Finite Group Databasebeta

← #31998024

Subgroups — #31998024

Order 18176.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590949C212yesmaximal
2#22588504C211yesmaximal, characteristic, complemented
3#22588500C211yesmaximal, characteristic, complemented
4#22588474C212yesmaximal, complemented
5#22588373C212yesmaximal
6#22588369C212yesmaximal
7#22588237C212yesmaximal, complemented
8C4×#14556062C212yesmaximal, complemented
9C2×#405×C71C211yesmaximal, characteristic
10C2×#14556062C2212yes—
11C2×#14556062C2212yes—
12C2×#14556062C2212yes—
13C2×#14556060C2212yes—
14#17985892—24no—
15#17985885—24no—
16#17985869C2212yes—
17#17985865C2212yes—
18#17985859C2214yes—
19C4×C4⋊Dic71C2214yes—
20#17985762C2214yes—
21#17985762C2214yes—
22C22×M16×C71C2211yescharacteristic
23C23×C8×C71C2211yescharacteristic
24C2×#117×C71C2211yescharacteristic
25#405×C71C418yescomplemented
26C2×#97×C71C2211yescharacteristic
27C2×#97×C71C2211yescharacteristic
28C2×C4×C8×C71C2211yescharacteristic
29C2×C4×C8×C71C2211yescharacteristic
30C2×C4⋊Dic71—24no—
31C2×C4⋊Dic71C2314yes—
32C2×C4⋊Dic71C2314yes—
33C2×C22⋊Dic71—24no—
34C2×C4×Dic71—24no—
35#14556062—24no—
36#14556062—24no—
37#14556062—24no—
38#14556060—24no—
39C71⋊#95—24no—
40C71⋊#95—24no—
41C71⋊#95—24no—
42C23×C4×C71C2311yescharacteristic
43C2×M16×C71—22no—
44C2×M16×C71—22no—
45C2×M16×C71C2×C414yes—
46C22×C8×C71—22no—
47C22×C8×C71—22no—
48C22×C8×C71C2×C414yes—
49C22×C8×C71C2311yescharacteristic
50C22×C8×C71C2311yescharacteristic
51C22×C8×C71C2311yescharacteristic
52C22×C8×C71C2311yescharacteristic
53#117×C71C2×C414yes—
54#117×C71Q814yes—
55C2×C4⋊C4×C71D412yes—
56C2×C22⋊C4×C71D412yes—
57C2×C42×C71C2311yescharacteristic
58C2×C42×C71C2311yescharacteristic
59#97×C71C2×C414yes—
60#97×C71C2×C414yes—
61C4×C8×C71C2×C414yes—
62C4×C8×C71C2×C414yes—
63C22×Dic71—24no—
64C22×Dic71—24no—
65C4⋊Dic71—24no—
66C4⋊Dic71—24no—
67C4⋊Dic71—24no—
68C4⋊Dic71—24no—
69C4⋊Dic71—84no—
70C22⋊Dic71—84no—
71C4×Dic71—84no—
72C24×C71C2×D411yescharacteristic
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71C22×C412yes—
78C22×C4×C71C2×D412yes—
79C22×C4×C71C2×D412yes—
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×Q812yes—
82C22×C4×C71C2411yescharacteristic
83M16×C71—28no—
84C2×C8×C71—21no—
85C2×C8×C71—21no—
86C2×C8×C71—21no—
87C2×C8×C71—21no—
88C2×C8×C71—28no—
89C2×C8×C71C22×C412yes—
90C2×C8×C71C22×C412yes—
91C2×C8×C71C22×C412yes—
92C2×C8×C71C22×C412yes—
93C2×C8×C71#4112yes—
94C2×C8×C71#4112yes—
95C2×C8×C71#4112yes—
96C2×C8×C71#4112yes—
97C4⋊C4×C71C4⋊C418yes—
98C22⋊C4×C71C4⋊C418yes—
99C42×C71C22×C412yes—
100C42×C71C22×C412yes—
101C42×C71C2×Q812yes—
102C42×C71C2×Q812yes—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—22no—
106C23×C71—22no—
107C23×C71C2×C4⋊C412yes—
108C23×C71C22×D411yescharacteristic
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—22no—
118C2×C4×C71—22no—
119C2×C4×C71C2×C4⋊C414yes—
120C2×C4×C71C2×C4⋊C414yes—
121C2×C4×C71C2×C4⋊C412yes—
122C2×C4×C71C23×C411yescharacteristic
123C2×C4×C71C22×Q811yescharacteristic
124C2×C4×C71C2×#4111yescharacteristic
125C2×C4×C71C2×#4111yescharacteristic
126C2×C4×C71C2×#4111yescharacteristic
127C2×C4×C71C2×#4111yescharacteristic
128C2×Dic71—44no—
129C2×Dic71—44no—
130C2×Dic71—44no—
131C2×Dic71—44no—
132C2×Dic71—44no—
133C2×Dic71—44no—
134C8×C71—22no—
135C8×C71—22no—
136C8×C71#11714yes—
137C8×C71#11714yes—
138C22×C71—24no—
139C22×C71—22no—
140C22×C71—42no—
141C22×C71—41no—
142C22×C71—41no—
143C22×C71C22×C4⋊C411yescharacteristic
144C22×C71#53211yescharacteristic
145C22×C71#53211yescharacteristic
146C22×C71#57612yes—
147C22×C71#57812yes—
148C4×C71—24no—
149C4×C71—24no—
150C4×C71—22no—
151C4×C71C2×#11711yescharacteristic
152C4×C71C2×#11711yescharacteristic
153C4×C71#53311yescharacteristic
154C4×C71#53311yescharacteristic
155Dic71—84no—
156Dic71—84no—
157#20821—711nomaximal, Sylow
158C2×C71—42no—
159C2×C71#228511yescharacteristic
160C2×C71#291312yes—
161C2×C71#317512yes—
162C2×C71#317611yescharacteristic
163C2×C71#317711yescharacteristic
164C2×#405D7111yescharacteristic
165#2314—712no—
