Finite Group Databasebeta

← #32160314

Subgroups — #32160314

Order 18176.

2410 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22591521C212yesmaximal, complemented
2#22591517C212yesmaximal, complemented
3#22591452C214yesmaximal, complemented
4#22591269C218yesmaximal, complemented
5#22591107C218yesmaximal, complemented
6#22590932C214yesmaximal, complemented
7#22590738C212yesmaximal, complemented
8C2×#532×C71C211yesmaximal, characteristic
9C2×#14556087C2214yes—
10C22×C4⋊Dic71C2214yes—
11#17985906C2218yes—
12#17985902C2218yes—
13#17985892C22116yes—
14#17985885C22116yes—
15#17985879C22116yes—
16#17985879C2218yes—
17#17985879C2218yes—
18#17985867C2214yes—
19#17985865C2214yes—
20#17985862C2214yes—
21#17985859C2214yes—
22#17985859C2218yes—
23C4×C4⋊Dic71C2214yes—
24C22⋊C4×Dic71C22116yes—
25C4×C22⋊Dic71C2214yes—
26#17985845C2214yes—
27C22×#41×C71C2212yes—
28#532×C71C4116yescomplemented
29C2×C4⋊Q8×C71C2211yescharacteristic
30C2×#127×C71C2211yescharacteristic
31C2×#124×C71C2212yes—
32C2×#121×C71C2214yes—
33C2×C4×D4×C71C2214yes—
34C22×C42×C71C2211yescharacteristic
35C23×Dic71C2318yes—
36#14556087—48no—
37C2×C4⋊Dic71—48no—
38C2×C4⋊Dic71C2318yes—
39C2×C4⋊Dic71C2318yes—
40C2×C22⋊Dic71C2318yes—
41C2×C22⋊Dic71C23116yes—
42C2×C22⋊Dic71C2318yes—
43C2×C4×Dic71C2318yes—
44C2×C4×Dic71C2318yes—
45C71⋊#95C2318yes—
46C71⋊#95C2318yes—
47C71⋊#95C2318yes—
48C71⋊#95C23116yes—
49C71⋊#95C2318yes—
50C2×#41×C71—44no—
51C2×#41×C71—44no—
52C2×#41×C71C2×C4116yescomplemented
53C22×Q8×C71C2312yes—
54C22×D4×C71C2314yes—
55C22×D4×C71C2312yes—
56C22×D4×C71C2314yes—
57C23×C4×C71C2311yescharacteristic
58C23×C4×C71C2314yes—
59C23×C4×C71C2312yes—
60C4⋊Q8×C71C2×C418yescomplemented
61#127×C71C2×C418yes—
62#124×C71C2×C4116yescomplemented
63#121×C71C2×C4132yescomplemented
64C4×D4×C71C2×C4132yescomplemented
65C2×C4⋊C4×C71C2314yes—
66C2×C22⋊C4×C71C2318yes—
67C2×C42×C71C2×C418yescomplemented
68C2×C42×C71C2311yescharacteristic
69C2×C42×C71C2312yes—
70C2×C42×C71C2311yescharacteristic
71C22×Dic71—416no—
72C22×Dic71—48no—
73C22×Dic71C24116yes—
74C4⋊Dic71—216no—
75C4⋊Dic71—48no—
76C22⋊Dic71—416no—
77C22⋊Dic71—48no—
78C22⋊Dic71—48no—
79C4×Dic71—48no—
80C4×Dic71—48no—
81C24×C71C2414yes—
82C24×C71C2411yescharacteristic
83#41×C71—432no—
84C2×Q8×C71—42no—
85C2×Q8×C71—42no—
86C2×Q8×C71C22×C418yes—
87C2×D4×C71—44no—
88C2×D4×C71—44no—
89C2×D4×C71—42no—
90C2×D4×C71—42no—
91C2×D4×C71—44no—
92C2×D4×C71—44no—
93C2×D4×C71C22×C4116yes—
94C2×D4×C71C22×C4116yes—
95C2×D4×C71C22×C418yes—
96C22×C4×C71—24no—
97C22×C4×C71—28no—
98C22×C4×C71—28no—
99C22×C4×C71—44no—
100C22×C4×C71C22×C414yes—
101C22×C4×C71C22×C4116yes—
102C22×C4×C71C22×C418yes—
103C22×C4×C71C2412yes—
104C22×C4×C71C2412yes—
105C22×C4×C71C2411yescharacteristic
106C22×C4×C71C2414yes—
