Finite Group Databasebeta

← #28349059

Subgroups — #28349059

Order 13696.

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

normal subgroups only

This group has 386 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#20293342C211yesmaximal, characteristic, complemented
2#20293280C211yesmaximal, characteristic, complemented
3#20293023C211yesmaximal, characteristic, complemented
4#20292954C212yesmaximal, complemented
5#20292944C212yesmaximal, complemented
6#20292918C211yesmaximal, characteristic, complemented
7#20292879C212yesmaximal, complemented
8C4×#13133803C211yesmaximal, characteristic, complemented
9#20292704C211yesmaximal, characteristic, complemented
10#20292668C212yesmaximal, complemented
11#474×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—21no—
13C2×#13133803C2211yescharacteristic
14C2×#13133801C2212yes—
15#16812109C2211yescharacteristic
16#16812095—22no—
17#16812094—22no—
18#16812085—21no—
19#16812084C2212yes—
20#16812083—21no—
21#16812080C2212yes—
22C4×D428C2211yescharacteristic
23C4×D428C2211yescharacteristic
24#16812068—21no—
25#16812067C2212yes—
26C4⋊C4×D107C2212yes—
27#16812042C2211yescharacteristic
28#16812041C2212yes—
29#16812040C2212yes—
30#16812040C2212yes—
31#16812031C2212yes—
32#16812030C2211yescharacteristic
33#16812029C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×D4×C107C2211yescharacteristic
38#107×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133819—44no—
42C2×Dic214—21no—
43C2×D428—22no—
44C2×D428D411yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—22no—
47C2×C4×D107—22no—
48C2×C4×D107—21no—
49C2×C4×D107C2312yes—
50#13133803—22no—
51#13133801—24no—
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C4⋊Dic107—21no—
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2312yes—
57C4⋊Dic107C2311yescharacteristic
58Dic107⋊C4—22no—
59C22⋊Dic107—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4D412yes—
64C4×Dic107—22no—
65C2×Q8×C107—21no—
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68C2×C8×C107C2312yes—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2312yes—
71C4⋊C4×C107C2311yescharacteristic
72C22⋊C4×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—21no—
75C22×D107—21no—
76C22×D107C2×D412yes—
77Q8×C107—42no—
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—82no—
87C2×Dic107—22no—
88C2×Dic107—21no—
89C2×Dic107C2×D412yes—
90D428—22no—
91D428—21no—
92D428—21no—
93D428—44no—
94C4×D107—22no—
95C4×D107—22no—
96C4×D107—44no—
97C4×D107—42no—
98Dic214—81no—
99C8×C107—22no—
100C107⋊C8—21no—
101C107⋊C8#4112yescomplemented
102C22×C107—41no—
103C22×C107—41no—
104C22×C107—41no—
105C22×C107C22×D411yescharacteristic
106C2×D107—22no—
107C2×D107—22no—
108C2×D107—22no—
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—41no—
112C4×C107—21no—
113C4×C107—42no—
114C4×C107—41no—
115C4×C107AGL(1,ℤ/8)12yes—
116C4×C107C2×#4111yescharacteristic
117C4×C1072−1+411yescharacteristic
118Dic107—22no—
119Dic107—41no—
120Dic107—42no—
121C2×C107—81no—
122C2×C107#54711yescharacteristic
123C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C107#57711yescharacteristic
125D107—42no—
126D107—81no—
127#3354—1071nomaximal, Sylow
128C107#335411yescharacteristic, complemented, Sylow
129#555—1071no—
130#547—1071no—
131#498—1071no—
132#482—1072no—
133#481—1072no—
134#476—1071no—
135#474D10711yescharacteristic, complemented
136#469—1072no—
137C4×D8—1071no—
138#436—1071no—
139#428—1072no—
140C2×#41—2141no—
141C2×D8—1071no—
142C2×M16—1072no—
143C4⋊Q8—1071no—
144C4⋊Q8C2×D10711yescharacteristic
145#124—2142no—
146#123—2142no—
147C22⋊Q8—2141no—
148#121—1072no—
149#121—1072no—
150#121—2141no—
151C4×D4—1071no—
152C4×D4—1071no—
153C4×D4C2×D10711yescharacteristic
154#117—1072no—
155#117—2141no—
156C2×C4⋊C4—1072no—
157#107—1071no—
158#107C2×D10712yes—
159#106—1072no—
160#106—1072no—
161#106—1072no—
162C4⋊C8C2×D10711yescharacteristic
163#102—1071no—
164#102—1072no—
165#102—1071no—
166#102C2×D10712yes—
167C22⋊C8—1072no—
168C4×C8—1071no—
169#41—4284no—
170C2×Q8—21no—
171C2×Q8—2141no—
172C2×D4—1071no—
173C2×D4—1071no—
174C2×D4—2142no—
175C2×D4—2142no—
176C2×D4C22×D10711yescharacteristic
177C22×C4—21no—
178C22×C4—1072no—
179C22×C4—2142no—
180C22×C4—2141no—
181D8—2142no—
182M16—2144no—
183C2×C8—1071no—
184C2×C8—1071no—
185C2×C8C22×D10712yes—
186C4⋊C4—21no—
187C4⋊C4—1071no—
188C4⋊C4—1072no—
189C4⋊C4—1071no—
190C4⋊C4—2142no—
191C4⋊C4—2141no—
192C4⋊C4C22×D10711yescharacteristic
193C4⋊C4C22×D10712yes—
194C22⋊C4—21no—
195C22⋊C4—1072no—
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2141no—
200C42—2142no—
201C42C22×D10711yescharacteristic
202C23—21no—
203C23—1072no—
204C23—2141no—
205Q8—42no—
206Q8—8561no—
207D4—21no—
208D4—21no—
209D4—8562no—
210D4—4284no—
211D4—2142no—
212D4—2141no—
213D4—2141no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—1072no—
218C2×C4—4284no—
219C2×C4—4282no—
220C2×C4—2142no—
221C2×C4—2141no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4D4×D10711yescharacteristic
225C2×C4D4×D10711yescharacteristic
226C2×C4C23×D10711yescharacteristic
227C8—22no—
228C8—1072no—
229C8—2141no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—4281no—
234C22—4281no—
235C22—4281no—
236C22—2142no—
237C22—2142no—
238C22—2142no—
239C22C2×D4×D10711yescharacteristic
240C4—21no—
241C4—41no—
242C4—42no—
243C4—4281no—
244C4—4282no—
245C4—2142no—
246C4#1681213312yes—
247C4#41×D10711yescharacteristic
248C4#1681215711yescharacteristic
249C2—81no—
250C2—8561no—
251C2—4282no—
252C2#2029327911yescharacteristic
253C2C2×#1681213311yescharacteristic
254C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#474×C107C2
socleC22×C107C22×D4
hypercenter#474D107
hypocenterC107#3354
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293280C2semidirect
3. #20293023C2semidirect
4. #20292954C2semidirect
5. #20292944C2semidirect
6. #20292918C2semidirect
7. #20292879C2semidirect
8. C4×#13133803C2semidirect
9. #20292704C2semidirect
10. #20292668C2semidirect
11. #474×C107C2semidirect
101. C107⋊C8#41semidirect
128. C107#3354semidirect
135. #474D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932802
#202930232
#202929542
#202929442
#202929182
#202928792
C4×#131338032
#202927042
#202926682
#474×C1072
#3354107