Finite Group Databasebeta

← #28349472

Subgroups — #28349472

Order 13696.

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

normal subgroups only

This group has 384 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#20293363C211yesmaximal, characteristic, complemented
2#20293362C211yesmaximal, characteristic, complemented
3#20293008C212yesmaximal, complemented
4#20293003C211yesmaximal, characteristic
5#20292844C212yesmaximal, complemented
6#20292841C212yesmaximal, complemented
7#20292828C211yesmaximal, characteristic, complemented
8#20292751C211yesmaximal, characteristic, complemented
9C4×#13133808C211yesmaximal, characteristic, complemented
10#20292707C212yesmaximal, complemented
11#460×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2211yescharacteristic
13C2×#13133807C2212yes—
14C2×#13133803C2212yes—
15C2×#13133803C2212yes—
16#16812106C2211yescharacteristic
17#16812098C2212yes—
18#16812097—21no—
19#16812096—21no—
20#16812096C2212yes—
21#16812094—22no—
22#16812090—22no—
23C4×Dic214C2211yescharacteristic
24C4×Dic214C2211yescharacteristic
25C4×D428—21no—
26D4×Dic107—21no—
27D4×Dic107C2212yes—
28#16812071C2212yes—
29#16812042C2211yescharacteristic
30Dic107⋊C8C2212yes—
31#16812033C2211yescharacteristic
32#16812032C2211yescharacteristic
33#16812031C2212yes—
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×SD16×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41C2×Dic214D411yescharacteristic
42C2×Dic214C2311yescharacteristic
43C2×D428—21no—
44C2×D428—21no—
45C2×D428C2312yes—
46C22×Dic107—22no—
47C2×C4×D107—22no—
48#13133808—22no—
49#13133807—42no—
50#13133803—24no—
51#13133803—42no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—21no—
57Dic107⋊C4D412yes—
58C22⋊Dic107—22no—
59C22⋊Dic107—22no—
60C22⋊Dic107—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107—21no—
65C4×Dic107C2312yes—
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×D107—22no—
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
89C2×Dic107—21no—
90C2×Dic107—21no—
91C2×Dic107—82no—
92C2×Dic107C2×D412yes—
93C2×Dic107C2×D412yes—
94D428—22no—
95D428—22no—
96D428—42no—
97D428—81no—
98C4×D107—82no—
99Dic214—22no—
100Dic214—21no—
101Dic214—21no—
102C8×C107—22no—
103C107⋊C8—21no—
104C107⋊C8#4112yes—
105C22×C107—42no—
106C22×C107—42no—
107C22×C107—42no—
108C22×C107C22×D411yescharacteristic
109C2×D107—42no—
110C2×D107—42no—
111C2×D107—42no—
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—41no—
121Dic107—41no—
122C2×C107—82no—
123C2×C107#55011yescharacteristic
124C2×C107C2×#13511yescharacteristic
125C2×C107#57611yescharacteristic
126D107—82no—
127#3396—1071nomaximal, Sylow
128C107#339611yescharacteristic, complemented, Sylow
129#559—1071no—
130#559—1071no—
131#495—1072no—
132#493—1071no—
133#463—1072no—
134#463—1072no—
135#460—1071no—
136#460D10711yescharacteristic, complemented
137C8×Q8—1071no—
138C4×Q16—1071no—
139C4×D8—1072no—
140C2×Q16—1071no—
141C2×SD16—1072no—
142C2×SD16C2×D10712yes—
143C2×D8—1072no—
144C2×D8—1072no—
145#127—1071no—
146#127C2×D10711yescharacteristic
147#124—1072no—
148#124—1072no—
149#124—2141no—
150#124—2141no—
151#123—2142no—
152#123—2142no—
153C4×Q8—1071no—
154C4×Q8—1071no—
155C4×Q8C2×D10711yescharacteristic
156C4×D4—1072no—
157C4×D4—1072no—
158C4×D4—2141no—
159C4×D4—2141no—
160#107—1071no—
161C4⋊C8—1072no—
162C4⋊C8C2×D10711yescharacteristic
163#103—1071no—
164#103—1071no—
165#102—1072no—
166#102C2×D10712yes—
167C4×C8—1072no—
168C4×C8—1071no—
169C2×Q8—1071no—
170C2×Q8—1071no—
171C2×Q8C22×D10711yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—1072no—
175C2×D4—2141no—
176C2×D4—2141no—
177C2×D4C22×D10712yes—
178C22×C4—2142no—
179C22×C4—2142no—
180Q16—2142no—
181SD16—42no—
182SD16—4282no—
183D8—4282no—
184D8—2144no—
185C2×C8—1071no—
186C2×C8—1071no—
187C2×C8C22×D10712yes—
188C4⋊C4—21no—
189C4⋊C4—1071no—
190C4⋊C4—1072no—
191C4⋊C4—1071no—
192C4⋊C4—2141no—
193C4⋊C4C22×D10711yescharacteristic
194C22⋊C4—2142no—
195C22⋊C4—2142no—
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2142no—
200C42—21no—
201C42—1072no—
202C42—2141no—
203C42C22×D10711yescharacteristic
204C23—22no—
205C23—2142no—
206Q8—21no—
207Q8—21no—
208Q8—2141no—
209Q8—2141no—
210Q8—2142no—
211D4—22no—
212D4—22no—
213D4—42no—
214D4—81no—
215D4—8561no—
216D4—4282no—
217D4—2142no—
218D4—2142no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—8562no—
222C2×C4—8562no—
223C2×C4—1072no—
224C2×C4—1072no—
225C2×C4—2141no—
226C2×C4—2141no—
227C2×C4D4×D10711yescharacteristic
228C2×C4D4×D10711yescharacteristic
229C2×C4C23×D10711yescharacteristic
230C8—22no—
231C8—1072no—
232C8—2141no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—4282no—
237C22—4282no—
238C22—4282no—
239C22C2×D4×D10711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—4281no—
244C4—4281no—
245C4—2142no—
246C4—2142no—
247C4#1681213012yes—
248C4#41×D10711yescharacteristic
249C4#1681215411yescharacteristic
250C2—82no—
251C2—8562no—
252C2#2029329511yescharacteristic
253C2C2×#1681213011yescharacteristic
254C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#460×C107C2
socleC22×C107C22×D4
prime radical#20293003C2
hypercenter#460D107
hypocenterC107#3396
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. #20293363C2semidirect
2. #20293362C2semidirect
3. #20293008C2semidirect
5. #20292844C2semidirect
6. #20292841C2semidirect
7. #20292828C2semidirect
8. #20292751C2semidirect
9. C4×#13133808C2semidirect
10. #20292707C2semidirect
11. #460×C107C2semidirect
128. C107#3396semidirect
136. #460D107semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933622
#202930082
#202930032
#202928442
#202928412
#202928282
#202927512
C4×#131338082
#202927072
#460×C1072
#3396107