166#1846—711no—
167#1845—711no—
168#1836—712no—
169#1809—712no—
170#1808—712no—
171#1776—712no—
172C4×#107—712no—
173C71#2082111yescharacteristic, complemented, Sylow
174C22×M16C2×D7111yescharacteristic
175C23×C8C2×D7111yescharacteristic
176C2×#117C2×D7111yescharacteristic
177C2×#107—712no—
178C2×#107—712no—
179C2×#107—712no—
180C2×#106—712no—
181#405Dic7118yescomplemented
182C2×#97C2×D7111yescharacteristic
183C2×#97C2×D7111yescharacteristic
184C2×C4×C8C2×D7111yescharacteristic
185C2×C4×C8C2×D7111yescharacteristic
186#388—1424no—
187#387—1424no—
188#384—712no—
189#382—712no—
190#380—714no—
191C4×C4⋊C4—714no—
192#340—714no—
193#340—714no—
194C23×C4C22×D7111yescharacteristic
195C2×M16—22no—
196C2×M16—22no—
197C2×M16C2×Dic7114yes—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C2×Dic7114yes—
201C22×C8C22×D7111yescharacteristic
202C22×C8C22×D7111yescharacteristic
203C22×C8C22×D7111yescharacteristic
204C22×C8C22×D7111yescharacteristic
205#117Dic14214yes—
206#117C2×Dic7114yes—
207C2×C4⋊C4—714no—
208C2×C4⋊C4—714no—
209C2×C4⋊C4—1424no—
210C2×C4⋊C4D28412yes—
211C2×C22⋊C4—1424no—
212C2×C22⋊C4D28412yes—
213C2×C42—1424no—
214C2×C42C22×D7111yescharacteristic
215C2×C42C22×D7111yescharacteristic
216#107—1424no—
217#107—1424no—
218#107—1424no—
219#106—1424no—
220#97C2×Dic7114yes—
221#97C2×Dic7114yes—
222C4×C8C2×Dic7114yes—
223C4×C8C2×Dic7114yes—
224#95—1424no—
225#95—1424no—
226#95—1424no—
227C24C2×D28411yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1424no—
233C22×C4—1424no—
234C22×C4C22×Dic7112yes—
235C22×C4C2×D28411yescharacteristic
236C22×C4C2×D28412yes—
237C22×C4C2×D28412yes—
238C22×C4C2×Dic14212yes—
239C22×C4C23×D7111yescharacteristic
240M16—28no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—28no—
246C2×C8C22×Dic7112yes—
247C2×C8C22×Dic7112yes—
248C2×C8C22×Dic7112yes—
249C2×C8C22×Dic7112yes—
250C2×C8#1176586512yes—
251C2×C8#1176586512yes—
252C2×C8#1176586512yes—
253C2×C8#1176586512yes—
254C4⋊C4—5684no—
255C4⋊C4—1424no—
256C4⋊C4—1424no—
257C4⋊C4—1424no—
258C4⋊C4—1424no—
259C4⋊C4C4⋊Dic7118yes—
260C22⋊C4—5684no—
261C22⋊C4C4⋊Dic7118yes—
262C42—5684no—
263C42C22×Dic7112yes—
264C42C22×Dic7112yes—
265C42C2×Dic14212yes—
266C42C2×Dic14212yes—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23C2×C4⋊Dic7112yes—
272C23C22×D28411yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—2844no—
284C2×C4—2844no—
285C2×C4—2844no—
286C2×C4—2844no—
287C2×C4—2844no—
288C2×C4—2844no—
289C2×C4C2×C4⋊Dic7114yes—
290C2×C4C2×C4⋊Dic7114yes—
291C2×C4C2×C4⋊Dic7112yes—
292C2×C4C23×Dic7111yescharacteristic
293C2×C4C22×Dic14211yescharacteristic
294C2×C4C2×#1176586511yescharacteristic
295C2×C4C2×#1176586511yescharacteristic
296C2×C4C2×#1176586511yescharacteristic
297C2×C4C2×#1176586511yescharacteristic
298C8—22no—
299C8—22no—
300C8#1455608714yes—
301C8#1455608714yes—
302C22—22no—
303C22—24no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22C22×C4⋊Dic7111yescharacteristic
308C22#1798654911yescharacteristic
309C22#1798654911yescharacteristic
310C22#1798680412yes—
311C22#1798681612yes—
312C4—22no—
313C4—24no—
314C4—24no—
315C4—5684no—
316C4—5684no—
317C4C2×#1455608711yescharacteristic
318C4C2×#1455608711yescharacteristic
319C4#1798655611yescharacteristic
320C4#1798655611yescharacteristic
321C2—42no—
322C2#2259074711yescharacteristic
323C2#2259694012yes—
324C2#2259897912yes—
325C2#2259899011yescharacteristic
326C2#2259899911yescharacteristic
subgroupG/H
centerC23C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#405×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenterC2×#405D71
hypocenterC71#20821
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #22588504C2semidirect
3. #22588500C2semidirect
4. #22588474C2semidirect
7. #22588237C2semidirect
8. C4×#14556062C2semidirect
25. #405×C71C4semidirect
173. C71#20821semidirect
181. #405Dic71semidirect

Maximal subgroups

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

subgroupindex
#225909492
#225885042
#225885002
#225884742
#225883732
#225883692
#225882372
C4×#145560622
C2×#405×C712
#2082171