Finite Group Databasebeta

← #28348186

Subgroups — #28348186

Order 13696.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812132C212yesmaximal, complemented
2#20293299C211yesmaximal, characteristic, complemented
3#20293280C211yesmaximal, characteristic, complemented
4#20293001C211yesmaximal, characteristic, complemented
5#20292859C212yesmaximal, complemented
6#20292823C211yesmaximal, characteristic, complemented
7#20292788C212yesmaximal, complemented
8#20292775C212yesmaximal, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10#20292692C211yesmaximal, characteristic, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133819C2212yes—
14#16812132—28no—
15C2×#13133805C2212yes—
16C2×#13133805C2212yes—
17C2×#13133805C2211yescharacteristic
18C2×D856C2212yes—
19C2×C8×D107C2212yes—
20#16812106—21no—
21#16812106C2211yescharacteristic
22#16812094—22no—
23#16812085C2212yes—
24#16812084—21no—
25#16812084—22no—
26#16812084C2212yes—
27C4×D428—21no—
28C4×D428C2211yescharacteristic
29C4×D428C2211yescharacteristic
30C4⋊C4×D107—21no—
31#16812040C2211yescharacteristic
32#16812033C2211yescharacteristic
33#16812031C2212yes—
34#16812029C2211yescharacteristic
35(C2×D107)⋊C8C2212yes—
36C4×C107⋊C8C2211yescharacteristic
37C2×Q16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39C4×Q8×C107C2211yescharacteristic
40C4⋊C8×C107C2211yescharacteristic
41#103×C107C2212yes—
42#13133819—24no—
43#13133819—24no—
44#13133819—42no—
45#13133819—42no—
46C2×D428—22no—
47C2×D428—22no—
48C2×D428—21no—
49C2×D428—22no—
50C2×D428—21no—
51C2×D428D411yescharacteristic
52C2×D428C2312yes—
53C2×D428C2311yescharacteristic
54C2×C4×D107—21no—
55C2×C4×D107—22no—
56C2×C4×D107—22no—
57C2×C4×D107C2312yes—
58#13133805—24no—
59#13133805—42no—
60#13133805D414yescomplemented
61D856—24no—
62C2×C107⋊C8C2311yescharacteristic
63C2×C107⋊C8C2311yescharacteristic
64C8×D107—24no—
65C4⋊Dic107D411yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—21no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4D412yes—
72C2×Q8×C107C2312yes—
73C2×Q8×C107C2311yescharacteristic
74Q16×C107—24no—
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2311yescharacteristic
80C42×C107—21no—
81C42×C107C2311yescharacteristic
82C22×D107—21no—
83C22×D107—22no—
84C22×D107C2×D412yes—
85Q8×C107—21no—
86Q8×C107—22no—
87Q8×C107—22no—
88Q8×C107C2×D412yes—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C2×Dic107—21no—
96C2×Dic107C2×D412yes—
97D428—22no—
98D428—22no—
99D428—21no—
100D428—22no—
101D428—44no—
102D428—44no—
103D428—42no—
104D428—81no—
105D428—82no—
106D428C2×D412yes—
107C4×D107—22no—
108C4×D107—22no—
109C4×D107—44no—
110C4×D107—44no—
111C4×D107—42no—
112C8×C107—22no—
113C107⋊C8—21no—
114C107⋊C8C2×D412yes—
115C22×C107C22×D411yescharacteristic
116C2×D107—22no—
117C2×D107—22no—
118C2×D107—22no—
119C2×D107—22no—
120C2×D107—41no—
121C2×D107—41no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107—42no—
125C4×C107—22no—
126C4×C107—21no—
127C4×C107—42no—
128C4×C107—41no—
129C4×C107#13512yescomplemented
130C4×C107C22×D411yescharacteristic
131C4×C107H(5,𝔽2)11yescharacteristic
132Dic107—22no—
133Dic107—41no—
134C2×C107D4211yescharacteristic
135C2×C107C2×#13511yescharacteristic
136C2×C107#57711yescharacteristic
137D107—42no—
138D107—42no—
139D107—82no—
140#3284—1071nomaximal, Sylow
