Finite Group Databasebeta

← #28347044

Subgroups — #28347044

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#20293181C212yesmaximal
2#20293031C211yesmaximal, characteristic, complemented
3#20293005C211yesmaximal, characteristic, complemented
4#20292976C212yesmaximal, complemented
5#20292869C211yesmaximal, characteristic
6#20292867C212yesmaximal
7#20292865C211yesmaximal, characteristic
8C4×Dic428C211yesmaximal, characteristic, complemented
9C4×#13133808C212yesmaximal, complemented
10C4×C8×D107C211yesmaximal, characteristic
11C4×Q16×C107C211yesmaximal, characteristic, complemented
12C2×Dic428C2211yescharacteristic
13C2×#13133808C2212yes—
14C2×C8×D107C2211yescharacteristic
15C2×C8×D107C2211yescharacteristic
16#16812108C2212yes—
17#16812107C2212yes—
18#16812099—22no—
19#16812089C2212yes—
20#16812087—22no—
21#16812085C2212yes—
22C4×Dic214—22no—
23C4×Dic214C2212yes—
24#16812066—22no—
25C42×D107C2211yescharacteristic
26#16812043C2211yescharacteristic
27#16812042C2212yes—
28#16812034C2212yes—
29#16812033C2212yes—
30#16812032C2212yes—
31C4×C107⋊C8C2211yescharacteristic
32C8×Dic107C2211yescharacteristic
33C8×Dic107C2211yescharacteristic
34C2×Q16×C107C2211yescharacteristic
35C4×Q8×C107C2212yes—
36#107×C107C2211yescharacteristic
37#103×C107C2212yes—
38C4×C8×C107C2211yescharacteristic
39C2×Dic214—22no—
40C2×Dic214C2312yes—
41C2×C4×D107D411yescharacteristic
42C2×C4×D107D411yescharacteristic
43C2×C4×D107C2311yescharacteristic
44Dic428—22no—
45#13133808—24no—
46C2×C107⋊C8C2311yescharacteristic
47C2×C107⋊C8C2311yescharacteristic
48C8×D107—22no—
49C8×D107—22no—
50C4⋊Dic107—22no—
51C4⋊Dic107C2312yes—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—22no—
59C4×Dic107—22no—
60C4×Dic107D411yescharacteristic
61C4×Dic107D411yescharacteristic
62C4×Dic107C2311yescharacteristic
63C2×Q8×C107C2312yes—
64Q16×C107—22no—
65C2×C8×C107C2311yescharacteristic
66C2×C8×C107C2311yescharacteristic
67C4⋊C4×C107—22no—
68C4⋊C4×C107C2312yes—
69C42×C107—22no—
70C42×C107C2311yescharacteristic
71C22×D107C2×D411yescharacteristic
72Q8×C107—22no—
73Q8×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2411yescharacteristic
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107C2×D411yescharacteristic
82C2×Dic107C2×D411yescharacteristic
83C2×Dic107C2×D411yescharacteristic
84C4×D107—21no—
85C4×D107—21no—
86C4×D107—22no—
87C4×D107Q1614yescomplemented
88Dic214—22no—
89Dic214—22no—
90Dic214—44no—
91C8×C107—21no—
92C8×C107#4112yes—
93C107⋊C8—21no—
94C107⋊C8#4112yes—
95C22×C107C22×D411yescharacteristic
96C2×D107—21no—
97C2×D107—21no—
98C2×D107C2×Q1612yes—
99C4×C107—21no—
100C4×C107—42no—
101C4×C107—42no—
102C4×C107C2×Q1612yes—
103C4×C107C2×#4111yescharacteristic
104C4×C107C2×#4111yescharacteristic
105Dic107—21no—
106Dic107—42no—
107Dic107—42no—
108Dic107C2×Q1612yes—
109Dic107#13512yes—
110C2×C107#53211yescharacteristic
111C2×C107C22×Q1611yescharacteristic
112C2×C107C2×#13511yescharacteristic
113D107—22no—
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—1071no—
122#467—1072no—
123C4×Q16—1072no—
124C4×Q16—1071no—
125C4×Q16D10711yescharacteristic, complemented
126C2×C4×C8—1071no—
127C2×Q16—1072no—
128C2×Q16—1071no—
129C2×Q16C2×D10711yescharacteristic
130C22×C8—1071no—
131C22×C8—1071no—
132C4⋊Q8—1072no—
133C4⋊Q8—1072no—
134#125—2142no—
135C22⋊Q8—1072no—
136C22⋊Q8—1072no—
137C22⋊Q8—2142no—
138C4×Q8—1072no—
139C4×Q8—2142no—
140C4×Q8C2×D10712yes—
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×C8C2×D10711yescharacteristic
154C2×Q8—1072no—
155C2×Q8—2142no—
156C2×Q8C22×D10712yes—
157C22×C4—1071no—
158C22×C4—1071no—
159C22×C4—1071no—
160Q16—22no—
161Q16—2144no—
162Q16—2142no—
163C2×C8—1071no—
164C2×C8—1071no—
165C2×C8—2142no—
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—2142no—
179C22⋊C4—2142no—
180C42—22no—
181C42—1071no—
182C42—1071no—
183C42—1071no—
184C42—2142no—
185C42C22×D10711yescharacteristic
186C23—1071no—
187Q8—22no—
188Q8—22no—
189Q8—4284no—
190Q8—2142no—
191Q8—2142no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—1071no—
195C2×C4—1071no—
196C2×C4—1074no—
197C2×C4—1071no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2141no—
202C2×C4—2141no—
203C2×C4D4×D10711yescharacteristic
204C2×C4D4×D10711yescharacteristic
205C2×C4C23×D10711yescharacteristic
206C8—21no—
207C8—1072no—
208C8—2141no—
209C8#1313381812yes—
210C22—1072no—
211C22—2141no—
212C22—2141no—
213C22C2×D4×D10711yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—1072no—
218C4—1072no—
219C4—4282no—
220C4—4282no—
221C4—2141no—
222C4Q16×D10712yes—
223C4#41×D10711yescharacteristic
224C4C2×#1313381811yescharacteristic
225C2—2142no—
226C2#2029317411yescharacteristic
227C2C2×Q16×D10711yescharacteristic
228C2#135×D10711yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×Q16×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenterC4×Q16D107
hypocenterC107#3170
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #20293031C2semidirect
3. #20293005C2semidirect
4. #20292976C2semidirect
8. C4×Dic428C2semidirect
9. C4×#13133808C2semidirect
11. C4×Q16×C107C2semidirect
87. C4×D107Q16semidirect
115. C107#3170semidirect
125. C4×Q16D107semidirect

Maximal subgroups

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

subgroupindex
#202931812
#202930312
#202930052
#202929762
#202928692
#202928672
#202928652
C4×Dic4282
C4×#131338082
C4×C8×D1072
C4×Q16×C1072
#3170107