107C22×C4×C71C2411yescharacteristic
108C4⋊C4×C71C22×C4116yes—
109C22⋊C4×C71C22×C4132yes—
110C42×C71C22×C418yes—
111C42×C71C22×C414yes—
112C42×C71C22×C414yes—
113C23×C71—22no—
114C23×C71—24no—
115C23×C71—24no—
116C23×C71—24no—
117C23×C71—28no—
118C23×C71—28no—
119C23×C71C23×C418yes—
120C23×C71C23×C412yes—
121C23×C71C2511yescharacteristic
122Q8×C71—48no—
123D4×C71—416no—
124D4×C71—48no—
125D4×C71—416no—
126C2×C4×C71—22no—
127C2×C4×C71—24no—
128C2×C4×C71—24no—
129C2×C4×C71—24no—
130C2×C4×C71—24no—
131C2×C4×C71—232no—
132C2×C4×C71—28no—
133C2×C4×C71—48no—
134C2×C4×C71C23×C414yes—
135C2×C4×C71C23×C414yes—
136C2×C4×C71C23×C412yes—
137C2×C4×C71C23×C418yes—
138C2×C4×C71C23×C412yes—
139C2×C4×C71H(5,𝔽2)14yes—
140C2×Dic71—216no—
141C2×Dic71—216no—
142C2×Dic71—216no—
143C2×Dic71—816no—
144C22×C71—22no—
145C22×C71—28no—
146C22×C71—24no—
147C22×C71—216no—
148C22×C71—44no—
149C22×C71—44no—
150C22×C71—48no—
151C22×C71—44no—
152C22×C71C24×C411yescharacteristic
153C22×C71C2×H(5,𝔽2)12yes—
154C22×C71#58514yes—
155C4×C71—22no—
156C4×C71—28no—
157C4×C71—24no—
158C4×C71—24no—
159C4×C71#51814yes—
160Dic71—416no—
161#34369—711nomaximal, Sylow
162C2×C71—42no—
163C2×C71—48no—
164C2×C71C2×#51811yescharacteristic
165C2×C71#342812yes—
166C2×C71#355114yes—
167C2×#532D7111yescharacteristic
168#2381—712no—
169#2380—712no—
170#2372—714no—
171#2352—718no—
172#2333—718no—
173#2311—714no—
174#2283—712no—
175C71#3436911yescharacteristic, complemented, Sylow
176C22×#41C2×D7112yes—
177#532Dic71116yescomplemented
178C2×C4⋊Q8C2×D7111yescharacteristic
179C2×#127C2×D7111yescharacteristic
180C2×#124C2×D7112yes—
181C2×#121C2×D7114yes—
182C2×C4×D4C2×D7114yes—
183C2×#117—714no—
184C22×C4⋊C4—714no—
185C22×C42C2×D7111yescharacteristic
186#391—718no—
187#390—718no—
188#388—7116no—
189#387—7116no—
190#386—718no—
191#386—718no—
192#386—7116no—
193#383—714no—
194#382—714no—
195#381—714no—
196#380—714no—
197#380—718no—
198C4×C4⋊C4—714no—
199C4×C22⋊C4—714no—
200C4×C22⋊C4—7116no—
201#376—714no—
202C2×#41—44no—
203C2×#41—44no—
204C2×#41C2×Dic71116yescomplemented
205C22×Q8C22×D7112yes—
206C22×D4C22×D7114yes—
207C22×D4C22×D7114yes—
208C22×D4C22×D7112yes—
209C23×C4—718no—
210C23×C4C22×D7114yes—
211C23×C4C22×D7111yescharacteristic
212C23×C4C22×D7112yes—
213C4⋊Q8C2×Dic7118yescomplemented
214#127C2×Dic7118yes—
215#124C2×Dic71116yescomplemented
216#121C2×Dic71132yescomplemented
217C4×D4C2×Dic71132yescomplemented
218#117—2848no—
219C2×C4⋊C4—2848no—
220C2×C4⋊C4—718no—
221C2×C4⋊C4—718no—
222C2×C4⋊C4C22×D7114yes—
223C2×C22⋊C4—718no—
224C2×C22⋊C4—718no—
225C2×C22⋊C4—7116no—
226C2×C22⋊C4C22×D7118yes—
227C2×C42—718no—
228C2×C42—718no—
229C2×C42C2×Dic7118yescomplemented
230C2×C42C22×D7112yes—
231C2×C42C22×D7111yescharacteristic
232C2×C42C22×D7111yescharacteristic
