Finite Group Databasebeta

← #33086556

Subgroups — #33086556

Order 18688.

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#22612540C212yesmaximal
2#22610095C211yesmaximal, characteristic, complemented
3#22610091C211yesmaximal, characteristic, complemented
4#22610065C212yesmaximal, complemented
5#22609964C212yesmaximal
6#22609960C212yesmaximal
7#22609828C212yesmaximal, complemented
8C4×#14556525C212yesmaximal, complemented
9C2×#405×C73C211yesmaximal, characteristic
10C2×#14556525C2212yes—
11C2×#14556525C2212yes—
12C2×#14556525C2212yes—
13C2×#14556523C2212yes—
14#17988480—24no—
15#17988473—24no—
16#17988457C2212yes—
17#17988453C2212yes—
18#17988447C2214yes—
19C4×C4⋊Dic73C2214yes—
20#17988350C2214yes—
21#17988350C2214yes—
22C22×M16×C73C2211yescharacteristic
23C23×C8×C73C2211yescharacteristic
24C2×#117×C73C2211yescharacteristic
25#405×C73C418yescomplemented
26C2×#97×C73C2211yescharacteristic
27C2×#97×C73C2211yescharacteristic
28C2×C4×C8×C73C2211yescharacteristic
29C2×C4×C8×C73C2211yescharacteristic
30C2×C4⋊Dic73—24no—
31C2×C4⋊Dic73C2314yes—
32C2×C4⋊Dic73C2314yes—
33C2×C22⋊Dic73—24no—
34C2×C4×Dic73—24no—
35#14556525—24no—
36#14556525—24no—
37#14556525—24no—
38#14556523—24no—
39#14556498—24no—
40#14556498—24no—
41#14556498—24no—
42C23×C4×C73C2311yescharacteristic
43C2×M16×C73—22no—
44C2×M16×C73—22no—
45C2×M16×C73C2×C414yes—
46C22×C8×C73—22no—
47C22×C8×C73—22no—
48C22×C8×C73C2×C414yes—
49C22×C8×C73C2311yescharacteristic
50C22×C8×C73C2311yescharacteristic
51C22×C8×C73C2311yescharacteristic
52C22×C8×C73C2311yescharacteristic
53#117×C73C2×C414yes—
54#117×C73Q814yes—
55C2×C4⋊C4×C73D412yes—
56C2×C22⋊C4×C73D412yes—
57C2×C42×C73C2311yescharacteristic
58C2×C42×C73C2311yescharacteristic
59#97×C73C2×C414yes—
60#97×C73C2×C414yes—
61C4×C8×C73C2×C414yes—
62C4×C8×C73C2×C414yes—
63C22×Dic73—24no—
64C22×Dic73—24no—
65C4⋊Dic73—24no—
66C4⋊Dic73—24no—
67C4⋊Dic73—24no—
68C4⋊Dic73—24no—
69C4⋊Dic73—84no—
70C22⋊Dic73—84no—
71C4×Dic73—84no—
72C24×C73C2×D411yescharacteristic
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73C22×C412yes—
78C22×C4×C73C2×D412yes—
79C22×C4×C73C2×D412yes—
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×Q812yes—
82C22×C4×C73C2411yescharacteristic
83M16×C73—28no—
84C2×C8×C73—21no—
85C2×C8×C73—21no—
86C2×C8×C73—21no—
87C2×C8×C73—21no—
88C2×C8×C73—28no—
89C2×C8×C73C22×C412yes—
90C2×C8×C73C22×C412yes—
91C2×C8×C73C22×C412yes—
92C2×C8×C73C22×C412yes—
93C2×C8×C73#4112yes—
94C2×C8×C73#4112yes—
95C2×C8×C73#4112yes—
96C2×C8×C73#4112yes—
97C4⋊C4×C73C4⋊C418yes—
98C22⋊C4×C73C4⋊C418yes—
99C42×C73C22×C412yes—
100C42×C73C22×C412yes—
101C42×C73C2×Q812yes—
102C42×C73C2×Q812yes—
103C23×C73—21no—
104C23×C73—21no—
105C23×C73—22no—
106C23×C73—22no—
107C23×C73C2×C4⋊C412yes—
108C23×C73C22×D411yescharacteristic
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—22no—
117C2×C4×C73—22no—
118C2×C4×C73—22no—
119C2×C4×C73C2×C4⋊C414yes—
120C2×C4×C73C2×C4⋊C414yes—
121C2×C4×C73C2×C4⋊C412yes—
122C2×C4×C73C23×C411yescharacteristic
123C2×C4×C73C22×Q811yescharacteristic
124C2×C4×C73C2×#4111yescharacteristic
125C2×C4×C73C2×#4111yescharacteristic
126C2×C4×C73C2×#4111yescharacteristic
127C2×C4×C73C2×#4111yescharacteristic
128C2×Dic73—44no—
129C2×Dic73—44no—
130C2×Dic73—44no—
131C2×Dic73—44no—
132C2×Dic73—44no—
133C2×Dic73—44no—
134C8×C73—22no—
135C8×C73—22no—
136C8×C73#11714yes—
137C8×C73#11714yes—
138C22×C73—24no—
139C22×C73—22no—
140C22×C73—42no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73C22×C4⋊C411yescharacteristic
144C22×C73#53211yescharacteristic
145C22×C73#53211yescharacteristic
146C22×C73#57612yes—
147C22×C73#57812yes—
148C4×C73—24no—
149C4×C73—24no—
150C4×C73—22no—
151C4×C73C2×#11711yescharacteristic
152C4×C73C2×#11711yescharacteristic
153C4×C73#53311yescharacteristic
154C4×C73#53311yescharacteristic
155Dic73—84no—
156Dic73—84no—
157#20821—731nomaximal, Sylow
158C2×C73—42no—
159C2×C73#228511yescharacteristic
160C2×C73#291312yes—
161C2×C73#317512yes—
162C2×C73#317611yescharacteristic
163C2×C73#317711yescharacteristic
