Finite Group Databasebeta

← #32864215

Subgroups — #32864215

Order 18176.

4854 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4854 conjugacy classes of subgroups — too many to draw. Only its 807 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#22601980C213yesmaximal, complemented
2#22597746C214yesmaximal, complemented
3#22597737C218yesmaximal, complemented
4#22597732C214yesmaximal, complemented
5C2×#17986189C218yesmaximal, complemented
6#22596801C213yesmaximal, complemented
7C71×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×#11765864C2211yescharacteristic, complemented
9#17986648—212no—
10#17986643—212no—
11#17986570C2216yescomplemented
12#17986558C22112yescomplemented
13#17986506C2216yescomplemented
14C2×#14556109C22112yescomplemented
15#17986474C2213yescomplemented
16#17986189C22116yescomplemented
17#17986189C22132yescomplemented
18#17986186C22116yescomplemented
19#17985973C22112yescomplemented
20C2×#14556043C22124yescomplemented
21#585×C71—212no—
22#585×C71C2214yes—
23C2×H(5,𝔽2)×C71—212no—
24C2×H(5,𝔽2)×C71C2218yes—
25C2×H(5,𝔽2)×C71C2213yes—
26C2×#11765864—212no—
27C22×Dic142C2311yescharacteristic, complemented
28C22×C22⋊D71C2312yescomplemented
29C23×Dic71C2312yescomplemented
30C22×C4×D71C2311yescharacteristic, complemented
31#14556126—26no—
32#14556115—26no—
33#14556111—212no—
34#14556109—224no—
35#14556109C23124yescomplemented
36#14556109C23112yescomplemented
37#14556107—212no—
38#14556104—26no—
39#14556100—224no—
40#14556099—26no—
41#14556097—212no—
42#14556096C23112yescomplemented
43D4×Dic71—212no—
44C4×C22⋊D71—212no—
45#14556087C2316yescomplemented
46C2×Dic71⋊C4—212no—
47C2×C22⋊Dic71—212no—
48C2×C22⋊Dic71C23112yescomplemented
49#14556043C23148yescomplemented
502−1+4×C71—216no—
512−1+4×C71—212no—
52H(5,𝔽2)×C71—248no—
53H(5,𝔽2)×C71—212no—
54H(5,𝔽2)×C71—248no—
55H(5,𝔽2)×C71—224no—
56H(5,𝔽2)×C71C23116yes—
57C2×#41×C71—248no—
58C2×#41×C71—224no—
59C2×#41×C71—224no—
60C2×#41×C71D4112yescomplemented
61C2×#41×C71C2316yes—
62C22×D4×C71—216no—
63C22×D4×C71—224no—
64C22×D4×C71D4112yescomplemented
65C22×D4×C71C23112yes—
66C22×D4×C71C2311yescharacteristic
67C23×D71C2411yescharacteristic
68#11765864—416no—
69C2×Dic142—26no—
70C2×C22⋊D71—212no—
71C2×C22⋊D71—212no—
72C22×Dic71—212no—
73C22×Dic71—212no—
74C22×Dic71C2411yescharacteristic
75C22×Dic71C2412yes—
76C2×C4×D71—26no—
77C4⋊Dic71—26no—
78Dic71⋊C4—212no—
79Dic71⋊C4—224no—
80C22⋊Dic71—212no—
81C22⋊Dic71—224no—
82C22⋊Dic71—212no—
83C22⋊Dic71C24112yes—
84(C2×D71)⋊C4—212no—
85C4×Dic71—26no—
86C24×C71—28no—
87C24×C71C2×D4112yes—
88C24×C71C2412yes—
89#41×C71—296no—
90#41×C71—248no—
91#41×C71—216no—
92#41×C71—248no—
93#41×C71—248no—
94#41×C71—224no—
95C2×Q8×C71—224no—
96C2×Q8×C71—28no—
97C2×Q8×C71C2×D416yes—
98C2×D4×C71—296no—
99C2×D4×C71—224no—
100C2×D4×C71—224no—
