Finite Group Databasebeta

← #33095224

Subgroups — #33095224

Order 18688.

1754 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D292C211yesmaximal, characteristic
2C2×#17988644C211yesmaximal, characteristic, complemented
3#22610729C214yesmaximal, complemented
4#22610728C214yesmaximal, complemented
5#22609840C214yesmaximal, complemented
6C2×#432×C73C211yesmaximal, characteristic, complemented
7C23×D292C411yescharacteristic
8C23×C4×D73C2212yes—
9C2×C4×D292—24no—
10C2×C4×D292—24no—
11C2×C4×D292C412yes—
12C2×C4×D292C412yes—
13C2×C4×D292C2212yes—
14C2×C4×D292C2212yes—
15C22×C4⋊Dic73C411yescharacteristic
16C22×#11924005C412yes—
17#17988644—24no—
18C2×C4⋊#10596831C2212yescomplemented
19C2×C22⋊#10596831C2212yes—
20C2×#14556503C22116yescomplemented
21C2×#14556502C2212yescomplemented
22C22×C42×C73C2211yescharacteristic
23#432×C73—24no—
24C2×C4⋊C8×C73C2212yescomplemented
25C2×C22⋊C8×C73C2212yes—
26C2×#97×C73C2212yescomplemented
27C24×D73C2×C412yes—
28C22×D292—28no—
29C22×D292—24no—
30C22×D292C2×C412yes—
31C22×D292C2×C412yes—
32C23×Dic73C2×C412yes—
33C22×C4×D73—24no—
34C22×C4×D73—24no—
35C22×C4×D73—48no—
36C22×C4×D73C2×C414yes—
37C22×C4×D73C2314yes—
38C22×#10596831C2314yes—
39C4×D292—28no—
40C4×D292—28no—
41C4×D292—24no—
42C4×D292—24no—
43C4×D292—24no—
44C4×D292—24no—
45C2×C4⋊Dic73—24no—
46C2×C4⋊Dic73C2×C412yes—
47C2×C4⋊Dic73C2×C412yes—
48C2×#11924005—28no—
49C2×#11924005C2×C414yes—
50C2×#11924005C2×C414yes—
51C4⋊#10596831—24no—
52C22⋊#10596831—24no—
53#14556503—416no—
54#14556502—24no—
55C23×C4×C73C2×C411yescharacteristic
56C23×C4×C73C2×C411yescharacteristic
57C23×C4×C73C2311yescharacteristic
58C22×C8×C73C2314yes—
59C2×C42×C73C2×C411yescharacteristic
60C2×C42×C73C2×C411yescharacteristic
61C2×C42×C73D414yes—
62C2×C42×C73D414yes—
63C2×C42×C73C2311yescharacteristic
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73—24no—
66C22⋊C8×C73—24no—
67#97×C73—24no—
68C23×D73—24no—
69C23×D73—24no—
70C23×D73—44no—
71C23×D73—44no—
72C23×D73M1618yes—
73C23×D73C22×C414yes—
74C2×D292—232no—
75C2×D292—28no—
76C2×D292—28no—
77C2×D292—24no—
78C2×D292—24no—
79C22×Dic73—24no—
80C22×Dic73—44no—
81C22×Dic73C22×C414yes—
82C2×C4×D73—28no—
83C2×C4×D73—24no—
84C2×C4×D73—24no—
85C2×C4×D73—216no—
86C2×C4×D73—48no—
87C2×C4×D73—416no—
88C2×C4×D73—416no—
89C2×#10596831—24no—
90C2×#10596831—28no—
91C4⋊Dic73—24no—
92C4⋊Dic73—24no—
93#11924005—28no—
94#11924005—28no—
95C24×C73C22×C411yescharacteristic
96C22×C4×C73—24no—
97C22×C4×C73—24no—
98C22×C4×C73—42no—
99C22×C4×C73C22⋊C414yes—
100C22×C4×C73C22⋊C414yes—
101C22×C4×C73C22×C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C22×C4×C73C22×C411yescharacteristic
104C22×C4×C73C22×C411yescharacteristic
105C22×C4×C73C22×C411yescharacteristic
106C22×C4×C73C2×D412yes—
107C22×C4×C73C2×D412yes—
108C22×C4×C73C2411yescharacteristic
109C2×C8×C73—24no—
110C2×C8×C73—28no—
111C42×C73C22⋊C414yes—
112C42×C73C22⋊C414yes—
113C42×C73C2×D412yes—
114C42×C73C2×D412yes—
115C42×C73C2×D412yes—
116C42×C73C2×D412yes—
