Finite Group Databasebeta

← #32864203

Subgroups — #32864203

Order 18176.

5128 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 5128 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#22601971C213yesmaximal, complemented
2#22597742C214yesmaximal, complemented
3#22597737C218yesmaximal, complemented
4#22597732C214yesmaximal, complemented
5C2×#17986186C218yesmaximal, complemented
6#22596801C213yesmaximal, complemented
7C71×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×D4×D71C2211yescharacteristic, complemented
9#17986637—212no—
10D4×C22⋊D71—212no—
11#17986563C2216yescomplemented
12#17986558C22112yescomplemented
13#17986506C2216yescomplemented
14C2×#14556096C22112yescomplemented
15#17986474C2213yescomplemented
16#17986189C22116yescomplemented
17#17986186C22116yescomplemented
18#17986186C22132yescomplemented
19#17985973C22112yescomplemented
20C2×#14556043C22124yescomplemented
21#585×C71—212no—
22#585×C71C2214yescomplemented
23C2×H(5,𝔽2)×C71—212no—
24C2×H(5,𝔽2)×C71C2213yes—
25C2×H(5,𝔽2)×C71C2218yescomplemented
26C24×D71C2312yescomplemented
27C22×D284C2311yescharacteristic, complemented
28C22×C22⋊D71—212no—
29C22×C22⋊D71C2312yescomplemented
30C2×D4×D71—212no—
31C22×C4×D71C2311yescharacteristic, complemented
32#14556124—26no—
33#14556117—26no—
34#14556111—212no—
35#14556109C23112yescomplemented
36#14556104—26no—
37#14556102—212no—
38#14556100—224no—
39#14556099—26no—
40#14556097—212no—
41#14556096—224no—
42#14556096C23124yescomplemented
43#14556096C23112yescomplemented
44D4×Dic71—212no—
45C4×C22⋊D71—212no—
46#14556087C2316yescomplemented
47C2×C22⋊Dic71C23112yescomplemented
48C2×(C2×D71)⋊C4—212no—
49#14556043C23148yescomplemented
502−1+4×C71—216no—
512−1+4×C71—212no—
52H(5,𝔽2)×C71—248no—
53H(5,𝔽2)×C71—224no—
54H(5,𝔽2)×C71—212no—
55H(5,𝔽2)×C71—248no—
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×D71—212no—
68C23×D71—84no—
69C23×D71C2411yescharacteristic
70C23×D71C2412yes—
71C2×D284—26no—
72C2×C22⋊D71—212no—
73C2×C22⋊D71—212no—
74C2×C22⋊D71—224no—
75C2×C22⋊D71—824no—
76D4×D71—416no—
77C22×Dic71—212no—
78C22×Dic71C2411yescharacteristic
79C2×C4×D71—26no—
80C4⋊Dic71—26no—
81Dic71⋊C4—212no—
82C22⋊Dic71—212no—
83C22⋊Dic71—212no—
84C22⋊Dic71C24112yes—
85(C2×D71)⋊C4—212no—
86(C2×D71)⋊C4—224no—
87C4×Dic71—26no—
88C24×C71—28no—
89C24×C71C2×D4112yescomplemented
90C24×C71C2412yes—
91#41×C71—296no—
92#41×C71—248no—
93#41×C71—216no—
94#41×C71—248no—
95#41×C71—248no—
96#41×C71—224no—
97C2×Q8×C71—224no—
98C2×Q8×C71—28no—
99C2×Q8×C71C2×D416yescomplemented
100C2×D4×C71—296no—
101C2×D4×C71—224no—
102C2×D4×C71—224no—
103C2×D4×C71—248no—
104C2×D4×C71—248no—
105C2×D4×C71—248no—
106C2×D4×C71C2×D4112yes—
107C2×D4×C71C2×D416yescomplemented
108C2×D4×C71C2×D4124yescomplemented
109C2×D4×C71C24112yes—
110C22×C4×C71—224no—
111C22×C4×C71—216no—
112C22×C4×C71C2×D4112yescomplemented
113C22×C4×C71C2×D4112yescomplemented
114C22×C4×C71C2411yescharacteristic
115C23×C71—216no—
116C23×C71—224no—
117C23×C71—82no—
118C23×C71—812no—
119C23×C71—168no—
120C23×C71C22×D4112yes—
