Finite Group Databasebeta

← #33091157

Subgroups — #33091157

Order 18688.

1238 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989144C211yesmaximal, characteristic
2C22×C4×#10596831C211yesmaximal, characteristic
3#22610891C212yesmaximal, complemented
4#22610732C212yesmaximal, complemented
5#22610352C214yesmaximal
6C4×#14556514C214yesmaximal, complemented
7C73×#2285C211yesmaximal, characteristic
8C23×#10596831C2212yes—
9#17989144—24no—
10#17989144C418yescomplemented
11C2×#14556591C2211yescharacteristic
12C2×#14556591C2211yescharacteristic
13C2×#14556583—21no—
14C2×#14556570—21no—
15C2×#14556570C2212yes—
16C2×C4×Dic146—21no—
17C2×C4×Dic146C2212yes—
18C2×#14556550—21no—
19C2×#14556514C2218yes—
20C2×C4×#10596831—24no—
21C2×C4×#10596831C2212yes—
22C2×C4×#10596831C2212yes—
23#17988349C2218yes—
24C2×#117×C73—21no—
25C22×C42×C73C2211yescharacteristic
26#384×C73C2212yes—
27#382×C73—21no—
28#380×C73—21no—
29#380×C73C2212yes—
30C4×C4⋊C4×C73—21no—
31C4×C4⋊C4×C73C2212yes—
32C22×Dic146—21no—
33C22×Dic146C2312yes—
34C22×#10596831—28no—
35C22×#10596831—24no—
36C22×#10596831C2314yes—
37#14556591—22no—
38#14556591—22no—
39#14556591C2×C414yes—
40#14556591C2×C414yes—
41#14556583—28no—
42#14556570—28no—
43#14556570—24no—
44#14556570C2×C418yes—
45C4×Dic146—28no—
46C4×Dic146—24no—
47C4×Dic146C2×C418yes—
48#14556550—28no—
49C2×C4⋊Dic73—21no—
50C2×C4⋊Dic73—21no—
51C2×C4⋊Dic73—21no—
52C2×C4⋊Dic73C2312yes—
53C2×#11924007—22no—
54C2×#11924007—22no—
55C2×C22⋊Dic73—22no—
56C2×C4×Dic73—22no—
57#14556514—216no—
58C4×#10596831—24no—
59C4×#10596831—24no—
60C23×C4×C73D411yescharacteristic
61C23×C4×C73D411yescharacteristic
62C23×C4×C73C2311yescharacteristic
63#117×C73—44no—
64C2×C4⋊C4×C73—22no—
65C2×C4⋊C4×C73—22no—
66C2×C4⋊C4×C73C2314yes—
67C2×C22⋊C4×C73—22no—
68C2×C42×C73—22no—
69C2×C42×C73C2×C414yes—
70C2×C42×C73D411yescharacteristic
71C2×C42×C73D411yescharacteristic
72C2×C42×C73D414yes—
73C2×C42×C73C2311yescharacteristic
74C2×C42×C73C2311yescharacteristic
75#95×C73—22no—
76#95×C73—22no—
77C2×Dic146—28no—
78C2×Dic146—24no—
79C2×Dic146—22no—
80C2×Dic146—44no—
81C2×Dic146C22×C414yes—
82C22×Dic73—22no—
83C22×Dic73—22no—
84C2×#10596831—216no—
85C2×#10596831—24no—
86C2×#10596831—24no—
87C2×#10596831#4118yes—
88C4⋊Dic73—24no—
89C4⋊Dic73—24no—
90C4⋊Dic73—24no—
91C4⋊Dic73—22no—
92C4⋊Dic73C22×C414yes—
93#11924007—28no—
94#11924007—28no—
95C22⋊Dic73—28no—
96C4×Dic73—28no—
97C24×C73C2×D411yescharacteristic
98C22×C4×C73—24no—
99C22×C4×C73—24no—
100C22×C4×C73—24no—
101C22×C4×C73C22⋊C414yes—
102C22×C4×C73C22⋊C414yes—
103C22×C4×C73SD1614yes—
104C22×C4×C73Q1614yes—
105C22×C4×C73C22×C412yes—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D412yes—
112C22×C4×C73C2411yescharacteristic
113C4⋊C4×C73—28no—
114C4⋊C4×C73—44no—
115C4⋊C4×C73—44no—
116C22⋊C4×C73—44no—
117C42×C73—44no—
118C42×C73C22⋊C414yes—
119C42×C73C22⋊C414yes—
120C42×C73C22×C412yes—
