Finite Group Databasebeta

← #28345696

Subgroups — #28345696

Order 13696.

738 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×Q8×D107C211yesmaximal, characteristic, complemented
2C2×#16812141C212yesmaximal, complemented
3#20293140C211yesmaximal, characteristic
4#20292782C211yesmaximal, characteristic, complemented
5#20292781C212yesmaximal, complemented
6#20292768C211yesmaximal, characteristic, complemented
7#20292767C212yesmaximal, complemented
8#20292766C211yesmaximal, characteristic, complemented
9#20292634C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#451×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C22×Dic214—21no—
14C22×Dic214C2211yescharacteristic
15C2×Q8×D107—28no—
16C2×Q8×D107—21no—
17C2×Q8×D107—21no—
18C2×Q8×D107C2212yes—
19C2×Q8×D107C2212yes—
20C22×C4×D107—21no—
21C22×C4×D107C2211yescharacteristic
22#16812141—28no—
23C2×#13133808C2212yescomplemented
24C2×#13133807C2212yescomplemented
25C2×#13133805C2212yescomplemented
26C2×#13133801C2212yes—
27#16812094—21no—
28#16812085C2211yescharacteristic
29#16812084C2211yescharacteristic
30#16812083C2211yescharacteristic
31#16812079—21no—
32#16812071—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067C2211yescharacteristic
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812032C2212yescomplemented
38C22⋊(C107⋊C8)C2211yescharacteristic
39(C2×D107)⋊C8C2212yes—
40C22×Q8×C107C2211yescharacteristic
41C2×Q16×C107C2212yescomplemented
42C22⋊Q8×C107C2211yescharacteristic
43#103×C107C2212yescomplemented
44C22⋊C8×C107C2211yescharacteristic
45C23×D107D411yescharacteristic
46#13133819—24no—
47C2×Dic214—28no—
48C2×Dic214—21no—
49C2×Dic214—22no—
50C2×Dic214—22no—
51C2×Dic214—24no—
52C2×Dic214C2312yes—
53Q8×D107—28no—
54Q8×D107—28no—
55Q8×D107—24no—
56Q8×D107—24no—
57Q8×D107—24no—
58Q8×D107D418yescomplemented
59C2×D428—21no—
60C2×D428—21no—
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C22×Dic107—21no—
64C22×Dic107D411yescharacteristic
65C2×C4×D107—28no—
66C2×C4×D107—22no—
67C2×C4×D107—22no—
68C2×C4×D107—24no—
69C2×C4×D107—21no—
70C2×C4×D107D411yescharacteristic
71C2×C4×D107D411yescharacteristic
72C2×C4×D107C2311yescharacteristic
73C2×C4×D107C2311yescharacteristic
74#13133808—24no—
75#13133807—24no—
76#13133805—24no—
77#13133801—24no—
78C2×C107⋊C8C2312yes—
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85C4×Dic107—21no—
86C2×Q8×C107—24no—
87C2×Q8×C107—21no—
88C2×Q8×C107C2311yescharacteristic
89C2×Q8×C107C2312yes—
90C22×C4×C107—21no—
91C22×C4×C107C2311yescharacteristic
92Q16×C107—24no—
93C2×C8×C107C2312yes—
94C4⋊C4×C107—21no—
95C4⋊C4×C107C2311yescharacteristic
96C22⋊C4×C107—21no—
97C23×C107C2×D411yescharacteristic
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—22no—
101C22×D107—21no—
102C22×D107C2×D411yescharacteristic
103C22×D107C2×D411yescharacteristic
104Q8×C107—24no—
105Q8×C107—22no—
106Q8×C107—22no—
107Q8×C107C2×D414yes—
108C2×C4×C107—24no—
109C2×C4×C107—22no—
110C2×C4×C107—22no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107C2×D411yescharacteristic
114C2×C4×C107C2411yescharacteristic
115C22⋊D107—42no—
116C2×Dic107—24no—
117C2×Dic107—22no—
118C2×Dic107—22no—
119C2×Dic107—21no—
120C2×Dic107C2×D411yescharacteristic
121C2×Dic107C2×D411yescharacteristic
122D428—22no—
123D428—42no—
124D428—42no—
125C4×D107—28no—
126C4×D107—28no—
127C4×D107—22no—
128C4×D107—22no—
129C4×D107—22no—
130C4×D107—42no—
131C4×D107C2×D414yes—
132Dic214—28no—
133Dic214—24no—
134Dic214—22no—
135Dic214—28no—
136Dic214C2×D414yes—
137C8×C107—22no—
138C107⋊C8—22no—
139C22×C107—21no—
140C22×C107—21no—
141C22×C107#13712yescomplemented
142C22×C107C22×D411yescharacteristic
143C2×D107—28no—
144C2×D107—21no—
145C2×D107—21no—
146C2×D107—21no—
147C2×D107—21no—
148C2×D107—41no—
149C2×D107—41no—
150C2×D107—41no—
151C2×D107#12012yes—
152C2×D107#13712yescomplemented
153C4×C107—24no—
154C4×C107—21no—
155C4×C107—41no—
156C4×C107—41no—
157C4×C107C22×D412yes—
158Dic107—21no—
159Dic107—24no—
160Dic107—41no—
161Dic107#12012yes—
162C2×C107—22no—
163C2×C107C2×#12011yescharacteristic
164C2×C107C2×#13711yescharacteristic
165C2×C107C2×#13711yescharacteristic
166D107—22no—
167D107—22no—
168D107—81no—
