Finite Group Databasebeta

← #28348399

Subgroups — #28348399

Order 13696.

396 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 115 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#20293351C211yesmaximal, characteristic, complemented
2#20293297C211yesmaximal, characteristic
3#20292949C212yesmaximal, complemented
4#20292909C211yesmaximal, characteristic, complemented
5#20292855C212yesmaximal, complemented
6#20292824C211yesmaximal, characteristic, complemented
7#20292790C212yesmaximal, complemented
8C4×Dic428C211yesmaximal, characteristic, complemented
9C8×D428C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal, complemented
11#486×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×Dic428C2211yescharacteristic
14C2×#13133806C2212yes—
15C2×C8×D107C2212yes—
16#16812105—21no—
17#16812104—22no—
18#16812098—21no—
19#16812089C2212yes—
20#16812081—22no—
21#16812080C2212yes—
22Q8×Dic107—21no—
23C4×Dic214C2211yescharacteristic
24C4×Dic214C2211yescharacteristic
25C4×D428C2211yescharacteristic
26D4×Dic107—21no—
27#16812067C2212yes—
28#16812043C2211yescharacteristic
29#16812040C2212yes—
30C4⋊(C107⋊C8)C2211yescharacteristic
31#16812034C2211yescharacteristic
32#16812034C2211yescharacteristic
33#16812032C2212yes—
34#16812030C2212yes—
35(C2×D107)⋊C8C2212yes—
36C4⋊Q8×C107C2211yescharacteristic
37#127×C107C2211yescharacteristic
38#102×C107C2212yes—
39#102×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133817—24no—
42#13133817—42no—
43C2×Dic214C2311yescharacteristic
44C2×Dic214C2311yescharacteristic
45C2×D428D411yescharacteristic
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C22×Dic107—22no—
49C2×C4×D107C2312yes—
50Dic428—22no—
51#13133806—42no—
52C2×C107⋊C8C2312yes—
53C8×D107—42no—
54C4⋊Dic107—22no—
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—21no—
59Dic107⋊C4—21no—
60C22⋊Dic107—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4D412yes—
64C4×Dic107—21no—
65C4×Dic107—21no—
66C4×Dic107—22no—
67C2×Q8×C107—21no—
68C2×D4×C107—21no—
69C2×D4×C107—21no—
70C2×D4×C107C2312yes—
71C2×C8×C107C2311yescharacteristic
72C2×C8×C107C2311yescharacteristic
73C4⋊C4×C107—21no—
74C4⋊C4×C107C2312yes—
75C42×C107C2311yescharacteristic
76C23×C107—22no—
77C22×D107C2×D412yes—
78Q8×C107—42no—
79D4×C107—22no—
80D4×C107—22no—
81D4×C107—42no—
82D4×C107—81no—
83C2×C4×C107—22no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2411yescharacteristic
87C22⋊D107—44no—
88C22⋊D107—82no—
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—44no—
94C2×Dic107C2×D412yes—
95D428—22no—
96C4×D107—22no—
97C4×D107—22no—
98Dic214—21no—
99Dic214—21no—
100Dic214—21no—
101Dic214#4112yes—
102C8×C107—21no—
103C8×C107—21no—
104C107⋊C8—22no—
105C22×C107—42no—
106C22×C107—42no—
107C22×C107—42no—
108C22×C107C22×D411yescharacteristic
109C2×D107—22no—
110C2×D107—22no—
111C2×D107—22no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107#13512yes—
115C4×C107C2×#4111yescharacteristic
116C4×C107H(5,𝔽2)11yescharacteristic
117Dic107—22no—
118Dic107—22no—
119Dic107—41no—
120Dic107—41no—
121Dic107—41no—
122C2×C107—82no—
123C2×C107#54611yescharacteristic
124C2×C107C2×#13511yescharacteristic