121C42×C73C22×C412yes—
122C42×C73C2×D412yes—
123C42×C73C2×D412yes—
124C23×C73—22no—
125C23×C73—22no—
126C23×C73C2×C22⋊C412yes—
127C23×C73C2×SD1612yes—
128C23×C73C2×Q1612yes—
129C23×C73C22×D411yescharacteristic
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73—24no—
133C2×C4×C73—24no—
134C2×C4×C73—28no—
135C2×C4×C73—44no—
136C2×C4×C73—44no—
137C2×C4×C73—44no—
138C2×C4×C73#103116yescomplemented
139C2×C4×C73C2×C22⋊C412yes—
140C2×C4×C73C2×C22⋊C412yes—
141C2×C4×C73C2×C22⋊C412yes—
142C2×C4×C73C2×C22⋊C412yes—
143C2×C4×C73C2×C22⋊C412yes—
144C2×C4×C73C2×SD1612yes—
145C2×C4×C73C2×SD1612yes—
146C2×C4×C73C2×Q1612yes—
147C2×C4×C73C2×Q1612yes—
148C2×C4×C73#13514yes—
149C2×C4×C73C23×C411yescharacteristic
150C2×C4×C73C22×D411yescharacteristic
151C2×C4×C73C2×#4114yes—
152C2×Dic73—24no—
153C2×Dic73—24no—
154C2×Dic73—44no—
155C2×Dic73—44no—
156Dic146—28no—
157Dic146—44no—
158Dic146—48no—
159#10596831—28no—
160C22×C73—22no—
161C22×C73—24no—
162C22×C73—24no—
163C22×C73—22no—
164C22×C73C2×#10314yes—
165C22×C73C22×C22⋊C411yescharacteristic
166C22×C73#53212yes—
167C22×C73C22×SD1611yescharacteristic
168C22×C73C22×Q1611yescharacteristic
169C22×C73C2×#13512yes—
170C4×C73—22no—
171C4×C73—22no—
172C4×C73—84no—
173C4×C73C2×#10314yes—
174C4×C73C2×#10314yes—
175C4×C73#41614yes—
176C4×C73C4×#4112yes—
177C4×C73#52512yes—
178Dic73—44no—
179Dic73—44no—
180#21400—731nomaximal, Sylow
181C2×C73—24no—
182C2×C73#229711yescharacteristic
183C2×C73C22×#10311yescharacteristic
184C2×C73C2×#41611yescharacteristic
185C2×C73#316712yes—
186C2×C73#317012yes—
187C2×#533—731no—
188C22×C4×C8—731no—
189#2285D7311yescharacteristic
190#1990—732no—
191#1956—732no—
192#1892—734no—
193C4×#103—734no—
194C73#2140011yescharacteristic, complemented, Sylow
195C23×C8—732no—
196#533—738no—
197#533—1464no—
198C2×C4⋊Q8—731no—
199C2×C4⋊Q8—731no—
200C2×#125—1461no—
201C2×C22⋊Q8—732no—
202C2×C22⋊Q8—1461no—
203C2×C4×Q8—732no—
204C2×C4×Q8—1461no—
205C2×#117—21no—
206C2×#117—1461no—
207C22×C42C2×D7311yescharacteristic
208C2×#103—738no—
209C2×C4×C8—732no—
210C2×C4×C8—732no—
211C2×C4×C8—1464no—
212#384C2×D7312yes—
213#382—21no—
214#380—21no—
215#380C2×D7312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2×D7312yes—
218#340—738no—
219C22×Q8—732no—
220C22×Q8—1461no—
221C23×C4C22⋊D7311yescharacteristic
222C23×C4C22⋊D7311yescharacteristic
223C23×C4C22×D7311yescharacteristic
224C22×C8—734no—
225C22×C8—1468no—
226C22×C8—1464no—
227C4⋊Q8—734no—
228C4⋊Q8—734no—
229C4⋊Q8—1462no—
230C4⋊Q8—1462no—
231#125—1468no—
232C22⋊Q8—738no—
233C22⋊Q8—1468no—
234C22⋊Q8—1464no—
235C4×Q8—738no—
236C4×Q8—1468no—
237C4×Q8—1464no—
238#117—44no—
239#117—1468no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—732no—
243C2×C4⋊C4—1461no—
244C2×C4⋊C4—1461no—
245C2×C4⋊C4—1462no—
246C2×C4⋊C4—1462no—
247C2×C4⋊C4—1461no—
248C2×C4⋊C4C22×D7314yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—1462no—
251C2×C42—22no—
