Finite Group Databasebeta

← #33084221

Subgroups — #33084221

Order 18688.

906 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
1C2×#17988648C211yesmaximal, characteristic
2#22612429C211yesmaximal, characteristic
3#22609803C212yesmaximal, complemented
4#22609798C212yesmaximal, complemented
5#22609712C212yesmaximal
6#22609707C212yesmaximal
7#22609684C211yesmaximal, characteristic, complemented
8#22609679C211yesmaximal, characteristic, complemented
9#22609617C211yesmaximal, characteristic, complemented
10C4×#14556501C211yesmaximal, characteristic, complemented
11C2×#405×C73C211yesmaximal, characteristic
12#17988648C418yescomplemented
13C2×C4⋊#10596831C2212yes—
14C2×C22⋊#10596831C2212yes—
15C2×#14556502C2211yescharacteristic
16C2×#14556501C2212yes—
17C2×#14556501C2212yes—
18C2×#14556501C2212yes—
19C2×C4×#10596831C2211yescharacteristic
20C2×C8×Dic73C2212yes—
21#17988480C414yes—
22#17988473C414yes—
23C4×C4⋊Dic73C2212yes—
24C4×C22⋊Dic73C2212yes—
25#17988433C2211yescharacteristic
26C42×Dic73C2211yescharacteristic
27#17988341C2212yes—
28#17988341C2212yes—
29#17988341C2212yes—
30#17988341C2212yes—
31C22×M16×C73C2211yescharacteristic
32C23×C8×C73C2211yescharacteristic
33C2×#117×C73C2211yescharacteristic
34#405×C73C418yescomplemented
35C2×#97×C73C2211yescharacteristic
36C2×#97×C73C2211yescharacteristic
37C2×C4×C8×C73C2211yescharacteristic
38C2×C4×C8×C73C2211yescharacteristic
39C22×#10596831C2312yes—
40C22×#10596831C2312yes—
41C2×C4⋊Dic73C2×C414yes—
42C2×C22⋊Dic73C2×C414yes—
43C2×C4×Dic73C2×C414yes—
44C2×C4×Dic73C2312yes—
45C2×C4×Dic73C2312yes—
46C4⋊#10596831C2×C418yes—
47C22⋊#10596831C2×C418yes—
48#14556502C2×C414yes—
49#14556501—24no—
50#14556501—24no—
51#14556501—24no—
52C4×#10596831C2×C414yes—
53C8×Dic73—24no—
54#14556498C2×C414yes—
55#14556498C2×C414yes—
56#14556498C2×C414yes—
57C23×C4×C73C2311yescharacteristic
58C2×M16×C73—24no—
59C2×M16×C73C2×C414yes—
60C22×C8×C73—24no—
61C22×C8×C73C2×C414yes—
62C22×C8×C73C2311yescharacteristic
63C22×C8×C73C2311yescharacteristic
64C22×C8×C73C2311yescharacteristic
65C22×C8×C73C2311yescharacteristic
66#117×C73C2×C414yes—
67#117×C73C2×C414yes—
68C2×C4⋊C4×C73C2×C412yes—
69C2×C22⋊C4×C73C2×C412yes—
70C2×C42×C73C2311yescharacteristic
71C2×C42×C73C2311yescharacteristic
72#97×C73C2×C414yes—
73#97×C73C2×C414yes—
74C4×C8×C73C2×C414yes—
75C4×C8×C73C2×C414yes—
76C22×Dic73C22×C414yes—
77C22×Dic73C22×C414yes—
78C2×#10596831—22no—
79C2×#10596831—22no—
80C2×#10596831—22no—
81C2×#10596831—22no—
82C2×#10596831C22×C414yes—
83C2×#10596831C22×C414yes—
84C4⋊Dic73—44no—
85C22⋊Dic73—44no—
86C4×Dic73—22no—
87C4×Dic73—22no—
88C4×Dic73—22no—
89C4×Dic73—22no—
90C4×Dic73—44no—
91C24×C73C22×C411yescharacteristic
92C22×C4×C73—22no—
93C22×C4×C73—22no—
94C22×C4×C73C22×C412yes—
95C22×C4×C73C22×C412yes—
96C22×C4×C73C22×C411yescharacteristic