117C23×C73—22no—
118C23×C73—22no—
119C23×C73C2×C22⋊C412yes—
120C23×C73C2×M1614yes—
121C23×C73C23×C411yescharacteristic
122C22×D73—24no—
123C22×D73—28no—
124C22×D73—24no—
125C22×D73—28no—
126C22×D73—216no—
127C22×D73—416no—
128C22×D73—48no—
129C22×D73—48no—
130C22×D73—416no—
131C22×D73C2×M1618yes—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—22no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—28no—
138C2×C4×C73—24no—
139C2×C4×C73—24no—
140C2×C4×C73—44no—
141C2×C4×C73C2×C22⋊C412yes—
142C2×C4×C73C2×C22⋊C412yes—
143C2×C4×C73C2×C22⋊C412yes—
144C2×C4×C73C2×C22⋊C412yes—
145C2×C4×C73C2×C22⋊C412yes—
146C2×C4×C73C22×D411yescharacteristic
147C2×C4×C73C22×D411yescharacteristic
148C2×C4×C73H(5,𝔽2)14yes—
149C2×Dic73—24no—
150C2×Dic73—28no—
151C2×Dic73—48no—
152D292—216no—
153D292—216no—
154C4×D73—416no—
155C4×D73—48no—
156C4×D73—48no—
157C8×C73—44no—
158#10596831—44no—
159C22×C73—22no—
160C22×C73—22no—
161C22×C73—28no—
162C22×C73#40714yes—
163C22×C73C22×C22⋊C411yescharacteristic
164C22×C73#51812yes—
165C22×C73C22×M1612yes—
166C22×C73#56812yes—
167C2×D73—28no—
168C2×D73—216no—
169C2×D73—432no—
170C2×D73—44no—
171C2×D73—44no—
172C2×D73—48no—
173C4×C73—22no—
174C4×C73—22no—
175C4×C73—22no—
176C4×C73—22no—
177C4×C73—22no—
178C4×C73#53512yes—
179C4×C73#53512yes—
180Dic73—44no—
181#21790—731nomaximal, Sylow
182C2×C73—24no—
183C2×C73#230911yescharacteristic
184C2×C73C2×#40711yescharacteristic
185C2×C73#287911yescharacteristic
186C2×C73#297314yes—
187D73—48no—
188C22×C4×D4—731no—
189C2×#432—731no—
190C2×#432D7311yescharacteristic, complemented
191#1955—734no—
192#1955—734no—
193#1779—734no—
194C73#2179011yescharacteristic, complemented, Sylow
195C23×D4—731no—
196C24×C4—732no—
197C2×C4×D4—732no—
198C2×C4×D4—732no—
199C2×C4×D4—732no—
200C2×C4×D4—732no—
201C2×C4×D4—1464no—
202C2×C4×D4—1464no—
203C22×C4⋊C4—731no—
204C22×C22⋊C4—732no—
205C22×C42C2×D7311yescharacteristic
206#432—24no—
207#432—1464no—
208C2×C4⋊C8—732no—
209C2×C4⋊C8C2×D7312yescomplemented
210C2×C22⋊C8—7316no—
211C2×C22⋊C8—732no—
212C2×C22⋊C8C2×D7312yes—
213C2×#97—732no—
214C2×#97C2×D7312yescomplemented
215C25—732no—
216C22×D4—732no—
217C22×D4—732no—
218C22×D4—1468no—
219C22×D4—1464no—
220C23×C4—2928no—
221C23×C4—732no—
222C23×C4—734no—
223C23×C4—734no—
224C23×C4—1464no—
225C23×C4—1464no—
226C23×C4C4×D7311yescharacteristic
227C23×C4C4×D7311yescharacteristic
228C23×C4C22×D7311yescharacteristic
229C22×C8—734no—
230C22×C8C22×D7314yes—
231C4×D4—1468no—
232C4×D4—1468no—
233C4×D4—1464no—
234C4×D4—1464no—
235C4×D4—1464no—
236C4×D4—1464no—
237C2×C4⋊C4—732no—
238C2×C4⋊C4—732no—
239C2×C4⋊C4—1464no—
240C2×C22⋊C4—734no—
241C2×C22⋊C4—734no—
242C2×C22⋊C4—1468no—
243C2×C42C4×D7311yescharacteristic
244C2×C42C4×D7311yescharacteristic
245C2×C42D29214yes—
246C2×C42C22⋊D7314yes—
247C2×C42C22×D7311yescharacteristic
248C2×C42C22×D7311yescharacteristic
249C4⋊C8—24no—
250C4⋊C8—1464no—
251C22⋊C8—24no—
252C22⋊C8—29216no—
253C22⋊C8—1464no—
254#97—24no—
255#97—1464no—
256C24—2924no—
257C24—2924no—
