Finite Group Databasebeta

← #28348194

Subgroups — #28348194

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×#16812128C212yesmaximal, complemented
2#20293297C211yesmaximal, characteristic, complemented
3#20293281C211yesmaximal, characteristic, complemented
4#20292999C211yesmaximal, characteristic, complemented
5#20292855C212yesmaximal, complemented
6#20292838C211yesmaximal, characteristic, complemented
7#20292790C212yesmaximal, complemented
8#20292769C212yesmaximal, complemented
9C4×#13133806C211yesmaximal, characteristic, complemented
10#20292692C211yesmaximal, characteristic, complemented
11#459×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×#13133817C2212yes—
14#16812128—28no—
15C2×Dic428C2212yes—
16C2×#13133806C2212yes—
17C2×#13133806C2212yes—
18C2×#13133806C2211yescharacteristic
19C2×C8×D107C2212yes—
20#16812109C2211yescharacteristic
21#16812105—21no—
22#16812093—22no—
23#16812089C2212yes—
24#16812083—21no—
25#16812081—22no—
26#16812080C2212yes—
27C4×Dic214C2211yescharacteristic
28C4×D428C2211yescharacteristic
29D4×Dic107—21no—
30C2×C4⋊Dic107—21no—
31#16812040C2211yescharacteristic
32#16812034C2212yes—
33#16812032C2211yescharacteristic
34#16812030C2211yescharacteristic
35(C2×D107)⋊C8C2212yes—
36C4×C107⋊C8C2211yescharacteristic
37C2×D8×C107C2212yes—
38#127×C107C2211yescharacteristic
39C4×D4×C107C2211yescharacteristic
40C4⋊C8×C107C2211yescharacteristic
41#102×C107C2212yes—
42#13133817—24no—
43#13133817—24no—
44#13133817—42no—
45#13133817—42no—
46C2×Dic214C2312yes—
47C2×Dic214C2311yescharacteristic
48C2×D428D411yescharacteristic
49C2×C22⋊D107—22no—
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52C22×Dic107—22no—
53C22×Dic107—22no—
54C2×C4×D107C2312yes—
55Dic428—24no—
56#13133806—24no—
57#13133806—42no—
58#13133806D414yescomplemented
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—24no—
62C4⋊Dic107—21no—
63C4⋊Dic107—21no—
64C4⋊Dic107—22no—
65C4⋊Dic107D411yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—21no—
68C22⋊Dic107—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4D412yes—
71C4×Dic107—21no—
72C2×D4×C107—21no—
73C2×D4×C107—21no—
74C2×D4×C107C2312yes—
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77D8×C107—24no—
78C2×C8×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C22⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—21no—
83C23×C107—22no—
84C22×D107C2×D412yes—
85D4×C107—22no—
86D4×C107—21no—
87D4×C107—22no—
88D4×C107—42no—
89D4×C107—81no—
90D4×C107C2×D412yes—
91C2×C4×C107—21no—
92C2×C4×C107—42no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2411yescharacteristic
96C22⋊D107—44no—
97C22⋊D107—44no—
98C22⋊D107—82no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101C2×Dic107—22no—
102C2×Dic107—44no—
103C2×Dic107—44no—
104C2×Dic107C2×D412yes—
105D428—22no—
106C4×D107—22no—
107C4×D107—22no—
108Dic214—21no—
109Dic214—22no—
110Dic214—22no—
111Dic214C2×D412yes—
112C8×C107—22no—
113C107⋊C8—21no—
114C107⋊C8C2×D412yes—
115C22×C107—22no—
116C22×C107—41no—
117C22×C107—41no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107C22×D411yescharacteristic
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C4×C107—21no—
126C4×C107—41no—
127C4×C107#13512yescomplemented
128C4×C107C22×D411yescharacteristic
129C4×C107H(5,𝔽2)11yescharacteristic
130Dic107—22no—
131Dic107—22no—
132Dic107—42no—
133Dic107—41no—
134C2×C107—42no—
