Finite Group Databasebeta

← #28337511

Subgroups — #28337511

Order 13696.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#16812134C211yesmaximal, characteristic, complemented
2#20293191C211yesmaximal, characteristic, complemented
3#20292644C211yesmaximal, characteristic, complemented
4#20292633C211yesmaximal, characteristic, complemented
5#20292601C211yesmaximal, characteristic, complemented
6#20292398C211yesmaximal, characteristic, complemented
7#575×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9#16812134—22no—
10#16812134—22no—
11C2×#13133805—21no—
12C2×#13133803—21no—
13C2×#13133802C2211yescharacteristic
14#16812105—21no—
15#16812098—21no—
16#16812083—22no—
17#16812077—22no—
18#16812068C2211yescharacteristic
19#16812068C2211yescharacteristic
20#16812036—22no—
21#16812031—22no—
22#16812024—22no—
23C2×#41×C107C2211yescharacteristic
24C2×#41×C107C2211yescharacteristic
25#137×C107—21no—
26#137×C107—21no—
27AGL(1,ℤ/8)×C107—21no—
28AGL(1,ℤ/8)×C107—21no—
29#135×C107—22no—
30C2×M16×C107C2211yescharacteristic
31C23×D107—21no—
32C2×D428—21no—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—21no—
36C2×D428—21no—
37C2×D428D412yes—
38C2×C22⋊D107—24no—
39#13133805—24no—
40#13133803—24no—
41#13133802—21no—
42#13133802D412yescomplemented
43C2×C107⋊C8—21no—
44C4⋊Dic107—21no—
45C4⋊Dic107—21no—
46C22⋊Dic107—21no—
47C22⋊Dic107D412yes—
48(C2×D107)⋊C4—24no—
49C4×Dic107—21no—
50C4×Dic107—21no—
51#41×C107—21no—
52#41×C107—21no—
53#41×C107—21no—
54#41×C107—21no—
55#41×C107—21no—
56#41×C107—21no—
57#41×C107—21no—
58#41×C107—21no—
59C2×Q8×C107D411yescharacteristic
60C2×Q8×C107D411yescharacteristic, complemented
61C2×D4×C107—21no—
62C2×D4×C107—21no—
63C2×D4×C107D411yescharacteristic
64C2×D4×C107D411yescharacteristic, complemented
65C22×C4×C107—21no—
66C22×C4×C107—21no—
67C22×C4×C107C2311yescharacteristic
68Q16×C107—22no—
69SD16×C107—22no—
70SD16×C107—22no—
71D8×C107—22no—
72M16×C107—21no—
73M16×C107—21no—
74C2×C8×C107D412yescomplemented
75C23×C107—21no—
76C23×C107—21no—
77C23×C107C2×D411yescharacteristic
78C22×D107—22no—
79C22×D107—41no—
80C22×D107—41no—
81C22×D107—82no—
82Q8×C107—21no—
83Q8×C107—21no—
84Q8×C107—21no—
85Q8×C107—21no—
86D4×C107—21no—
87D4×C107—21no—
88D4×C107—21no—
89D4×C107—21no—
90D4×C107—41no—
91D4×C107—41no—
92D4×C107—41no—
93D4×C107—41no—
94C2×C4×C107—21no—
95C2×C4×C107—21no—
96C2×C4×C107—41no—
97C2×C4×C107—41no—
98C2×C4×C107—41no—
99C2×C4×C107—41no—
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D411yescharacteristic
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107#4111yescharacteristic
106C22⋊D107—84no—
107C2×Dic107—22no—
108C2×Dic107—41no—
109D428—22no—
110D428—22no—
111D428—41no—
112D428—41no—
113C8×C107—22no—
114C107⋊C8—22no—
115C22×C107—41no—
116C22×C107—42no—
117C22×C107—41no—
118C22×C107—41no—
119C22×C107—81no—
120C22×C107#12011yescharacteristic
121C22×C107#12111yescharacteristic
122C22×C107#12711yescharacteristic
123C2×D107—44no—
124C2×D107—42no—
125C2×D107—82no—
126C2×D107—82no—
127C4×C107—41no—
128C4×C107—41no—
129C4×C107#12011yescharacteristic
130C4×C107#12011yescharacteristic
131C4×C107#12111yescharacteristic
132C4×C107#12111yescharacteristic
133Dic107—42no—
134Dic107—81no—
135C2×C107—21no—
136C2×C107—21no—
137C2×C107—21no—
138C2×C107—81no—
139C2×C107—81no—
