Finite Group Databasebeta

← #33084792

Subgroups — #33084792

Order 18688.

1006 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988641C211yesmaximal, characteristic
2C22×#14556520C211yesmaximal, characteristic
3C4×#14556547C211yesmaximal, characteristic
4#22609805C212yesmaximal, complemented
5#22609802C212yesmaximal, complemented
6#22609783C212yesmaximal, complemented
7C4×#14556520C212yesmaximal, complemented
8#22609710C211yesmaximal, characteristic
9#22609707C211yesmaximal, characteristic
10C22⋊C4×#10596831C211yesmaximal, characteristic
11C8×C22⋊C4×C73C211yesmaximal, characteristic
12C23×#10596831C2211yescharacteristic
13C2×#14556547C412yes—
14C22×C4×Dic73C2211yescharacteristic
15#17988641C418yescomplemented
16C2×#14556520C418yescomplemented
17C2×#14556520C2211yescharacteristic
18C2×#14556520C2211yescharacteristic
19C2×#14556520C2211yescharacteristic
20C2×#14556520C2211yescharacteristic
21C2×#14556520C2212yes—
22C2×C22⋊#10596831C2211yescharacteristic
23C2×#14556501C2212yes—
24C2×#14556501C2212yes—
25C2×#14556501C2212yes—
26C2×C4×#10596831C2212yes—
27C4×#11924007C2212yes—
28#17988438C412yes—
29C22⋊C4×Dic73C412yes—
30C4×C22⋊Dic73C2211yescharacteristic
31#17988432C412yes—
32C42×Dic73C2212yes—
33#17988342C2212yes—
34#17988341C2212yes—
35#17988341C2212yes—
36C23×C8×C73C2211yescharacteristic
37C2×C22⋊C8×C73C2211yescharacteristic
38C2×C4×C8×C73C2212yes—
39C4×C22⋊C4×C73C2211yescharacteristic
40#336×C73C2212yes—
41C23×Dic73C2×C411yescharacteristic
42C23×Dic73C2×C411yescharacteristic
43C22×#10596831—42no—
44C22×#10596831C2×C414yes—
45C22×#10596831C2311yescharacteristic
46C22×#10596831C2312yes—
47C22×#10596831C2311yescharacteristic
48#14556547—28no—
49C2×#11924007C2×C414yes—
50C2×C22⋊Dic73C2×C412yes—
51C2×C4×Dic73C2×C414yes—
52C2×C4×Dic73C2×C414yes—
53C2×C4×Dic73C2×C411yescharacteristic
54C2×C4×Dic73C2×C411yescharacteristic
55C2×C4×Dic73C2×C414yes—
56C2×C4×Dic73C2×C414yes—
57C2×C4×Dic73C2312yes—
58C2×C4×Dic73C2311yescharacteristic
59C2×C4×Dic73C2311yescharacteristic
60#14556520C2×C418yes—
61#14556520C2×C414yes—
62#14556520C2×C414yes—
63#14556520C2×C414yes—
64#14556520C2×C414yes—
65C22⋊#10596831C2×C414yes—
66#14556501—42no—
67#14556501—42no—
68#14556501C2×C418yes—
69C4×#10596831—42no—
70#14556498C2×C412yes—
71#14556498C2×C414yes—
72C23×C4×C73C2311yescharacteristic
73C22×C8×C73—42no—
74C22×C8×C73C2×C414yes—
75C22×C8×C73C2311yescharacteristic
76C22×C8×C73C2312yes—
77C22×C8×C73C2311yescharacteristic
78C2×C22⋊C4×C73C2×C412yes—
79C2×C42×C73C2312yes—
80C22⋊C8×C73C2×C414yes—
81C4×C8×C73—42no—
82#95×C73C2×C412yes—
83C22×Dic73—24no—
84C22×Dic73—24no—
85C22×Dic73C4214yes—
86C22×Dic73C4214yes—
87C22×Dic73C22×C414yes—
88C22×Dic73C22×C411yescharacteristic
89C22×Dic73C22×C411yescharacteristic
90C22×Dic73C22×C411yescharacteristic
91C22×Dic73C22×C411yescharacteristic
92C2×#10596831—22no—
93C2×#10596831—22no—
94C2×#10596831—22no—
95C2×#10596831—22no—
96C2×#10596831—44no—
97C2×#10596831C22×C414yes—
98C2×#10596831C22×C412yes—
99C2×#10596831C22×C412yes—
100#11924007—28no—
101C22⋊Dic73—24no—
102C4×Dic73—28no—
103C4×Dic73—22no—
104C4×Dic73—22no—
