Finite Group Databasebeta

← #28347006

Subgroups — #28347006

Order 13696.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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#20293183C211yesmaximal, characteristic
2C4⋊Q8×D107C211yesmaximal, characteristic
3#20293037C211yesmaximal, characteristic, complemented
4#20293036C211yesmaximal, characteristic, complemented
5#20293033C211yesmaximal, characteristic, complemented
6#20293031C212yesmaximal, complemented
7#20293024C212yesmaximal, complemented
8#20292668C212yesmaximal
9#20292666C212yesmaximal
10#20292579C211yesmaximal, characteristic
11#501×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107—21no—
13C2×#13133801C2212yes—
14#16812109C2211yescharacteristic
15#16812108—21no—
16#16812108C2212yes—
17#16812108C2211yescharacteristic
18#16812108C2211yescharacteristic
19#16812107—21no—
20#16812101C2211yescharacteristic
21#16812087—21no—
22#16812087—21no—
23#16812074—21no—
24#16812074—21no—
25#16812067C2211yescharacteristic
26#16812067C2211yescharacteristic
27C4⋊C4×D107—21no—
28C4⋊C4×D107C2212yes—
29C42×D107C2211yescharacteristic
30#16812043C2212yes—
31#16812042C2212yes—
32#16812042C2212yes—
33#16812041C2212yes—
34#16812040C2212yes—
35#16812040C2212yes—
36#16812020C2211yescharacteristic
37C8×Dic107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39#125×C107C2211yescharacteristic
40#107×C107C2212yes—
41#106×C107C2212yes—
42#97×C107C2211yescharacteristic
43C2×Dic214—21no—
44C2×Dic214—21no—
45C2×Dic214—21no—
46C2×Dic214—22no—
47Q8×D107—44no—
48C2×C4×D107—22no—
49C2×C4×D107D411yescharacteristic
50C2×C4×D107D411yescharacteristic
51C2×C4×D107C2311yescharacteristic
52#13133801Q818yescomplemented
53C2×C107⋊C8C2312yes—
54C4⋊Dic107—21no—
55C4⋊Dic107C2311yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—21no—
59Dic107⋊C4—21no—
60Dic107⋊C4—22no—
61Dic107⋊C4—22no—
62Dic107⋊C4—21no—
63Dic107⋊C4—21no—
64(C2×D107)⋊C4—21no—
65(C2×D107)⋊C4—21no—
66C4×Dic107—21no—
67C4×Dic107—21no—
68C4×Dic107D411yescharacteristic
69C4×Dic107D411yescharacteristic
70C4×Dic107C2311yescharacteristic
71C2×Q8×C107—21no—
72C2×C8×C107C2312yes—
73C4⋊C4×C107—21no—
74C4⋊C4×C107—21no—
75C4⋊C4×C107—21no—
76C4⋊C4×C107C2311yescharacteristic
77C4⋊C4×C107C2312yes—
78C4⋊C4×C107C2311yescharacteristic
79C42×C107C2311yescharacteristic
80C22×D107C2×D411yescharacteristic
81Q8×C107—42no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2411yescharacteristic
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107C2×D411yescharacteristic
92C2×Dic107C2×D411yescharacteristic
93C2×Dic107C2×D411yescharacteristic
94C4×D107—22no—
95C4×D107—22no—
96C4×D107—44no—
97C4×D107C2×Q814yes—
98Dic214—42no—
99Dic214—44no—
100Dic214—42no—
101Dic214—42no—
102C8×C107C2×Q814yescomplemented
103C107⋊C8C2×Q814yescomplemented
104C22×C107C22×D411yescharacteristic
105C2×D107—21no—
106C2×D107—21no—
107C2×D107C4⋊Q812yes—
108C4×C107—21no—
109C4×C107—21no—
110C4×C107—42no—
111C4×C107—41no—
112C4×C107—41no—
113C4×C107C22×Q812yes—
114Dic107—42no—
115Dic107—41no—
116Dic107—41no—
117Dic107C4⋊Q812yes—
118Dic107AGL(1,ℤ/8)12yes—
119Dic107#13712yes—
120C2×C107C2×C4⋊Q811yescharacteristic