101C2×D4×C71—248no—
102C2×D4×C71—248no—
103C2×D4×C71—248no—
104C2×D4×C71C2×D416yes—
105C2×D4×C71C2×D4124yes—
106C2×D4×C71C2×D4112yes—
107C2×D4×C71C24112yes—
108C22×C4×C71—224no—
109C22×C4×C71—216no—
110C22×C4×C71C2×D4112yes—
111C22×C4×C71C2×D4112yes—
112C22×C4×C71C2411yescharacteristic
113C23×C71—216no—
114C23×C71—224no—
115C23×C71—82no—
116C23×C71—812no—
117C23×C71—168no—
118C23×C71C22×D4112yes—
119C23×C71C22×D4112yes—
120C23×C71C2511yescharacteristic
121C22×D71—23no—
122C22×D71—81no—
123Q8×C71—216no—
124Q8×C71—212no—
125D4×C71—248no—
126D4×C71—248no—
127D4×C71—224no—
128D4×C71—212no—
129D4×C71#120116yes—
130C2×C4×C71—248no—
131C2×C4×C71—224no—
132C2×C4×C71—224no—
133C2×C4×C71C22×D4112yes—
134C2×C4×C71C22×D416yes—
135C22⋊D71—48no—
136C22⋊D71—812no—
137C2×Dic71—212no—
138C2×Dic71—23no—
139C2×Dic71—26no—
140C2×Dic71—48no—
141C2×Dic71—812no—
142C4×D71—44no—
143Dic142—44no—
144C22×C71—212no—
145C22×C71—412no—
146C22×C71—41no—
147C22×C71—412no—
148C22×C71—824no—
149C22×C71—816no—
150C22×C71C2×#12018yes—
151C22×C71C23×D413yes—
152C2×D71—46no—
153C2×D71—41no—
154C4×C71—212no—
155C4×C71C2×#12014yes—
156Dic71—412no—
157Dic71—42no—
158Dic71—41no—
159#60401—711nomaximal, Sylow
160C2×C71—23no—
161C2×C71—28no—
162C2×C71—412no—
163C2×C71C22×#12011yescharacteristic
164D71—81no—
165H(7,𝔽2)D7111yescharacteristic, complemented
166#3485—713no—
167#3028—714no—
168#3026—718no—
169#3025—714no—
170C2×#458—718no—
171#2883—713no—
172C71#6040111yescharacteristic, complemented, Sylow
173#585—212no—
174#585C2×D7114yes—
175C2×H(5,𝔽2)—212no—
176C2×H(5,𝔽2)C2×D7118yes—
177C2×H(5,𝔽2)C2×D7113yes—
178C22×#41—711no—
179#547—14212no—
180#546—14212no—
181#536—716no—
182#534—7112no—
183#525—716no—
184C2×#123—7112no—
185#518—713no—
186#458—7116no—
187#458—7132no—
188B(4,𝔽2)—7116no—
189#410—7112no—
190C2×#99—7124no—
1912−1+4—216no—
1922−1+4—212no—
193H(5,𝔽2)—248no—
194H(5,𝔽2)—212no—
195H(5,𝔽2)—248no—
196H(5,𝔽2)—224no—
197H(5,𝔽2)C22×D71116yes—
198C2×#41—248no—
199C2×#41—224no—
200C2×#41—224no—
201C2×#41—14212no—
202C2×#41C22⋊D71112yescomplemented
203C2×#41C22×D7116yes—
204C22×Q8—711no—
205C22×D4—216no—
206C22×D4—224no—
207C22×D4—712no—
208C22×D4C22⋊D71112yescomplemented
209C22×D4C22×D71112yes—
210C22×D4C22×D7111yescharacteristic
211C23×C4—712no—
212C23×C4—711no—
213C4⋊Q8—1426no—
214#124—1426no—
215#123—7112no—
216#123—7124no—
217#123—14224no—
218#123—14212no—
219C22⋊Q8—1426no—
220C22⋊Q8—14212no—
221#121—1426no—
222#121—14224no—
223#120—7112no—
224#120—14212no—
225C4×D4—14212no—
226C4×D4—14212no—
227#117—716no—
228C2×C4⋊C4—14212no—
229C2×C22⋊C4—7112no—
230C2×C22⋊C4—14212no—
231#99—7148no—
232C24—28no—
233C24—711no—
234C24C2×C22⋊D71112yes—
235C24C23×D7112yes—
236#41—296no—
237#41—248no—
238#41—216no—
239#41—248no—
240#41—248no—
241#41—224no—
242#41—28416no—
243C2×Q8—224no—
244C2×Q8—28no—