105C4×Dic73—28no—
106C4×Dic73C4214yes—
107C4×Dic73C4214yes—
108C4×Dic73C22×C412yes—
109C4×Dic73C22×C412yes—
110C4×Dic73C22×C412yes—
111C4×Dic73C22×C412yes—
112C24×C73C22×C411yescharacteristic
113C22×C4×C73—22no—
114C22×C4×C73—22no—
115C22×C4×C73C22×C411yescharacteristic
116C22×C4×C73C22×C412yes—
117C22×C4×C73C22×C412yes—
118C22×C4×C73C22×C414yes—
119C22×C4×C73C2411yescharacteristic
120C2×C8×C73—22no—
121C2×C8×C73—22no—
122C2×C8×C73—22no—
123C2×C8×C73—22no—
124C2×C8×C73—44no—
125C2×C8×C73C22×C414yes—
126C2×C8×C73C22×C412yes—
127C2×C8×C73C22×C412yes—
128C22⋊C4×C73M1618yes—
129C42×C73—22no—
130C42×C73—22no—
131C23×C73—22no—
132C23×C73—22no—
133C23×C73C2×C4212yes—
134C23×C73C2×M1614yes—
135C23×C73C23×C411yescharacteristic
136C2×C4×C73—24no—
137C2×C4×C73—22no—
138C2×C4×C73—24no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—24no—
142C2×C4×C73C2×C4212yes—
143C2×C4×C73C2×M1618yes—
144C2×C4×C73C23×C411yescharacteristic
145C2×C4×C73C23×C411yescharacteristic
146C2×C4×C73H(5,𝔽2)12yes—
147C2×C4×C732−1+412yes—
148C2×Dic73—22no—
149C2×Dic73—22no—
150C2×Dic73—22no—
151C2×Dic73—28no—
152C2×Dic73—28no—
153C2×Dic73—24no—
154C2×Dic73—24no—
155C2×Dic73—24no—
156C2×Dic73C2×C4212yes—
157C2×Dic73C2×C4212yes—
158C2×Dic73C2×C4212yes—
159C2×Dic73C2×C4212yes—
160C2×Dic73#13114yes—
161C8×C73—22no—
162C8×C73—22no—
163C8×C73—24no—
164#10596831—22no—
165#10596831—22no—
166#10596831—24no—
167C22×C73—22no—
168C22×C73—28no—
169C22×C73—22no—
170C22×C73C4×M1614yes—
171C22×C73C22×C4211yescharacteristic
172C22×C73#52012yes—
173C22×C73C22×M1612yes—
174C22×C73C2×#13112yes—
175C4×C73—28no—
176C4×C73—22no—
177C4×C73#51811yescharacteristic
178C4×C73#51911yescharacteristic
179C4×C73#52012yes—
180Dic73—22no—
181Dic73—22no—
182Dic73—22no—
183Dic73—44no—
184Dic73#40514yes—
185#20546—731nomaximal, Sylow
186C2×C73—24no—
187C2×C73#227411yescharacteristic
188C2×C73C2×C4×M1611yescharacteristic
189C2×C73C2×#40511yescharacteristic
190C2×C73#299014yes—
191C2×#432—731no—
192C22×C4⋊C8—731no—
193C4×#117—731no—
194#1771—732no—
195#1771—732no—
196#1767—732no—
197C4×C4⋊C8—732no—
198#1752—731no—
199#1752—731no—
200C8×C22⋊C4—731no—
201C8×C22⋊C4D7311yescharacteristic
202C73#2054611yescharacteristic, complemented, Sylow
203C23×C8—731no—
204C23×C8C2×D7311yescharacteristic
205C2×#117—732no—
206C22×C42—731no—
207#432—738no—
208C2×C4⋊C8—738no—
209C2×C4⋊C8—731no—
210C2×C4⋊C8—731no—
211C2×C4⋊C8—731no—
212C2×C4⋊C8—731no—
213C2×C4⋊C8—732no—
214C2×C22⋊C8—731no—
215C2×C22⋊C8C2×D7311yescharacteristic
216C2×#97—732no—
217C2×#97—732no—
218C2×#97—732no—
219C2×C4×C8—732no—
220C2×C4×C8C2×D7312yes—
221C4×C4⋊C4—732no—
222C4×C4⋊C4—732no—
223C4×C22⋊C4—731no—
224C4×C22⋊C4—732no—
225C4×C22⋊C4C2×D7311yescharacteristic
226#376—732no—
227C43—732no—
228#336—732no—
229#336—732no—
230#336—732no—
231#336C2×D7312yes—
232C23×C4—731no—
233C23×C4—731no—
234C23×C4C22×D7311yescharacteristic
235C22×C8—42no—
236C22×C8—2922no—
237C22×C8—734no—
238C22×C8—731no—
239C22×C8—731no—
240C22×C8—732no—
241C22×C8C4×D7314yes—
242C22×C8C22×D7311yescharacteristic
243C22×C8C22×D7311yescharacteristic
244C22×C8C22×D7312yes—
245#117—1468no—
