Finite Group Databasebeta

← #28349167

Subgroups — #28349167

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#20293338C211yesmaximal, characteristic, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20292990C212yesmaximal, complemented
4#20292975C211yesmaximal, characteristic, complemented
5#20292926C212yesmaximal, complemented
6#20292823C211yesmaximal, characteristic, complemented
7#20292821C212yesmaximal, complemented
8#20292754C211yesmaximal, characteristic, complemented
9#20292738C212yesmaximal, complemented
10C4×#13133803C211yesmaximal, characteristic, complemented
11#476×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C2×D856C2212yes—
14C2×#13133803C2211yescharacteristic
15#16812106C2211yescharacteristic
16#16812105C2212yes—
17#16812100C2212yes—
18#16812089—21no—
19#16812085—22no—
20#16812083—21no—
21#16812082—22no—
22C4×Dic214C2211yescharacteristic
23C4×D428C2211yescharacteristic
24#16812071C2212yes—
25#16812071C2212yes—
26C4⋊C4×D107—21no—
27#16812042C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812031C2212yes—
30#16812030C2211yescharacteristic
31#16812029C2212yes—
32#16812029C2211yescharacteristic
33#16812029C2212yes—
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×D4×C107C2211yescharacteristic
38#106×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133818—44no—
42C2×Dic214D411yescharacteristic
43C2×D428—21no—
44C2×D428—21no—
45C2×D428C2312yes—
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C4×D107—21no—
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52D856—24no—
53#13133803—22no—
54C2×C107⋊C8C2311yescharacteristic
55C2×C107⋊C8C2311yescharacteristic
56C4⋊Dic107—22no—
57C4⋊Dic107D411yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—22no—
60Dic107⋊C4D412yes—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107C2312yes—
65C2×Q8×C107—21no—
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2311yescharacteristic
71C4⋊C4×C107C2312yes—
72C22⋊C4×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—21no—
75C22×D107—21no—
76C22×D107—22no—
77Q8×C107—81no—
78D4×C107—21no—
79D4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—42no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—44no—
87C22⋊D107—82no—
88C2×Dic107—21no—
89C2×Dic107C2×D412yes—
90C2×Dic107C2×D412yes—
91D428—22no—
92D428—21no—
93D428—22no—
94D428—42no—
95D428—81no—
96D428#4112yescomplemented
97C4×D107—44no—
98C4×D107—42no—
99C4×D107—82no—
100Dic214—22no—
101C8×C107—22no—
102C107⋊C8—21no—
103C107⋊C8—21no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107C22×D411yescharacteristic
108C2×D107—22no—
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—42no—
112C2×D107—42no—
113C2×D107—42no—
114C4×C107—21no—
115C4×C107—41no—
116C4×C107—42no—
117C4×C107AGL(1,ℤ/8)12yes—
118C4×C107C2×#4111yescharacteristic
119C4×C1072−1+411yescharacteristic
120Dic107—22no—
121Dic107—22no—
122Dic107—41no—
123C2×C107—81no—
124C2×C107#54811yescharacteristic
125C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C107#57611yescharacteristic
127D107—42no—
128D107—82no—
129#3366—1071nomaximal, Sylow
130C107#336611yescharacteristic, complemented, Sylow
131#554—1071no—
132#550—1071no—
133#490—1072no—
134#486—1071no—
135#478—1072no—
136#476D10711yescharacteristic, complemented
137#459—1071no—
138#459—1072no—
139#446—1071no—
140#442—1072no—
141C4×D8—1071no—
142C2×#41—2141no—
143C2×D8—1071no—
144C2×D8—1072no—
145C4⋊Q8C2×D10711yescharacteristic
146#127—1072no—
147#127—1071no—
148#125—1072no—
149C22⋊Q8—2141no—
150C22⋊Q8—2142no—
151#121—2142no—
152#121—2141no—
153C4×Q8—1071no—
154C4×D4—1072no—
155C4×D4—1072no—
156C4×D4—1071no—
157C4×D4C2×D10711yescharacteristic
158C2×C4⋊C4—2141no—
159#107—1071no—
160#106C2×D10712yes—
161C4⋊C8—1072no—
162C4⋊C8C2×D10711yescharacteristic
163#102—1071no—
164#102—1072no—
165#102—1071no—
166#102—1072no—
167#102—1072no—
168#102C2×D10712yes—
169#97—1072no—
170C4×C8—1071no—
171#41—4284no—
172C2×Q8—21no—
173C2×Q8—1071no—
174C2×D4—1071no—
175C2×D4—1072no—
176C2×D4—2142no—
177C2×D4—2142no—
178C2×D4—2141no—
179C2×D4—2141no—
180C2×D4C22×D10711yescharacteristic
181C22×C4—21no—
182C22×C4—2142no—
183C22×C4—2142no—
184C22×C4—2141no—
185D8—2144no—
186D8—2142no—
187C2×C8—1071no—
188C2×C8—1071no—
189C2×C8C22×D10712yes—
190C4⋊C4—21no—
191C4⋊C4—1071no—
192C4⋊C4—1072no—
193C4⋊C4—1071no—
194C4⋊C4—2142no—
195C4⋊C4—2142no—
196C4⋊C4C22×D10712yes—
197C4⋊C4C22×D10711yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—2142no—
200C22⋊C4—2142no—
201C22⋊C4—2141no—
202C42—1072no—
203C42C22×D10711yescharacteristic
204C23—21no—
205C23—2142no—
206C23—2141no—
207Q8—81no—
208Q8—2142no—
209D4—21no—
210D4—21no—
211D4—8561no—
212D4—8562no—
213D4—1072no—
214D4—4284no—
215D4—4282no—
216D4—2142no—
217D4—2142no—
218D4—2141no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—8562no—
223C2×C4—1072no—
224C2×C4—1072no—
225C2×C4—4284no—
226C2×C4—4282no—
227C2×C4—2141no—
228C2×C4D4×D10711yescharacteristic
229C2×C4D4×D10711yescharacteristic
230C2×C4C23×D10711yescharacteristic
231C8—22no—
232C8—2141no—
233C8—2141no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—4282no—
238C22—4282no—
239C22—4282no—
240C22—4281no—
241C22—4281no—
242C22—2142no—
243C22C2×D4×D10711yescharacteristic
244C4—21no—
245C4—41no—
246C4—42no—
247C4—4281no—
248C4—2142no—
249C4—2142no—
250C4#1681213612yes—
251C4#41×D10711yescharacteristic
252C4#1681215711yescharacteristic
253C2—81no—
254C2—8562no—
255C2—4282no—
256C2#2029328711yescharacteristic
257C2C2×#1681213611yescharacteristic
258C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#476×C107C2
socleC22×C107C22×D4
hypercenter#476D107
hypocenterC107#3366
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #20293338C2semidirect
2. #20293298C2semidirect
3. #20292990C2semidirect
4. #20292975C2semidirect
5. #20292926C2semidirect
6. #20292823C2semidirect
7. #20292821C2semidirect
8. #20292754C2semidirect
9. #20292738C2semidirect
10. C4×#13133803C2semidirect
11. #476×C107C2semidirect
96. D428#41semidirect
130. C107#3366semidirect
136. #476D107semidirect

Maximal subgroups

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

subgroupindex
#202933382
#202932982
#202929902
#202929752
#202929262
#202928232
#202928212
#202927542
#202927382
C4×#131338032
#476×C1072
#3366107