—42no—
101C4×C107—42no—
102C4×C107C2×#4111yescharacteristic
103C4×C107C2×#4111yescharacteristic
104Dic107—21no—
105Dic107—42no—
106Dic107—42no—
107Dic107C2×Q1612yes—
108Dic107C2×Q1612yes—
109Dic107#13512yes—
110C2×C107—22no—
111C2×C107#53211yescharacteristic
112C2×C107C22×Q1611yescharacteristic
113C2×C107C2×#13511yescharacteristic
114#3170—1071nomaximal, Sylow
115C107#317011yescharacteristic, complemented, Sylow
116#533—1072no—
117#500—1071no—
118#494—1071no—
119#487—1072no—
120#467—1071no—
121#467—1072no—
122#467D10711yescharacteristic
123C4×Q16—1071no—
124C4×Q16—1071no—
125C4×Q16—1072no—
126C2×C4×C8—1071no—
127C2×Q16—1072no—
128C2×Q16—1071no—
129C2×Q16C2×D10711yescharacteristic
130C22×C8—1071no—
131C22×C8C2×D10711yescharacteristic
132C4⋊Q8—1072no—
133C4⋊Q8—1072no—
134#125—2142no—
135C22⋊Q8—1072no—
136C22⋊Q8—2142no—
137C22⋊Q8C2×D10712yes—
138C4×Q8—1072no—
139C4×Q8—1072no—
140C4×Q8—2142no—
141#117—2142no—
142C2×C42—1071no—
143#107—1071no—
144#107—1072no—
145#107C2×D10711yescharacteristic
146#103—1072no—
147#103—1072no—
148#103—1072no—
149#103C2×D10712yes—
150C4×C8—1071no—
151C4×C8—1071no—
152C4×C8—1071no—
153C4×C8—1071no—
154C2×Q8—1072no—
155C2×Q8—2142no—
156C2×Q8C22×D10712yes—
157C22×C4—1071no—
158C22×C4—1071no—
159C22×C4C22×D10711yescharacteristic
160Q16—22no—
161Q16—2144no—
162Q16—2142no—
163C2×C8—22no—
164C2×C8—1071no—
165C2×C8—1071no—
166C2×C8—2142no—
167C2×C8C22×D10711yescharacteristic
168C2×C8C22×D10711yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—1072no—
171C4⋊C4—2142no—
172C4⋊C4—2142no—
173C4⋊C4—2142no—
174C4⋊C4—2142no—
175C4⋊C4—2142no—
176C4⋊C4—2142no—
177C4⋊C4C22×D10712yes—
178C22⋊C4—22no—
179C22⋊C4—2142no—
180C42—1071no—
181C42—1071no—
182C42—1071no—
183C42—1071no—
184C42—2142no—
185C42—2142no—
186C23D4×D10711yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—4284no—
190Q8—2142no—
191Q8—2142no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—1074no—
197C2×C4—1071no—
198C2×C4—1071no—
199C2×C4—1071no—
200C2×C4—1071no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4D4×D10711yescharacteristic
205C2×C4C23×D10711yescharacteristic
206C8—21no—
207C8—1072no—
208C8—2141no—
209C8#1313381712yes—
210C22—21no—
211C22—21no—
212C22Q16×D10712yes—
213C22C2×D4×D10711yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—1072no—
218C4—1072no—
219C4—1072no—
220C4—4282no—
221C4—4282no—
222C4—2141no—
223C4#41×D10711yescharacteristic
224C4C2×#1313381711yescharacteristic
225C2—22no—
226C2#2029317211yescharacteristic
227C2C2×Q16×D10711yescharacteristic
228C2#135×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#467D107
hypocenterC107#3170
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #20293031C2semidirect
3. #20293005C2semidirect
4. #20292976C2semidirect
7. Q16×Dic107C2semidirect
8. #20292721C2semidirect
9. #20292719C2semidirect
83. C2×Dic107Q16semidirect
115. C107#3170semidirect

Maximal subgroups

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

subgroupindex
#202931792
#202930312
#202930052
#202929762
#202928682
#202928662
Q16×Dic1072
#202927212
#202927192
C2×C8×Dic1072
#467×C1072
#3170107