164C2×#405D7311yescharacteristic
165#2314—732no—
166#1846—731no—
167#1845—731no—
168#1836—732no—
169#1809—732no—
170#1808—732no—
171#1776—732no—
172C4×#107—732no—
173C73#2082111yescharacteristic, complemented, Sylow
174C22×M16C2×D7311yescharacteristic
175C23×C8C2×D7311yescharacteristic
176C2×#117C2×D7311yescharacteristic
177C2×#107—732no—
178C2×#107—732no—
179C2×#107—732no—
180C2×#106—732no—
181#405Dic7318yescomplemented
182C2×#97C2×D7311yescharacteristic
183C2×#97C2×D7311yescharacteristic
184C2×C4×C8C2×D7311yescharacteristic
185C2×C4×C8C2×D7311yescharacteristic
186#388—1464no—
187#387—1464no—
188#384—732no—
189#382—732no—
190#380—734no—
191C4×C4⋊C4—734no—
192#340—734no—
193#340—734no—
194C23×C4C22×D7311yescharacteristic
195C2×M16—22no—
196C2×M16—22no—
197C2×M16C2×Dic7314yes—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C2×Dic7314yes—
201C22×C8C22×D7311yescharacteristic
202C22×C8C22×D7311yescharacteristic
203C22×C8C22×D7311yescharacteristic
204C22×C8C22×D7311yescharacteristic
205#117Dic14614yes—
206#117C2×Dic7314yes—
207C2×C4⋊C4—734no—
208C2×C4⋊C4—734no—
209C2×C4⋊C4—1464no—
210C2×C4⋊C4D29212yes—
211C2×C22⋊C4—1464no—
212C2×C22⋊C4D29212yes—
213C2×C42—1464no—
214C2×C42C22×D7311yescharacteristic
215C2×C42C22×D7311yescharacteristic
216#107—1464no—
217#107—1464no—
218#107—1464no—
219#106—1464no—
220#97C2×Dic7314yes—
221#97C2×Dic7314yes—
222C4×C8C2×Dic7314yes—
223C4×C8C2×Dic7314yes—
224#95—1464no—
225#95—1464no—
226#95—1464no—
227C24C2×D29211yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1464no—
233C22×C4—1464no—
234C22×C4C22×Dic7312yes—
235C22×C4C2×D29211yescharacteristic
236C22×C4C2×D29212yes—
237C22×C4C2×D29212yes—
238C22×C4C2×Dic14612yes—
239C22×C4C23×D7311yescharacteristic
240M16—28no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—28no—
246C2×C8C22×Dic7312yes—
247C2×C8C22×Dic7312yes—
248C2×C8C22×Dic7312yes—
249C2×C8C22×Dic7312yes—
250C2×C8#1192402812yes—
251C2×C8#1192402812yes—
252C2×C8#1192402812yes—
253C2×C8#1192402812yes—
254C4⋊C4—5844no—
255C4⋊C4—1464no—
256C4⋊C4—1464no—
257C4⋊C4—1464no—
258C4⋊C4—1464no—
259C4⋊C4C4⋊Dic7318yes—
260C22⋊C4—5844no—
261C22⋊C4C4⋊Dic7318yes—
262C42—5844no—
263C42C22×Dic7312yes—
264C42C22×Dic7312yes—
265C42C2×Dic14612yes—
266C42C2×Dic14612yes—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23C2×C4⋊Dic7312yes—
272C23C22×D29211yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—2924no—
284C2×C4—2924no—
285C2×C4—2924no—
286C2×C4—2924no—
287C2×C4—2924no—
288C2×C4—2924no—
289C2×C4C2×C4⋊Dic7314yes—
290C2×C4C2×C4⋊Dic7314yes—
291C2×C4C2×C4⋊Dic7312yes—
292C2×C4C23×Dic7311yescharacteristic
293C2×C4C22×Dic14611yescharacteristic
294C2×C4C2×#1192402811yescharacteristic
295C2×C4C2×#1192402811yescharacteristic
296C2×C4C2×#1192402811yescharacteristic
297C2×C4C2×#1192402811yescharacteristic
298C8—22no—
299C8—22no—
300C8#1455655014yes—
301C8#1455655014yes—
302C22—22no—
303C22—24no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22C22×C4⋊Dic7311yescharacteristic
308C22#1798913711yescharacteristic
309C22#1798913711yescharacteristic
310C22#1798939212yes—
311C22#1798940412yes—
312C4—22no—
313C4—24no—
314C4—24no—
315C4—5844no—
316C4—5844no—
317C4C2×#1455655011yescharacteristic
318C4C2×#1455655011yescharacteristic
319C4#1798914411yescharacteristic
320C4#1798914411yescharacteristic
321C2—42no—
322C2#2261233811yescharacteristic
323C2#2261853112yes—
324C2#2262057012yes—
325C2#2262058111yescharacteristic
326C2#2262059011yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#405×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC2×#405D73
hypocenterC73#20821
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #22610095C2semidirect
3. #22610091C2semidirect
4. #22610065C2semidirect
7. #22609828C2semidirect
8. C4×#14556525C2semidirect
25. #405×C73C4semidirect
173. C73#20821semidirect
181. #405Dic73semidirect

Maximal subgroups

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

subgroupindex
#226125402
#226100952
#226100912
#226100652
#226099642
#226099602
#226098282
C4×#145565252
C2×#405×C732
#2082173