121C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C107C2×#13711yescharacteristic
123D107—22no—
124#3165—1071nomaximal, Sylow
125C107#316511yescharacteristic, complemented, Sylow
126#533—1071no—
127C2×C4⋊Q8—1071no—
128#501—1071no—
129#501—1071no—
130#501—1071no—
131#501D10711yescharacteristic, complemented
132#500—1072no—
133#498—1072no—
134#428—1072no—
135#428—1072no—
136C4×M16—1071no—
137C22×Q8—2141no—
138C2×M16—1072no—
139C4⋊Q8—1072no—
140C4⋊Q8—1071no—
141C4⋊Q8—1071no—
142C4⋊Q8—1071no—
143C4⋊Q8—2141no—
144C4⋊Q8—2141no—
145C4⋊Q8C2×D10711yescharacteristic
146#125—1071no—
147#125C2×D10711yescharacteristic
148C22⋊Q8—2141no—
149C22⋊Q8—2141no—
150C4×Q8—2141no—
151C4×Q8—2141no—
152#117—1071no—
153#117—1071no—
154C2×C4⋊C4—1072no—
155C2×C4⋊C4—2141no—
156C2×C42—1071no—
157#107—1072no—
158#107—1072no—
159#107—1072no—
160#107C2×D10712yes—
161#106—1072no—
162#106—1072no—
163#106—1072no—
164#106C2×D10712yes—
165#97—1071no—
166#97C2×D10711yescharacteristic
167C4×C8—1072no—
168C2×Q8—21no—
169C2×Q8—4284no—
170C2×Q8—2141no—
171C2×Q8—2141no—
172C2×Q8—2142no—
173C2×Q8—2141no—
174C22×C4—1071no—
175C22×C4—1071no—
176C22×C4—1071no—
177C22×C4—2142no—
178M16—1078no—
179C2×C8—1072no—
180C2×C8C22×D10712yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—1071no—
185C4⋊C4—1072no—
186C4⋊C4—1071no—
187C4⋊C4—2142no—
188C4⋊C4—2142no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C4⋊C4—2141no—
192C4⋊C4—2141no—
193C4⋊C4—2141no—
194C4⋊C4C22×D10711yescharacteristic
195C4⋊C4C22×D10712yes—
196C4⋊C4C22×D10711yescharacteristic
197C22⋊C4—2141no—
198C22⋊C4—2141no—
199C42—1071no—
200C42—1071no—
201C42—1071no—
202C42—2141no—
203C42—2141no—
204C42C22×D10711yescharacteristic
205C23—1071no—
206Q8—42no—
207Q8—4282no—
208Q8—4284no—
209Q8—4282no—
210Q8—4282no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—1071no—
215C2×C4—1071no—
216C2×C4—1071no—
217C2×C4—1074no—
218C2×C4—4284no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4—2141no—
222C2×C4—2141no—
223C2×C4—2142no—
224C2×C4D4×D10711yescharacteristic
225C2×C4D4×D10711yescharacteristic
226C2×C4C23×D10711yescharacteristic
227C8—1074no—
228C8Q8×D10714yescomplemented
229C22—1072no—
230C22—2141no—
231C22—2141no—
232C22C2×D4×D10711yescharacteristic
233C4—21no—
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—1072no—
239C4—1072no—
240C4—1072no—
241C4—4282no—
242C4—4281no—
243C4—4281no—
244C4C2×Q8×D10712yes—
245C2—2142no—
246C2C4⋊Q8×D10711yescharacteristic
247C2AGL(1,ℤ/8)×D10711yescharacteristic
248C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#501×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#501D107
hypocenterC107#3165
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #20293037C2semidirect
4. #20293036C2semidirect
5. #20293033C2semidirect
6. #20293031C2semidirect
7. #20293024C2semidirect
11. #501×C107C2semidirect
52. #13133801Q8semidirect
102. C8×C107C2×Q8semidirect
103. C107⋊C8C2×Q8semidirect
125. C107#3165semidirect
131. #501D107semidirect
228. C8Q8×D107semidirect

Maximal subgroups

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

subgroupindex
#202931832
C4⋊Q8×D1072
#202930372
#202930362
#202930332
#202930312
#202930242
#202926682
#202926662
#202925792
#501×C1072
#3165107