121C23×C71C22×D4112yes—
122C23×C71C2511yescharacteristic
123C22×D71—23no—
124C22×D71—26no—
125C22×D71—42no—
126C22×D71—424no—
127C22×D71—48no—
128C22×D71—82no—
129C22×D71—812no—
130Q8×C71—216no—
131Q8×C71—212no—
132D4×C71—248no—
133D4×C71—248no—
134D4×C71—224no—
135D4×C71—212no—
136D4×C71#120116yes—
137C2×C4×C71—248no—
138C2×C4×C71—224no—
139C2×C4×C71—224no—
140C2×C4×C71C22×D4112yes—
141C2×C4×C71C22×D416yes—
142C22⋊D71—424no—
143C22⋊D71—48no—
144C22⋊D71—824no—
145C2×Dic71—212no—
146C2×Dic71—23no—
147C2×Dic71—812no—
148D284—44no—
149C4×D71—44no—
150C22×C71—212no—
151C22×C71—412no—
152C22×C71—41no—
153C22×C71—412no—
154C22×C71—824no—
155C22×C71—816no—
156C22×C71C2×#12018yes—
157C22×C71C23×D413yes—
158C2×D71—26no—
159C2×D71—43no—
160C2×D71—41no—
161C2×D71—412no—
162C2×D71—42no—
163C2×D71—816no—
164C4×C71—212no—
165C4×C71C2×#12014yes—
166Dic71—412no—
167Dic71—41no—
168#60399—711nomaximal, Sylow
169C2×C71—23no—
170C2×C71—28no—
171C2×C71—412no—
172C2×C71C22×#12011yescharacteristic
173D71—42no—
174D71—82no—
175H(7,𝔽2)D7111yescharacteristic, complemented
176#3484—713no—
177#3027—714no—
178#3026—718no—
179#3025—714no—
180C2×B(4,𝔽2)—718no—
181#2883—713no—
182C71#6039911yescharacteristic, complemented, Sylow
183#585—212no—
184#585C2×D7114yescomplemented
185C2×H(5,𝔽2)—212no—
186C2×H(5,𝔽2)C2×D7113yes—
187C2×H(5,𝔽2)C2×D7118yescomplemented
188C23×D4—711no—
189#546—14212no—
190D42—14212no—
191#535—716no—
192#534—7112no—
193#525—716no—
194C2×#120—7112no—
195#518—713no—
196#458—7116no—
197B(4,𝔽2)—7116no—
198B(4,𝔽2)—7132no—
199#410—7112no—
200C2×#99—7124no—
201C25—712no—
2022−1+4—216no—
2032−1+4—212no—
204H(5,𝔽2)—248no—
205H(5,𝔽2)—224no—
206H(5,𝔽2)—212no—
207H(5,𝔽2)—248no—
208H(5,𝔽2)C22×D71116yes—
209C2×#41—248no—
210C2×#41—224no—
211C2×#41—224no—
212C2×#41C22⋊D71112yescomplemented
213C2×#41C22×D7116yes—
214C22×D4—216no—
215C22×D4—224no—
216C22×D4—712no—
217C22×D4—711no—
218C22×D4—14212no—
219C22×D4—14212no—
220C22×D4C22⋊D71112yescomplemented
221C22×D4C22×D71112yes—
222C22×D4C22×D7111yescharacteristic
223C23×C4—711no—
224#127—1426no—
225#124—1426no—
226#123—7112no—
227#123—14212no—
228C22⋊Q8—1426no—
229#121—1426no—
230#121—14224no—
231#121—14212no—
232#120—7112no—
233#120—7124no—
234#120—14212no—
235#120—14224no—
236C4×D4—14212no—
237C4×D4—14212no—
238#117—716no—
239C2×C22⋊C4—7112no—
240C2×C22⋊C4—14212no—
241#99—7148no—
242C24—28no—
243C24—5684no—
244C24—711no—
245C24—712no—
246C24—14212no—
247C24C2×C22⋊D71112yescomplemented
248C24C23×D7112yes—
249#41—296no—
250#41—248no—
251#41—216no—
252#41—248no—
253#41—248no—
254#41—224no—
255C2×Q8—224no—
256C2×Q8—28no—
257C2×Q8C2×C22⋊D7116yescomplemented
258C2×D4—296no—
259C2×D4—224no—
260C2×D4—224no—
261C2×D4—248no—
262C2×D4—248no—
263C2×D4—248no—
264C2×D4—28416no—
265C2×D4—56824no—
266C2×D4—14212no—
267C2×D4—14212no—
268C2×D4—14224no—
269C2×D4—1426no—
270C2×D4C2×C22⋊D71112yes—
271C2×D4C2×C22⋊D71124yescomplemented
272C2×D4C2×C22⋊D7116yescomplemented
273C2×D4C23×D71112yes—
274C22×C4—224no—
275C22×C4—216no—
276C22×C4—711no—
277C22×C4—1426no—