97C22×C4×C73C22×C412yes—
98C22×C4×C73C22×C412yes—
99C22×C4×C73C2411yescharacteristic
100M16×C73—28no—
101C2×C8×C73—22no—
102C2×C8×C73—22no—
103C2×C8×C73—28no—
104C2×C8×C73C22×C412yes—
105C2×C8×C73C22×C412yes—
106C2×C8×C73C22×C412yes—
107C2×C8×C73C22×C412yes—
108C2×C8×C73#4114yes—
109C2×C8×C73#4114yes—
110C4⋊C4×C73C4218yes—
111C22⋊C4×C73C4218yes—
112C42×C73C22×C412yes—
113C42×C73C22×C412yes—
114C42×C73C22×C412yes—
115C42×C73C22×C412yes—
116C23×C73—22no—
117C23×C73—24no—
118C23×C73C2×C4212yes—
119C23×C73C23×C411yescharacteristic
120C2×C4×C73—22no—
121C2×C4×C73—24no—
122C2×C4×C73—24no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—24no—
126C2×C4×C73C2×C4214yes—
127C2×C4×C73C2×C4214yes—
128C2×C4×C73C2×C4212yes—
129C2×C4×C73C23×C411yescharacteristic
130C2×C4×C73C23×C411yescharacteristic
131C2×C4×C73C2×#4112yes—
132C2×C4×C73C2×#4112yes—
133C2×Dic73—24no—
134C2×Dic73—24no—
135C2×Dic73—24no—
136C2×Dic73—24no—
137C2×Dic73—24no—
138C2×Dic73—24no—
139C8×C73—22no—
140C8×C73—22no—
141C8×C73#11714yes—
142C8×C73#11714yes—
143#10596831—24no—
144#10596831—24no—
145C22×C73—24no—
146C22×C73—22no—
147C22×C73—44no—
148C22×C73C22×C4211yescharacteristic
149C22×C73C4×#4112yes—
150C22×C73#56814yes—
151C4×C73—24no—
152C4×C73—24no—
153C4×C73—22no—
154C4×C73C2×#11711yescharacteristic
155C4×C73C2×#11711yescharacteristic
156C4×C73C4×#4112yes—
157Dic73—44no—
158Dic73—44no—
159#20475—731nomaximal, Sylow
160C2×C73—42no—
161C2×C73C4×#11711yescharacteristic
162C2×C73#287512yes—
163C2×C73#296714yes—
164C2×#433—731no—
165C2×#405D7311yescharacteristic
166#2298—731no—
167#1771—732no—
168#1770—732no—
169#1753—732no—
170#1752—732no—
171#1746—731no—
172#1745—731no—
173#1726—731no—
174C4×#97—731no—
175C73#2047511yescharacteristic, complemented, Sylow
176C22×M16C2×D7311yescharacteristic
177C23×C8C2×D7311yescharacteristic
178C2×#117C2×D7311yescharacteristic
179#433—738no—
180C2×C4⋊C8—732no—
181C2×C22⋊C8—732no—
182#405Dic7318yescomplemented
183C2×#97—731no—
184C2×#97—732no—
185C2×#97—732no—
186C2×#97—732no—
187C2×#97C2×D7311yescharacteristic
188C2×#97C2×D7311yescharacteristic
189C2×C4×C8—731no—
190C2×C4×C8—732no—
191C2×C4×C8C2×D7311yescharacteristic
192C2×C4×C8C2×D7311yescharacteristic
193#388—734no—
194#387—734no—
195C4×C4⋊C4—732no—
196C4×C22⋊C4—732no—
197#376—731no—
198C43—731no—
199#336—732no—
200#336—732no—
201#336—732no—
202#336—732no—
203C23×C4C22×D7311yescharacteristic
204C2×M16—24no—
205C2×M16C2×Dic7314yes—
206C22×C8—24no—
207C22×C8—732no—
208C22×C8—732no—
209C22×C8C2×Dic7314yes—
210C22×C8C22×D7311yescharacteristic
211C22×C8C22×D7311yescharacteristic
212C22×C8C22×D7311yescharacteristic
213C22×C8C22×D7311yescharacteristic
214#117C4×D7314yes—
215#117C2×Dic7314yes—
216C2×C4⋊C4—734no—
217C2×C4⋊C4C4×D7312yes—
218C2×C22⋊C4—734no—
219C2×C22⋊C4C4×D7312yes—