245C2×Q8—1426no—
246C2×Q8C2×C22⋊D7116yes—
247C2×D4—296no—
248C2×D4—224no—
249C2×D4—224no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—248no—
253C2×D4—14212no—
254C2×D4—14212no—
255C2×D4C2×C22⋊D71124yes—
256C2×D4C2×C22⋊D7116yes—
257C2×D4C2×C22⋊D71112yes—
258C2×D4C23×D71112yes—
259C22×C4—224no—
260C22×C4—216no—
261C22×C4—711no—
262C22×C4—712no—
263C22×C4—14212no—
264C22×C4—1426no—
265C22×C4—14212no—
266C22×C4C2×C22⋊D71112yes—
267C22×C4C2×C22⋊D71112yes—
268C22×C4C23×D7111yescharacteristic
269C4⋊C4—14212no—
270C4⋊C4—14224no—
271C4⋊C4—1426no—
272C22⋊C4—7112no—
273C22⋊C4—14212no—
274C22⋊C4—14212no—
275C22⋊C4—14224no—
276C22⋊C4—14212no—
277C42—1426no—
278C23—216no—
279C23—224no—
280C23—82no—
281C23—812no—
282C23—168no—
283C23—5681no—
284C23—1423no—
285C23C22×C22⋊D71112yes—
286C23C22×C22⋊D71112yes—
287C23C24×D7111yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—2844no—
291D4—248no—
292D4—224no—
293D4—248no—
294D4—212no—
295D4—2848no—
296D4—56812no—
297D4#14556097116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—2848no—
302C2×C4—2844no—
303C2×C4—56812no—
304C2×C4—14212no—
305C2×C4—1423no—
306C2×C4—1426no—
307C2×C4C22×C22⋊D71112yes—
308C2×C4C22×C22⋊D7116yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—824no—
314C22—816no—
315C22—2846no—
316C22—2841no—
317C22C2×#1455609718yes—
318C22C23×C22⋊D7113yes—
319C4—212no—
320C4—28412no—
321C4—2841no—
322C4—2842no—
323C4C2×#1455609714yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—5681no—
328C2C22×#1455609711yescharacteristic
subgroupG/H
centerC2C22×#14556097
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC71×H(7,𝔽2)C2
socleC2×C71C22×#120
hypercenterH(7,𝔽2)D71
hypocenterC71#60401
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22601980C2semidirect
2. #22597746C2semidirect
3. #22597737C2semidirect
4. #22597732C2semidirect
5. C2×#17986189C2semidirect
6. #22596801C2semidirect
7. C71×H(7,𝔽2)C2semidirect
8. C22×#11765864C22semidirect
11. #17986570C22semidirect
12. #17986558C22semidirect
13. #17986506C22semidirect
14. C2×#14556109C22semidirect
15. #17986474C22semidirect
16. #17986189C22semidirect
17. #17986189C22semidirect
18. #17986186C22semidirect
19. #17985973C22semidirect
20. C2×#14556043C22semidirect
27. C22×Dic142C23semidirect
28. C22×C22⋊D71C23semidirect
29. C23×Dic71C23semidirect
30. C22×C4×D71C23semidirect
35. #14556109C23semidirect
36. #14556109C23semidirect
42. #14556096C23semidirect
45. #14556087C23semidirect
48. C2×C22⋊Dic71C23semidirect
49. #14556043C23semidirect
60. C2×#41×C71D4semidirect
64. C22×D4×C71D4semidirect
165. H(7,𝔽2)D71semidirect
172. C71#60401semidirect
202. C2×#41C22⋊D71semidirect
208. C22×D4C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
#226019802
#225977462
#225977372
#225977322
C2×#179861892
#225968012
C71×H(7,𝔽2)2
#6040171