233#95—718no—
234#95—718no—
235#95—7116no—
236#95—718no—
237#95—718no—
238C24C23×D7111yescharacteristic
239C24C23×D7114yes—
240#41—432no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C22×Dic7118yes—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4C22×Dic71116yes—
251C2×D4C22×Dic71116yes—
252C2×D4C22×Dic7118yes—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—44no—
257C22×C4—2848no—
258C22×C4—28416no—
259C22×C4—7116no—
260C22×C4C22×Dic7114yes—
261C22×C4C22×Dic71116yes—
262C22×C4C22×Dic7118yes—
263C22×C4C23×D7112yes—
264C22×C4C23×D7111yescharacteristic
265C22×C4C23×D7112yes—
266C22×C4C23×D7114yes—
267C22×C4C23×D7111yescharacteristic
268C4⋊C4—2848no—
269C4⋊C4—14216no—
270C4⋊C4C22×Dic71116yes—
271C22⋊C4—28416no—
272C22⋊C4—2848no—
273C22⋊C4—2848no—
274C22⋊C4C22×Dic71132yes—
275C42—2848no—
276C42—2848no—
277C42C22×Dic7118yes—
278C42C22×Dic7114yes—
279C42C22×Dic7114yes—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—28no—
285C23—28no—
286C23C23×Dic7118yes—
287C23C23×Dic7112yes—
288C23C24×D7111yescharacteristic
289Q8—48no—
290D4—48no—
291D4—416no—
292D4—416no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—28no—
299C2×C4—232no—
300C2×C4—48no—
301C2×C4—56816no—
302C2×C4—14216no—
303C2×C4—14216no—
304C2×C4—14216no—
305C2×C4C23×Dic7114yes—
306C2×C4C23×Dic7114yes—
307C2×C4C23×Dic7112yes—
308C2×C4C23×Dic7118yes—
309C2×C4C23×Dic7112yes—
310C2×C4#1455617314yes—
311C22—22no—
312C22—28no—
313C22—24no—
314C22—216no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—44no—
319C22C24×Dic7111yescharacteristic
320C22C2×#1455617312yes—
321C22#1798683514yes—
322C4—22no—
323C4—28no—
324C4—24no—
325C4—24no—
326C4—28416no—
327C4#1798647414yes—
328C2—42no—
329C2—48no—
330C2C2×#1798647411yescharacteristic
331C2#2260154312yes—
332C2#2260276714yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
Fitting, prime radicalC2×#532×C71C2
socle, cosocleC23×C71C25
hypercenterC2×#532D71
hypocenterC71#34369
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. #22591521C2semidirect
2. #22591517C2semidirect
3. #22591452C2semidirect
4. #22591269C2semidirect
5. #22591107C2semidirect
6. #22590932C2semidirect
7. #22590738C2semidirect
28. #532×C71C4semidirect
52. C2×#41×C71C2×C4semidirect
60. C4⋊Q8×C71C2×C4semidirect
62. #124×C71C2×C4semidirect
63. #121×C71C2×C4semidirect
64. C4×D4×C71C2×C4semidirect
67. C2×C42×C71C2×C4semidirect
175. C71#34369semidirect
177. #532Dic71semidirect
204. C2×#41C2×Dic71semidirect
213. C4⋊Q8C2×Dic71semidirect
215. #124C2×Dic71semidirect
216. #121C2×Dic71semidirect
217. C4×D4C2×Dic71semidirect
229. C2×C42C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225915212
#225915172
#225914522
#225912692
#225911072
#225909322
#225907382
C2×#532×C712
#3436971