278C22×C4—14212no—
279C22×C4C2×C22⋊D71112yescomplemented
280C22×C4C2×C22⋊D71112yescomplemented
281C22×C4C23×D7111yescharacteristic
282C4⋊C4—14212no—
283C4⋊C4—1426no—
284C22⋊C4—7112no—
285C22⋊C4—14212no—
286C22⋊C4—14212no—
287C22⋊C4—14212no—
288C22⋊C4—14224no—
289C42—1426no—
290C23—216no—
291C23—224no—
292C23—82no—
293C23—812no—
294C23—168no—
295C23—2842no—
296C23—28424no—
297C23—2848no—
298C23—5682no—
299C23—56812no—
300C23—1426no—
301C23—1423no—
302C23C22×C22⋊D71112yes—
303C23C22×C22⋊D71112yes—
304C23C24×D7111yescharacteristic
305Q8—216no—
306Q8—212no—
307D4—248no—
308D4—224no—
309D4—248no—
310D4—212no—
311D4—28424no—
312D4—2848no—
313D4—2844no—
314D4—56824no—
315D4#14556097116yes—
316C2×C4—248no—
317C2×C4—224no—
318C2×C4—224no—
319C2×C4—2844no—
320C2×C4—56812no—
321C2×C4—14212no—
322C2×C4—1423no—
323C2×C4C22×C22⋊D71112yes—
324C2×C4C22×C22⋊D7116yes—
325C22—212no—
326C22—412no—
327C22—41no—
328C22—412no—
329C22—824no—
330C22—816no—
331C22—2843no—
332C22—28412no—
333C22—2841no—
334C22—2842no—
335C22—56816no—
336C22—1426no—
337C22C2×#1455609718yes—
338C22C23×C22⋊D7113yes—
339C4—212no—
340C4—28412no—
341C4—2841no—
342C4C2×#1455609714yes—
343C2—23no—
344C2—28no—
345C2—412no—
346C2—2842no—
347C2—5682no—
348C2C22×#1455609711yescharacteristic
subgroupG/H
centerC2C22×#14556097
commutator, cosocleC23×C71C25
FrattiniC23C24×D71
FittingC71×H(7,𝔽2)C2
socleC2×C71C22×#120
hypercenterH(7,𝔽2)D71
hypocenterC71#60399
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. #22601971C2semidirect
2. #22597742C2semidirect
3. #22597737C2semidirect
4. #22597732C2semidirect
5. C2×#17986186C2semidirect
6. #22596801C2semidirect
7. C71×H(7,𝔽2)C2semidirect
8. C22×D4×D71C22semidirect
11. #17986563C22semidirect
12. #17986558C22semidirect
13. #17986506C22semidirect
14. C2×#14556096C22semidirect
15. #17986474C22semidirect
16. #17986189C22semidirect
17. #17986186C22semidirect
18. #17986186C22semidirect
19. #17985973C22semidirect
20. C2×#14556043C22semidirect
22. #585×C71C22semidirect
25. C2×H(5,𝔽2)×C71C22semidirect
26. C24×D71C23semidirect
27. C22×D284C23semidirect
29. C22×C22⋊D71C23semidirect
31. C22×C4×D71C23semidirect
35. #14556109C23semidirect
42. #14556096C23semidirect
43. #14556096C23semidirect
46. #14556087C23semidirect
47. C2×C22⋊Dic71C23semidirect
49. #14556043C23semidirect
60. C2×#41×C71D4semidirect
64. C22×D4×C71D4semidirect
89. C24×C71C2×D4semidirect
99. C2×Q8×C71C2×D4semidirect
107. C2×D4×C71C2×D4semidirect
108. C2×D4×C71C2×D4semidirect
112. C22×C4×C71C2×D4semidirect
113. C22×C4×C71C2×D4semidirect
175. H(7,𝔽2)D71semidirect
182. C71#60399semidirect
184. #585C2×D71semidirect
187. C2×H(5,𝔽2)C2×D71semidirect
212. C2×#41C22⋊D71semidirect
220. C22×D4C22⋊D71semidirect
247. C24C2×C22⋊D71semidirect
257. C2×Q8C2×C22⋊D71semidirect
271. C2×D4C2×C22⋊D71semidirect
272. C2×D4C2×C22⋊D71semidirect
279. C22×C4C2×C22⋊D71semidirect
280. C22×C4C2×C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
#226019712
#225977422
#225977372
#225977322
C2×#179861862
#225968012
C71×H(7,𝔽2)2
#6039971