135C2×C107—82no—
136C2×C107D4211yescharacteristic
137C2×C107C2×#13511yescharacteristic
138C2×C107#57711yescharacteristic
139D107—42no—
140#3284—1071nomaximal, Sylow
141C107#328411yescharacteristic, complemented, Sylow
142C2×#135—1072no—
143#550—1071no—
144#547—1071no—
145#492—1071no—
146#465—1072no—
147#462—1071no—
148#459D10711yescharacteristic, complemented
149#452—1072no—
150#449—1072no—
151C4×SD16—1071no—
152C8×D4—1071no—
153C2×#41—1072no—
154C2×#41—1072no—
155#135—2148no—
156C2×Q16—1072no—
157C2×SD16—1071no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×D8C2×D10712yes—
161C22×C8—1072no—
162C4⋊Q8—1071no—
163#127—2141no—
164#127C2×D10711yescharacteristic
165#123—2142no—
166C22⋊Q8—1072no—
167#121—1072no—
168#121—2142no—
169#121—2141no—
170C4×Q8—1071no—
171C4×D4—1071no—
172C4×D4—2141no—
173C4×D4C2×D10711yescharacteristic
174C2×C4⋊C4—2141no—
175#106—1071no—
176C4⋊C8C2×D10711yescharacteristic
177#103—1072no—
178#103—1071no—
179#102—1071no—
180#102C2×D10712yes—
181C22⋊C8—1072no—
182C4×C8—1071no—
183#41—4282no—
184#41—4282no—
185#41—2144no—
186#41—2144no—
187C2×Q8—1071no—
188C2×Q8—1072no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—1071no—
192C2×D4—2142no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4C22×D10712yes—
196C2×D4C22×D10711yescharacteristic
197C22×C4—21no—
198C22×C4—1072no—
199C22×C4—2142no—
200C22×C4—2142no—
201Q16—2144no—
202SD16—1074no—
203SD16—4282no—
204SD16—2144no—
205D8—24no—
206C2×C8—1071no—
207C2×C8—1071no—
208C2×C8—2144no—
209C2×C8C22×D10712yes—
210C4⋊C4—1071no—
211C4⋊C4—1071no—
212C4⋊C4—2142no—
213C4⋊C4—2141no—
214C4⋊C4—2141no—
215C4⋊C4—2141no—
216C4⋊C4C22×D10711yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1072no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C42—2141no—
222C42C22×D10711yescharacteristic
223C23—21no—
224C23—22no—
225C23—1072no—
226Q8—1072no—
227Q8—2142no—
228Q8—2142no—
229Q8—2141no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—8562no—
236D4—4284no—
237D4—4284no—
238D4—2142no—
239D4D4×D10712yes—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—1072no—
243C2×C4—4284no—
244C2×C4—4284no—
245C2×C4—2141no—
246C2×C4—2142no—
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—22no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—2142no—
263C22—2142no—
264C22—2142no—
265C22C2×D4×D10711yescharacteristic
266C4—21no—
267C4—41no—
268C4—4281no—
269C4—4282no—
270C4—2142no—
271C4—2142no—
272C4#1681212812yescomplemented
273C4C2×D4×D10711yescharacteristic
274C4#1681215511yescharacteristic
275C2—42no—
276C2—82no—
277C2—4282no—
278C2#2029326111yescharacteristic
279C2C2×#1681212811yescharacteristic
280C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3284
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812128C2semidirect
2. #20293297C2semidirect
3. #20293281C2semidirect
4. #20292999C2semidirect
5. #20292855C2semidirect
6. #20292838C2semidirect
7. #20292790C2semidirect
8. #20292769C2semidirect
9. C4×#13133806C2semidirect
10. #20292692C2semidirect
11. #459×C107C2semidirect
58. #13133806D4semidirect
127. C4×C107#135semidirect
141. C107#3284semidirect
148. #459D107semidirect
272. C4#16812128semidirect

Maximal subgroups

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

subgroupindex
C2×#168121282
#202932972
#202932812
#202929992
#202928552
#202928382
#202927902
#202927692
C4×#131338062
#202926922
#459×C1072
#3284107