246C2×C4⋊C4—734no—
247C2×C22⋊C4—732no—
248C2×C22⋊C4C4×D7312yes—
249C2×C42—731no—
250C2×C42—731no—
251C2×C42—734no—
252C2×C42—734no—
253C2×C42—731no—
254C2×C42—731no—
255C2×C42—734no—
256C2×C42—734no—
257C2×C42—732no—
258C2×C42C22×D7312yes—
259C4⋊C8—734no—
260C4⋊C8—734no—
261C4⋊C8—734no—
262C4⋊C8—734no—
263C4⋊C8—738no—
264C22⋊C8—734no—
265C22⋊C8C4×D7314yes—
266#97—2922no—
267#97—2922no—
268#97—738no—
269C4×C8—42no—
270C4×C8—2922no—
271#95—734no—
272#95—732no—
273#95C4×D7312yes—
274C24C2×C4×D7311yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—731no—
278C22×C4—734no—
279C22×C4—734no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—734no—
284C22×C4—1464no—
285C22×C4—1464no—
286C22×C4C2×C4×D7311yescharacteristic
287C22×C4C2×C4×D7312yes—
288C22×C4C2×C4×D7312yes—
289C22×C4C2×C4×D7314yes—
290C22×C4C23×D7311yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—44no—
296C2×C8—2924no—
297C2×C8—734no—
298C2×C8—732no—
299C2×C8—732no—
300C2×C8—1462no—
301C2×C8—1462no—
302C2×C8—1462no—
303C2×C8—1462no—
304C2×C8C2×C4×D7314yes—
305C2×C8C2×C4×D7312yes—
306C2×C8C2×C4×D7312yes—
307C4⋊C4—1468no—
308C22⋊C4—1464no—
309C22⋊C4#1192401118yes—
310C42—22no—
311C42—22no—
312C42—734no—
313C42—734no—
314C42—732no—
315C42—732no—
316C42—732no—
317C42—732no—
318C42—1468no—
319C42—1468no—
320C42—1462no—
321C42—1462no—
322C23—22no—
323C23—22no—
324C23C42×D7312yes—
325C23C2×#1192401114yes—
326C23C22×C4×D7311yescharacteristic
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—732no—
334C2×C4—734no—
335C2×C4—732no—
336C2×C4—732no—
337C2×C4—732no—
338C2×C4—1464no—
339C2×C4—1464no—
340C2×C4—1464no—
341C2×C4—1468no—
342C2×C4—1468no—
343C2×C4—1462no—
344C2×C4—1462no—
345C2×C4—1462no—
346C2×C4C42×D7312yes—
347C2×C4C2×#1192401118yes—
348C2×C4C22×C4×D7311yescharacteristic
349C2×C4C22×C4×D7311yescharacteristic
350C2×C4#1455663612yes—
351C2×C4#1455663812yes—
352C8—22no—
353C8—22no—
354C8—24no—
355C8—1462no—
356C8—1462no—
357C8—1464no—
358C22—22no—
359C22—28no—
360C22—22no—
361C22C4×#1192401114yes—
362C22C2×C42×D7311yescharacteristic
363C22#1798906612yes—
364C22C22×#1192401112yes—
365C22#131×D7312yes—
366C4—22no—
367C4—28no—
368C4—2924no—
369C4—734no—
370C4—1462no—
371C4—1462no—
372C4—1462no—
373C4#1798906011yescharacteristic
374C4#1798906411yescharacteristic
375C4#1798906612yes—
376C2—24no—
377C2#2261228611yescharacteristic
378C2C2×C4×#1192401111yescharacteristic
379C2#405×D7311yescharacteristic
380C2#2261905414yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C22×C42
FittingC8×C22⋊C4×C73C2
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenterC8×C22⋊C4D73
hypocenterC73#20546
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #22609805C2semidirect
5. #22609802C2semidirect
6. #22609783C2semidirect
7. C4×#14556520C2semidirect
15. #17988641C4semidirect
16. C2×#14556520C4semidirect
202. C73#20546semidirect

Maximal subgroups

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

subgroupindex
C2×#179886412
C22×#145565202
C4×#145565472
#226098052
#226098022
#226097832
C4×#145565202
#226097102
#226097072
C22⋊C4×#105968312
C8×C22⋊C4×C732
#2054673