125C2×C107#57711yescharacteristic
126D107—42no—
127#3304—1071nomaximal, Sylow
128C107#330411yescharacteristic, complemented, Sylow
129#556—1071no—
130#550—1071no—
131#486D10711yescharacteristic, complemented
132#482—1072no—
133#475—1071no—
134#465—1072no—
135#460—1071no—
136#452—1072no—
137C4×Q16—1071no—
138C8×D4—1071no—
139#416—1072no—
140C2×#41—1072no—
141C2×Q16—1071no—
142C2×SD16—1072no—
143C22×C8—1072no—
144C4⋊Q8C2×D10711yescharacteristic
145#127—2141no—
146#127C2×D10711yescharacteristic
147#126—2142no—
148#124—2141no—
149C22⋊Q8—1072no—
150#121—1072no—
151#121—2142no—
152C4×Q8—1071no—
153C4×Q8—1071no—
154C4×Q8—2141no—
155C4×D4—1071no—
156C4×D4—2141no—
157#117—1072no—
158#107—1071no—
159#106—1072no—
160C4⋊C8—1071no—
161#103—1071no—
162#103—1071no—
163#103—1072no—
164#102—1072no—
165#102C2×D10712yes—
166#102C2×D10712yes—
167C22⋊C8—1072no—
168C4×C8C2×D10711yescharacteristic
169#41—4282no—
170#41—2144no—
171C2×Q8—21no—
172C2×Q8—1071no—
173C2×Q8—1071no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—1071no—
177C2×D4—2142no—
178C2×D4—2142no—
179C2×D4C22×D10712yes—
180C22×C4—1072no—
181C22×C4—2142no—
182Q16—2142no—
183SD16—4282no—
184C2×C8—1072no—
185C2×C8—4282no—
186C2×C8C22×D10711yescharacteristic
187C2×C8C22×D10711yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—1071no—
190C4⋊C4—1071no—
191C4⋊C4—1071no—
192C4⋊C4—2142no—
193C4⋊C4—2141no—
194C4⋊C4—2141no—
195C4⋊C4C22×D10712yes—
196C22⋊C4—1072no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2142no—
200C42—2142no—
201C42—2141no—
202C42—2141no—
203C42C22×D10711yescharacteristic
204C23—22no—
205C23—1072no—
206Q8—42no—
207Q8—1072no—
208Q8—2141no—
209Q8—2141no—
210Q8—2141no—
211D4—22no—
212D4—22no—
213D4—42no—
214D4—81no—
215D4—8562no—
216D4—4284no—
217D4—2142no—
218C2×C4—22no—
219C2×C4—1072no—
220C2×C4—4284no—
221C2×C4—2141no—
222C2×C4—2141no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4D4×D10711yescharacteristic
228C2×C4D4×D10711yescharacteristic
229C2×C4C23×D10711yescharacteristic
230C8—21no—
231C8—21no—
232C8—2142no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—2142no—
237C22—2142no—
238C22—2142no—
239C22C2×D4×D10711yescharacteristic
240C4—21no—
241C4—42no—
242C4—4281no—
243C4—4281no—
244C4—4281no—
245C4—2142no—
246C4—2142no—
247C4#1681212812yes—
248C4#41×D10711yescharacteristic
249C4#1681215511yescharacteristic
250C2—82no—
251C2—4282no—
252C2#2029327011yescharacteristic
253C2C2×#1681212811yescharacteristic
254C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#486×C107C2
socleC22×C107C22×D4
prime radical#20293297C2
hypercenter#486D107
hypocenterC107#3304
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20293351C2semidirect
3. #20292949C2semidirect
4. #20292909C2semidirect
5. #20292855C2semidirect
6. #20292824C2semidirect
7. #20292790C2semidirect
8. C4×Dic428C2semidirect
9. C8×D428C2semidirect
10. #20292620C2semidirect
11. #486×C107C2semidirect
128. C107#3304semidirect
131. #486D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202932972
#202929492
#202929092
#202928552
#202928242
#202927902
C4×Dic4282
C8×D4282
#202926202
#486×C1072
#3304107