141C107#328411yescharacteristic, complemented, Sylow
142C2×#135—1072no—
143#550—1071no—
144#547—1071no—
145#492—1071no—
146#465—1072no—
147#462D10711yescharacteristic, complemented
148#459—1071no—
149#452—1072no—
150#449—1072no—
151C4×SD16—1071no—
152C8×D4—1071no—
153C2×#41—1072no—
154C2×#41—1072no—
155#135—2148no—
156C2×Q16C2×D10712yes—
157C2×SD16—1071no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×D8—1072no—
161C22×C8—1072no—
162C4⋊Q8C2×D10711yescharacteristic
163#127—1071no—
164#127—2141no—
165#123—2142no—
166C22⋊Q8—1072no—
167#121—1072no—
168#121—2142no—
169#121—2141no—
170C4×Q8C2×D10711yescharacteristic
171C4×D4—1071no—
172C4×D4—1071no—
173C4×D4—2141no—
174C2×C4⋊C4—2141no—
175#106—1071no—
176C4⋊C8C2×D10711yescharacteristic
177#103—1071no—
178#103C2×D10712yes—
179#102—1072no—
180#102—1071no—
181C22⋊C8—1072no—
182C4×C8—1071no—
183#41—4282no—
184#41—4282no—
185#41—2144no—
186#41—2144no—
187C2×Q8C22×D10712yes—
188C2×Q8C22×D10711yescharacteristic
189C2×D4—1071no—
190C2×D4—1071no—
191C2×D4—1072no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4—2141no—
196C2×D4—2141no—
197C22×C4—1072no—
198C22×C4—2142no—
199C22×C4—2142no—
200C22×C4—2141no—
201Q16—24no—
202SD16—1074no—
203SD16—4282no—
204SD16—2144no—
205D8—2144no—
206C2×C8—1071no—
207C2×C8—1071no—
208C2×C8—2144no—
209C2×C8C22×D10712yes—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—1071no—
214C4⋊C4—1071no—
215C4⋊C4—2142no—
216C4⋊C4C22×D10711yescharacteristic
217C22⋊C4—1072no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C22⋊C4—2141no—
221C42—21no—
222C42C22×D10711yescharacteristic
223C23—1072no—
224C23—2142no—
225C23—2141no—
226Q8—21no—
227Q8—22no—
228Q8—22no—
229Q8D4×D10712yes—
230D4—8561no—
231D4—8562no—
232D4—1072no—
233D4—4284no—
234D4—4284no—
235D4—4282no—
236D4—2142no—
237D4—2142no—
238D4—2142no—
239D4—2141no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—1072no—
244C2×C4—4284no—
245C2×C4—4284no—
246C2×C4—4282no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4D4×D10711yescharacteristic
251C2×C4D4×D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—22no—
254C8—1072no—
255C8—2141no—
256C22—4282no—
257C22—4282no—
258C22—4282no—
259C22—4281no—
260C22—4281no—
261C22—2142no—
262C22—2142no—
263C22—2142no—
264C22—2142no—
265C22C2×D4×D10711yescharacteristic
266C4—21no—
267C4—22no—
268C4—41no—
269C4—42no—
270C4—4281no—
271C4—2142no—
272C4#1681213212yescomplemented
273C4C2×D4×D10711yescharacteristic
274C4#1681215511yescharacteristic
275C2—8562no—
276C2—4282no—
277C2—4282no—
278C2#2029326111yescharacteristic
279C2C2×#1681213211yescharacteristic
280C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
hypercenter#462D107
hypocenterC107#3284
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. #20293299C2semidirect
3. #20293280C2semidirect
4. #20293001C2semidirect
5. #20292859C2semidirect
6. #20292823C2semidirect
7. #20292788C2semidirect
8. #20292775C2semidirect
9. C4×#13133805C2semidirect
10. #20292692C2semidirect
11. #462×C107C2semidirect
60. #13133805D4semidirect
129. C4×C107#135semidirect
141. C107#3284semidirect
147. #462D107semidirect
272. C4#16812132semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
#202932992
#202932802
#202930012
#202928592
#202928232
#202927882
#202927752
C4×#131338052
#202926922
#462×C1072
#3284107