140C2×C107#39211yescharacteristic
141D107—82no—
142#2008—1071nomaximal, Sylow
143C107#200811yescharacteristic, complemented, Sylow
144#575D10711yescharacteristic, complemented
145C2×AGL(1,ℤ/8)—1071no—
146#535—1071no—
147#421—1071no—
148#418—1071no—
149#410—1071no—
150#344—1071no—
151C2×#41C2×D10711yescharacteristic
152C2×#41C2×D10711yescharacteristic
153C22×D4—1071no—
154#137—21no—
155#137—21no—
156AGL(1,ℤ/8)—21no—
157AGL(1,ℤ/8)—21no—
158AGL(1,ℤ/8)—2142no—
159AGL(1,ℤ/8)—2142no—
160#135—22no—
161C2×SD16—2141no—
162C2×D8—2141no—
163C2×M16—1071no—
164C2×M16C2×D10711yescharacteristic
165#127—2141no—
166#124—2141no—
167#121—2142no—
168#120—2142no—
169#117—1071no—
170#117—1071no—
171#104—2142no—
172#102—2142no—
173#99—2142no—
174C24—2141no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183C2×Q8C22⋊D10711yescharacteristic
184C2×Q8C22⋊D10711yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—1072no—
188C2×D4—2144no—
189C2×D4—2141no—
190C2×D4—2141no—
191C2×D4—2141no—
192C2×D4—2141no—
193C2×D4—2141no—
194C2×D4C22⋊D10711yescharacteristic
195C2×D4C22⋊D10711yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—21no—
198C22×C4C22×D10711yescharacteristic
199Q16—22no—
200SD16—22no—
201SD16—22no—
202SD16—2144no—
203D8—22no—
204D8—2144no—
205M16—21no—
206M16—21no—
207M16—1072no—
208M16—2141no—
209C2×C8—2141no—
210C2×C8C22⋊D10712yescomplemented
211C4⋊C4—2141no—
212C4⋊C4—2141no—
213C22⋊C4—1072no—
214C22⋊C4—2144no—
215C22⋊C4—2141no—
216C42—2141no—
217C42—2141no—
218C23—21no—
219C23—21no—
220C23—8562no—
221C23—4281no—
222C23—4281no—
223C23—2142no—
224C23D4×D10711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—21no—
228Q8—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—8564no—
238D4—4281no—
239D4—4281no—
240D4—2142no—
241D4—2142no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—4281no—
249C2×C4—2142no—
250C2×C4D4×D10711yescharacteristic
251C2×C4D4×D10711yescharacteristic
252C2×C4C2×C22⋊D10711yescharacteristic
253C2×C4C2×C22⋊D10711yescharacteristic
254C2×C4C2×C22⋊D10711yescharacteristic
255C2×C4#1313381711yescharacteristic
256C8—22no—
257C8—2142no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22—8562no—
264C22—8562no—
265C22—4284no—
266C22—4282no—
267C22#1681207711yescharacteristic
268C22#1681208111yescharacteristic
269C22#1681210511yescharacteristic
270C4—41no—
271C4—41no—
272C4—8561no—
273C4—4282no—
274C4#1681207711yescharacteristic
275C4#1681207811yescharacteristic
276C4#1681208011yescharacteristic
277C4#1681208111yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—81no—
283C2—8562no—
284C2#2029253511yescharacteristic
subgroupG/H
centerC2#20292535
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#575×C107C2
socleC2×C107#392
hypercenter#575D107
hypocenterC107#2008
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. #20293191C2semidirect
3. #20292644C2semidirect
4. #20292633C2semidirect
5. #20292601C2semidirect
6. #20292398C2semidirect
7. #575×C107C2semidirect
42. #13133802D4semidirect
60. C2×Q8×C107D4semidirect
64. C2×D4×C107D4semidirect
74. C2×C8×C107D4semidirect
143. C107#2008semidirect
144. #575D107semidirect
184. C2×Q8C22⋊D107semidirect
195. C2×D4C22⋊D107semidirect
210. C2×C8C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
#202931912
#202926442
#202926332
#202926012
#202923982
#575×C1072
#2008107