258C24—738no—
259C24—734no—
260C24—1464no—
261C24—1464no—
262C24C2×C4×D7311yescharacteristic
263C2×D4—1468no—
264C2×D4—1468no—
265C2×D4—14632no—
266C2×D4—1464no—
267C2×D4—1464no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—29216no—
272C22×C4—29216no—
273C22×C4—2928no—
274C22×C4—2924no—
275C22×C4—734no—
276C22×C4—1464no—
277C22×C4—1468no—
278C22×C4—1464no—
279C22×C4—1464no—
280C22×C4—14616no—
281C22×C4#1192400514yes—
282C22×C4#1192400514yes—
283C22×C4C2×C4×D7311yescharacteristic
284C22×C4C2×C4×D7311yescharacteristic
285C22×C4C2×C4×D7311yescharacteristic
286C22×C4C2×C4×D7311yescharacteristic
287C22×C4C2×C4×D7311yescharacteristic
288C22×C4C2×C22⋊D7312yes—
289C22×C4C2×D29212yes—
290C22×C4C23×D7311yescharacteristic
291C2×C8—24no—
292C2×C8—28no—
293C2×C8—1464no—
294C2×C8—1468no—
295C4⋊C4—1464no—
296C4⋊C4—1464no—
297C22⋊C4—1468no—
298C22⋊C4—1468no—
299C42#1192400514yes—
300C42#1192400514yes—
301C42C2×C22⋊D7312yes—
302C42C2×C22⋊D7312yes—
303C42C2×D29212yes—
304C42C2×D29212yes—
305C23—22no—
306C23—22no—
307C23—2928no—
308C23—2928no—
309C23—29216no—
310C23—29216no—
311C23—738no—
312C23—1464no—
313C23—1468no—
314C23—14616no—
315C23—1464no—
316C23—1468no—
317C23C2×#1192400512yes—
318C23M16×D7314yes—
319C23C22×C4×D7311yescharacteristic
320D4—14616no—
321D4—14616no—
322C2×C4—22no—
323C2×C4—28no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—44no—
331C2×C4—29216no—
332C2×C4—2928no—
333C2×C4—2928no—
334C2×C4—2928no—
335C2×C4—1464no—
336C2×C4—1468no—
337C2×C4C2×#1192400512yes—
338C2×C4C2×#1192400512yes—
339C2×C4C2×#1192400512yes—
340C2×C4C2×#1192400512yes—
341C2×C4C2×#1192400512yes—
342C2×C4C22×C22⋊D7311yescharacteristic
343C2×C4C22×D29211yescharacteristic
344C2×C4#1455663714yes—
345C8—44no—
346C8—2924no—
347C22—22no—
348C22—22no—
349C22—28no—
350C22—29232no—
351C22—2924no—
352C22—2924no—
353C22—2928no—
354C22—14616no—
355C22—1468no—
356C22#1798855014yes—
357C22C22×#1192400511yescharacteristic
358C22#1798906112yes—
359C22C2×M16×D7312yes—
360C22#1798935312yes—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—2924no—
367C4#1798915112yes—
368C4#1798915612yes—
369C2—24no—
370C2—2928no—
371C2#2261251111yescharacteristic
372C2C2×#1798855011yescharacteristic
373C2#2261837511yescharacteristic
374C2#2261895914yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC2×#432×C73C2
prime radicalC23×D292C4
cosocleC22×C4×C73C24
hypercenterC2×#432D73
hypocenterC73#21790
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
2. C2×#17988644C2semidirect
3. #22610729C2semidirect
4. #22610728C2semidirect
5. #22609840C2semidirect
6. C2×#432×C73C2semidirect
18. C2×C4⋊#10596831C22semidirect
20. C2×#14556503C22semidirect
21. C2×#14556502C22semidirect
24. C2×C4⋊C8×C73C22semidirect
26. C2×#97×C73C22semidirect
190. C2×#432D73semidirect
194. C73#21790semidirect
209. C2×C4⋊C8C2×D73semidirect
214. C2×#97C2×D73semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2922
C2×#179886442
#226107292
#226107282
#226098402
C2×#432×C732
#2179073