Finite Group Databasebeta

← #28349058

Subgroups — #28349058

Order 13696.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293282C211yesmaximal, characteristic, complemented
3#20293025C211yesmaximal, characteristic, complemented
4#20292950C212yesmaximal, complemented
5#20292944C212yesmaximal, complemented
6#20292917C211yesmaximal, characteristic, complemented
7#20292907C211yesmaximal, characteristic, complemented
8#20292877C212yesmaximal, complemented
9#20292706C211yesmaximal, characteristic, complemented
10#20292668C212yesmaximal, complemented
11C4×D8×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C2×#13133801C2212yes—
14#16812109C2211yescharacteristic
15#16812109C2211yescharacteristic
16#16812095—22no—
17#16812091—22no—
18#16812088—21no—
19#16812083—21no—
20#16812080C2212yes—
21#16812080C2212yes—
22C4×D428C2211yescharacteristic
23#16812068—21no—
24#16812067C2212yes—
25C4⋊C4×D107C2212yes—
26#16812042C2212yes—
27#16812041C2212yes—
28#16812041C2212yes—
29#16812040C2212yes—
30C4⋊(C107⋊C8)C2211yescharacteristic
31#16812030C2212yes—
32#16812030C2212yes—
33(C2×D107)⋊C8C2212yes—
34C2×D8×C107C2211yescharacteristic
35C4×D4×C107C2211yescharacteristic
36C4×D4×C107C2211yescharacteristic
37#107×C107C2211yescharacteristic
38#102×C107C2211yescharacteristic
39#102×C107C2211yescharacteristic
40C4×C8×C107C2211yescharacteristic
41#13133818—44no—
42C2×Dic214—21no—
43C2×Dic214—21no—
44C2×D428D411yescharacteristic
45C2×C22⋊D107—22no—
46C2×C22⋊D107—22no—
47C2×C4×D107—22no—
48C2×C4×D107C2312yes—
49#13133801—24no—
50C2×C107⋊C8C2312yes—
51C4⋊Dic107—21no—
52C4⋊Dic107—21no—
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107C2312yes—
55C4⋊Dic107C2312yes—
56Dic107⋊C4—22no—
57C22⋊Dic107—22no—
58C22⋊Dic107—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4D412yes—
61C4×Dic107—22no—
62C2×D4×C107C2311yescharacteristic
63C2×D4×C107C2311yescharacteristic
64C22×C4×C107—21no—
65C22×C4×C107—21no—
66D8×C107—22no—
67C2×C8×C107C2311yescharacteristic
68C2×C8×C107C2311yescharacteristic
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2311yescharacteristic
71C22⋊C4×C107—21no—
72C22⋊C4×C107—21no—
73C42×C107C2311yescharacteristic
74C23×C107—21no—
75C23×C107—21no—
76C22×D107C2×D412yes—
77D4×C107—21no—
78D4×C107—21no—
79D4×C107—21no—
80D4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—42no—
84C2×C4×C107—42no—
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2411yescharacteristic
88C22⋊D107—44no—
89C22⋊D107—82no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107C2×D412yes—
93D428—22no—
94C4×D107—22no—
95C4×D107—22no—
96C4×D107—44no—
97Dic214—42no—
98Dic214—81no—
99C8×C107—21no—
100C8×C107#4112yescomplemented
101C107⋊C8—22no—
102C22×C107—41no—
103C22×C107—41no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—41no—
108C22×C107C22×D411yescharacteristic
109C2×D107—22no—
110C2×D107—22no—
111C2×D107—22no—
112C4×C107—21no—
113C4×C107—41no—
114C4×C107—41no—
115C4×C107AGL(1,ℤ/8)12yes—
116C4×C107C2×#4111yescharacteristic
117C4×C1072−1+411yescharacteristic
118Dic107—22no—
119Dic107—42no—
120Dic107—42no—
121C2×C107—81no—
122C2×C107—81no—
123C2×C107#54711yescharacteristic
124C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
125C2×C107#57711yescharacteristic
126D107—42no—
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#474—1071no—
136#469—1072no—
137C4×D8D10711yescharacteristic, complemented
138#436—1071no—
139#428—1072no—
140C2×#41—2141no—
141C2×D8C2×D10711yescharacteristic
142C2×M16—1072no—
143C4⋊Q8—1071no—
144C4⋊Q8—1071no—
145#124—2142no—
146#123—2142no—
147C22⋊Q8—2141no—
148#121—1072no—
149#121—1072no—
150#121—2141no—
151C4×D4—1071no—
152C4×D4C2×D10711yescharacteristic
153C4×D4C2×D10711yescharacteristic
154#117—1072no—
155#117—2141no—
156C2×C4⋊C4—1072no—
157#107—1072no—
158#107C2×D10711yescharacteristic
159#106—1072no—
160#106—1072no—
161#106—1072no—
162C4⋊C8—1071no—
163#102—1072no—
164#102—1072no—
165#102C2×D10711yescharacteristic
166#102C2×D10711yescharacteristic
167C22⋊C8—1072no—
168C4×C8C2×D10711yescharacteristic
169#41—4284no—
170C2×Q8—2141no—
171C2×Q8—2141no—
172C2×D4—1071no—
173C2×D4—2142no—
174C2×D4—2142no—
175C2×D4C22×D10711yescharacteristic
176C2×D4C22×D10711yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—1072no—
180C22×C4—2142no—
181D8—22no—
182M16—2144no—
183C2×C8—1072no—
184C2×C8C22×D10711yescharacteristic
185C2×C8C22×D10711yescharacteristic
186C4⋊C4—1071no—
187C4⋊C4—1072no—
188C4⋊C4—1072no—
189C4⋊C4—2142no—
190C4⋊C4—2141no—
191C4⋊C4—2141no—
192C4⋊C4C22×D10711yescharacteristic
193C4⋊C4C22×D10711yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—1072no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2142no—
200C42—2142no—
201C42C22×D10711yescharacteristic
202C23—21no—
203C23—21no—
204C23—1072no—
205Q8—8561no—
206Q8—4282no—
207D4—21no—
208D4—21no—
209D4—21no—
210D4—21no—
211D4—8562no—
212D4—4284no—
213D4—2142no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—1072no—
219C2×C4—4284no—
220C2×C4—2142no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4D4×D10711yescharacteristic
225C2×C4D4×D10711yescharacteristic
226C2×C4C23×D10711yescharacteristic
227C8—21no—
228C8—2142no—
229C8#1313381812yescomplemented
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—2142no—
237C22—2142no—
238C22—2142no—
239C22C2×D4×D10711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—4282no—
244C4—4282no—
245C4—2142no—
246C4#1681213312yes—
247C4C2×#1313381811yescharacteristic
248C4#1681215611yescharacteristic
249C2—81no—
250C2—81no—
251C2—4282no—
252C2#2029328211yescharacteristic
253C2C2×#1681213311yescharacteristic
254C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×D8×C107C2
socleC22×C107C22×D4
hypercenterC4×D8D107
hypocenterC107#3354
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20293342C2semidirect
2. #20293282C2semidirect
3. #20293025C2semidirect
4. #20292950C2semidirect
5. #20292944C2semidirect
6. #20292917C2semidirect
7. #20292907C2semidirect
8. #20292877C2semidirect
9. #20292706C2semidirect
10. #20292668C2semidirect
11. C4×D8×C107C2semidirect
100. C8×C107#41semidirect
128. C107#3354semidirect
137. C4×D8D107semidirect
229. C8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202932822
#202930252
#202929502
#202929442
#202929172
#202929072
#202928772
#202927062
#202926682
C4×D8×C1072
#3354107