169#3004—1071nomaximal, Sylow
170C107#300411yescharacteristic, complemented, Sylow
171C23×Q8—1071no—
172C2×#137—1072no—
173#525—1071no—
174#451—1072no—
175#451—1071no—
176#451D10711yescharacteristic, complemented
177#448—1071no—
178#448—1072no—
179#448—1071no—
180#419—1072no—
181#407—1071no—
182C2×#41—1071no—
183C22×Q8—1072no—
184C22×Q8—1072no—
185C22×Q8—1071no—
186C22×Q8—2141no—
187C22×Q8—2141no—
188C22×Q8—2141no—
189C22×Q8—2148no—
190C22×Q8C2×D10711yescharacteristic
191C23×C4—1071no—
192C23×C4—2141no—
193#137—2148no—
194C2×Q16—1072no—
195C2×Q16C2×D10712yescomplemented
196C2×SD16—1072no—
197C2×SD16—1072no—
198C2×M16—1072no—
199#123—2141no—
200C22⋊Q8—1071no—
201C22⋊Q8C2×D10711yescharacteristic
202#121—1071no—
203#121—1071no—
204#120—2141no—
205C4×D4—2141no—
206C4×D4—2141no—
207#117—1071no—
208#103—1072no—
209#103—1072no—
210#103—1072no—
211#103C2×D10712yescomplemented
212C22⋊C8—1071no—
213C22⋊C8—1072no—
214C22⋊C8C2×D10711yescharacteristic
215C24—1071no—
216#41—2144no—
217C2×Q8—24no—
218C2×Q8—21no—
219C2×Q8—1078no—
220C2×Q8—1072no—
221C2×Q8—2142no—
222C2×Q8—2142no—
223C2×Q8—2144no—
224C2×Q8—2148no—
225C2×Q8—2144no—
226C2×Q8—2144no—
227C2×Q8—2148no—
228C2×Q8—2148no—
229C2×Q8—2144no—
230C2×Q8—2141no—
231C2×Q8C22×D10711yescharacteristic
232C2×Q8C22×D10712yes—
233C2×D4—1071no—
234C2×D4—2141no—
235C2×D4—2141no—
236C2×D4—2141no—
237C22×C4—21no—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1071no—
241C22×C4—1071no—
242C22×C4—1071no—
243C22×C4—2141no—
244C22×C4—2144no—
245C22×C4—2142no—
246C22×C4—2142no—
247C22×C4—2148no—
248C22×C4—2141no—
249C22×C4C22×D10711yescharacteristic
250Q16—24no—
251Q16—2144no—
252SD16—2144no—
253SD16—2144no—
254M16—2144no—
255C2×C8—1072no—
256C2×C8C22×D10712yes—
257C4⋊C4—21no—
258C4⋊C4—1071no—
259C4⋊C4—2141no—
260C4⋊C4C22×D10711yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2141no—
263C22⋊C4—2141no—
264C22⋊C4—2141no—
265C22⋊C4—2141no—
266C42—2141no—
267C23—1071no—
268C23—1071no—
269C23—2142no—
270C23—2142no—
271C23—2141no—
272C23—2142no—
273C23D4×D10711yescharacteristic
274Q8—24no—
275Q8—22no—
276Q8—22no—
277Q8—1074no—
278Q8—2148no—
279Q8—2148no—
280Q8—2144no—
281Q8—2142no—
282Q8D4×D10714yes—
283D4—4282no—
284D4—4282no—
285D4—4282no—
286D4—2142no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—1071no—
293C2×C4—1074no—
294C2×C4—1071no—
295C2×C4—4282no—
296C2×C4—2142no—
297C2×C4—2142no—
298C2×C4—2148no—
299C2×C4—2148no—
300C2×C4—2142no—
301C2×C4—2141no—
302C2×C4—2144no—
303C2×C4—2142no—
304C2×C4—2142no—
305C2×C4D4×D10711yescharacteristic
306C2×C4C23×D10711yescharacteristic
307C8—22no—
308C8—2142no—
309C22—21no—
310C22—21no—
311C22—1072no—
312C22—1072no—
313C22—4281no—
314C22—4281no—
315C22—4281no—
316C22—2148no—
317C22—2141no—
318C22—2141no—
319C22—2141no—
320C22—2141no—
321C22#1681214112yescomplemented
322C22C2×D4×D10711yescharacteristic
323C4—24no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—1072no—
328C4—4281no—
329C4—2141no—
330C4—2144no—
331C4C2×D4×D10712yes—
332C2—22no—
333C2—8561no—
334C2—2142no—
335C2—2142no—
336C2#120×D10711yescharacteristic
337C2#137×D10711yescharacteristic
338C2C2×#1681214111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#451×C107C2
socleC22×C107C22×D4
prime radical#20293140C2
hypercenter#451D107
hypocenterC107#3004
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×#16812141C2semidirect
4. #20292782C2semidirect
5. #20292781C2semidirect
6. #20292768C2semidirect
7. #20292767C2semidirect
8. #20292766C2semidirect
9. #20292634C2semidirect
10. #20292589C2semidirect
11. #451×C107C2semidirect
23. C2×#13133808C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133805C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812032C22semidirect
41. C2×Q16×C107C22semidirect
43. #103×C107C22semidirect
58. Q8×D107D4semidirect
141. C22×C107#137semidirect
152. C2×D107#137semidirect
170. C107#3004semidirect
176. #451D107semidirect
195. C2×Q16C2×D107semidirect
211. #103C2×D107semidirect
321. C22#16812141semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×#168121412
#202931402
#202927822
#202927812
#202927682
#202927672
#202927662
#202926342
#202925892
#451×C1072
#3004107