Finite Group Databasebeta

← #28348403

Subgroups — #28348403

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#20293350C211yesmaximal, characteristic, complemented
2#20293297C211yesmaximal, characteristic
3#20292973C211yesmaximal, characteristic, complemented
4#20292950C212yesmaximal, complemented
5#20292913C211yesmaximal, characteristic, complemented
6#20292853C212yesmaximal, complemented
7#20292786C212yesmaximal, complemented
8C4×#13133808C211yesmaximal, characteristic, complemented
9#20292692C211yesmaximal, characteristic, complemented
10#20292620C212yesmaximal, complemented
11#460×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×#13133808C2211yescharacteristic
14C2×C8×D107C2212yes—
15#16812109C2211yescharacteristic
16#16812105—21no—
17#16812104—22no—
18#16812097—21no—
19#16812085C2212yes—
20#16812081—22no—
21#16812080C2212yes—
22C4×Dic214—21no—
23C4×Dic214C2211yescharacteristic
24C4×D428C2211yescharacteristic
25D4×Dic107—21no—
26#16812067C2212yes—
27#16812042C2211yescharacteristic
28#16812041C2212yes—
29#16812034C2212yes—
30#16812033C2211yescharacteristic
31#16812032C2211yescharacteristic
32#16812030C2212yes—
33#16812030C2212yes—
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×SD16×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133817—24no—
42#13133817—42no—
43C2×Dic214—21no—
44C2×Dic214C2311yescharacteristic
45C2×D428D411yescharacteristic
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C22×Dic107—22no—
49C2×C4×D107C2312yes—
50#13133808—22no—
51C2×C107⋊C8C2311yescharacteristic
52C2×C107⋊C8C2311yescharacteristic
53C8×D107—42no—
54C4⋊Dic107—21no—
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2311yescharacteristic
57C4⋊Dic107C2312yes—
58Dic107⋊C4—21no—
59Dic107⋊C4—22no—
60C22⋊Dic107—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4D412yes—
64C4×Dic107—21no—
65C4×Dic107—22no—
66C2×Q8×C107C2311yescharacteristic
67C2×D4×C107—21no—
68C2×D4×C107—21no—
69C2×D4×C107C2312yes—
70SD16×C107—42no—
71C2×C8×C107C2312yes—
72C4⋊C4×C107—21no—
73C4⋊C4×C107C2311yescharacteristic
74C42×C107—21no—
75C42×C107C2311yescharacteristic
76C23×C107—22no—
77C22×D107C2×D412yes—
78Q8×C107—21no—
79Q8×C107—21no—
80D4×C107—22no—
81D4×C107—22no—
82D4×C107—42no—
83D4×C107—81no—
84C2×C4×C107—21no—
85C2×C4×C107—21no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2411yescharacteristic
89C22⋊D107—44no—
90C22⋊D107—82no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—21no—
94C2×Dic107—44no—
95C2×Dic107C2×D412yes—
96D428—22no—
97C4×D107—22no—
98C4×D107—22no—
99Dic214—21no—
100Dic214—42no—
101Dic214#4112yes—
102C8×C107—22no—
103C107⋊C8—21no—
104C107⋊C8—21no—
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—41no—
114C4×C107—41no—
115C4×C107#13512yes—
116C4×C107C2×#4111yescharacteristic
117C4×C107H(5,𝔽2)11yescharacteristic
118Dic107—22no—
119Dic107—22no—
120Dic107—42no—
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#486—1071no—
132#482—1072no—
133#475—1071no—
134#465—1072no—
135#460D10711yescharacteristic, complemented
136#452—1072no—
137C4×Q16—1071no—
138C8×D4—1071no—
139#416—1072no—
140C2×#41—1072no—
141C2×Q16—1071no—
142C2×SD16C2×D10712yes—
143C22×C8—1072no—
144C4⋊Q8—1071no—
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—2141no—
154C4×Q8C2×D10711yescharacteristic
155C4×D4—1071no—
156C4×D4—2141no—
157#117—1072no—
158#107—1071no—
159#106—1072no—
160C4⋊C8C2×D10711yescharacteristic
161#103—1071no—
162#103—1072no—
163#103—1071no—
164#102—1072no—
165#102—1072no—
166#102C2×D10712yes—
167C22⋊C8—1072no—
168C4×C8—1071no—
169#41—4282no—
170#41—2144no—
171C2×Q8—1071no—
172C2×Q8—2141no—
173C2×Q8C22×D10711yescharacteristic
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—42no—
184C2×C8—1071no—
185C2×C8—1071no—
186C2×C8—4282no—
187C2×C8C22×D10712yes—
188C4⋊C4—21no—
189C4⋊C4—1071no—
190C4⋊C4—1072no—
191C4⋊C4—1071no—
192C4⋊C4—2142no—
193C4⋊C4—2141no—
194C4⋊C4—2141no—
195C4⋊C4C22×D10711yescharacteristic
196C22⋊C4—1072no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2142no—
200C42—21no—
201C42—2142no—
202C42—2141no—
203C42C22×D10711yescharacteristic
204C23—22no—
205C23—1072no—
206Q8—21no—
207Q8—21no—
208Q8—1072no—
209Q8—4282no—
210Q8—2141no—
211D4—22no—
212D4—22no—
213D4—42no—
214D4—81no—
215D4—8562no—
216D4—4284no—
217D4—2142no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—1072no—
221C2×C4—4284no—
222C2×C4—2142no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4D4×D10711yescharacteristic
228C2×C4D4×D10711yescharacteristic
229C2×C4C23×D10711yescharacteristic
230C8—22no—
231C8—2141no—
232C8—2141no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—2142no—
237C22—2142no—
238C22—2142no—
239C22C2×D4×D10711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—4282no—
244C4—4281no—
245C4—2142no—
246C4—2142no—
247C4#1681213012yes—
248C4#41×D10711yescharacteristic
249C4#1681215411yescharacteristic
250C2—82no—
251C2—4282no—
252C2#2029327411yescharacteristic
253C2C2×#1681213011yescharacteristic
254C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293297C2
hypercenter#460D107
hypocenterC107#3304
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #20293350C2semidirect
3. #20292973C2semidirect
4. #20292950C2semidirect
5. #20292913C2semidirect
6. #20292853C2semidirect
7. #20292786C2semidirect
8. C4×#13133808C2semidirect
9. #20292692C2semidirect
10. #20292620C2semidirect
11. #460×C107C2semidirect
128. C107#3304semidirect
135. #460D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202932972
#202929732
#202929502
#202929132
#202928532
#202927862
C4×#131338082
#202926922
#202926202
#460×C1072
#3304107