220C2×C42—734no—
221C2×C42—732no—
222C2×C42—732no—
223C2×C42C22×D7311yescharacteristic
224C2×C42C22×D7311yescharacteristic
225C4⋊C8—738no—
226C22⋊C8—738no—
227#97—734no—
228#97—1464no—
229#97—1464no—
230#97—1464no—
231#97C2×Dic7314yes—
232#97C2×Dic7314yes—
233C4×C8—734no—
234C4×C8—1464no—
235C4×C8C2×Dic7314yes—
236C4×C8C2×Dic7314yes—
237#95—734no—
238#95—734no—
239#95—734no—
240C24C2×C4×D7311yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—734no—
244C22×C4—734no—
245C22×C4C2×C4×D7311yescharacteristic
246C22×C4C2×C4×D7312yes—
247C22×C4C2×C4×D7312yes—
248C22×C4C2×C4×D7312yes—
249C22×C4C22×Dic7312yes—
250C22×C4C23×D7311yescharacteristic
251M16—28no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—28no—
255C2×C8—734no—
256C2×C8—734no—
257C2×C8—1462no—
258C2×C8—1462no—
259C2×C8—1462no—
260C2×C8—1462no—
261C2×C8C22×Dic7312yes—
262C2×C8C22×Dic7312yes—
263C2×C8C22×Dic7312yes—
264C2×C8C22×Dic7312yes—
265C2×C8#1192402814yes—
266C2×C8#1192402814yes—
267C4⋊C4—2924no—
268C4⋊C4C4×Dic7318yes—
269C22⋊C4—2924no—
270C22⋊C4C4×Dic7318yes—
271C42—2924no—
272C42—1462no—
273C42—1462no—
274C42—1462no—
275C42—1462no—
276C42C2×C4×D7312yes—
277C42C2×C4×D7312yes—
278C42C22×Dic7312yes—
279C42C22×Dic7312yes—
280C23—22no—
281C23—24no—
282C23C2×C4×Dic7312yes—
283C23C22×C4×D7311yescharacteristic
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—1464no—
291C2×C4—1464no—
292C2×C4—1464no—
293C2×C4—1464no—
294C2×C4—1464no—
295C2×C4—1464no—
296C2×C4C2×C4×Dic7314yes—
297C2×C4C2×C4×Dic7314yes—
298C2×C4C2×C4×Dic7312yes—
299C2×C4C22×C4×D7311yescharacteristic
300C2×C4C23×Dic7311yescharacteristic
301C2×C4C2×#1192402812yes—
302C2×C4C2×#1192402812yes—
303C8—22no—
304C8—22no—
305C8—1464no—
306C8—1464no—
307C8#1455655014yes—
308C8#1455655014yes—
309C22—22no—
310C22—24no—
311C22—44no—
312C22C22×C4×Dic7311yescharacteristic
313C22C4×#1192402812yes—
314C22#1798935214yes—
315C4—22no—
316C4—24no—
317C4—24no—
318C4—2924no—
319C4—2924no—
320C4C2×#1455655011yescharacteristic
321C4C2×#1455655011yescharacteristic
322C4C4×#1192402812yes—
323C2—42no—
324C2C4×#1455655011yescharacteristic
325C2#2261835712yes—
326C2#2261891214yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C22×C42
FittingC2×#405×C73C2
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenterC2×#405D73
hypocenterC73#20475
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
3. #22609803C2semidirect
4. #22609798C2semidirect
7. #22609684C2semidirect
8. #22609679C2semidirect
9. #22609617C2semidirect
10. C4×#14556501C2semidirect
12. #17988648C4semidirect
34. #405×C73C4semidirect
175. C73#20475semidirect
182. #405Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886482
#226124292
#226098032
#226097982
#226097122
#226097072
#226096842
#226096792
#226096172
C4×#145565012
C2×#405×C732
#2047573