252C2×C42—1462no—
253C2×C42C4×D7314yes—
254C2×C42D29214yes—
255C2×C42C22⋊D7311yescharacteristic
256C2×C42C22⋊D7311yescharacteristic
257C2×C42C22×D7311yescharacteristic
258C2×C42C22×D7311yescharacteristic
259#103—14616no—
260C4×C8—1464no—
261C4×C8—1464no—
262#95—22no—
263#95—22no—
264C24C2×C22⋊D7311yescharacteristic
265C2×Q8—2924no—
266C2×Q8—734no—
267C2×Q8—1468no—
268C2×Q8—1464no—
269C2×Q8—1462no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—1462no—
274C22×C4—1462no—
275C22×C4#1192400514yes—
276C22×C4#1192400514yes—
277C22×C4#1192401614yes—
278C22×C4#1192401814yes—
279C22×C4C2×C4×D7312yes—
280C22×C4C2×C22⋊D7311yescharacteristic
281C22×C4C2×C22⋊D7311yescharacteristic
282C22×C4C2×C22⋊D7311yescharacteristic
283C22×C4C2×C22⋊D7311yescharacteristic
284C22×C4C2×C22⋊D7311yescharacteristic
285C22×C4C2×D29212yes—
286C22×C4C23×D7311yescharacteristic
287C2×C8—738no—
288C2×C8—14616no—
289C2×C8—1464no—
290C2×C8—1464no—
291C4⋊C4—28no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—734no—
295C4⋊C4—1468no—
296C4⋊C4—1468no—
297C4⋊C4—1464no—
298C4⋊C4—1464no—
299C4⋊C4—1464no—
300C4⋊C4—1462no—
301C22⋊C4—44no—
302C22⋊C4—1468no—
303C42—44no—
304C42—1468no—
305C42#1192400514yes—
306C42#1192400514yes—
307C42C2×C4×D7312yes—
308C42C2×C4×D7312yes—
309C42C2×D29212yes—
310C42C2×D29212yes—
311C23—22no—
312C23—22no—
313C23C2×#1192400512yes—
314C23C2×#1192401612yes—
315C23C2×#1192401812yes—
316C23C22×C22⋊D7311yescharacteristic
317Q8—2928no—
318Q8—2924no—
319Q8—1468no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—2924no—
329C2×C4—2924no—
330C2×C4—1464no—
331C2×C4—1464no—
332C2×C4#14556514116yescomplemented
333C2×C4C2×#1192400512yes—
334C2×C4C2×#1192400512yes—
335C2×C4C2×#1192400512yes—
336C2×C4C2×#1192400512yes—
337C2×C4C2×#1192400512yes—
338C2×C4C2×#1192401612yes—
339C2×C4C2×#1192401612yes—
340C2×C4C2×#1192401812yes—
341C2×C4C2×#1192401812yes—
342C2×C4#1455661114yes—
343C2×C4C22×C4×D7311yescharacteristic
344C2×C4C22×D29211yescharacteristic
345C2×C4#41×D7314yes—
346C8—1468no—
347C22—22no—
348C22—24no—
349C22—24no—
350C22—22no—
351C22C2×#1455651414yes—
352C22C22×#1192400511yescharacteristic
353C22#1798913312yes—
354C22C22×#1192401611yescharacteristic
355C22C22×#1192401811yescharacteristic
356C22C2×#1455661112yes—
357C4—22no—
358C4—22no—
359C4—84no—
360C4—2924no—
361C4—2924no—
362C4C2×#1455651414yes—
363C4C2×#1455651414yes—
364C4#1798858114yes—
365C4#1798905512yes—
366C4#1798909912yes—
367C2—24no—
368C2#2261241911yescharacteristic
369C2C22×#1455651411yescharacteristic
370C2C2×#1798858111yescharacteristic
371C2#2262048612yes—
372C2#2262051312yes—
subgroupG/H
centerC22×C4C2×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2285C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenter#2285D73
hypocenterC73#21400
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
3. #22610891C2semidirect
4. #22610732C2semidirect
6. C4×#14556514C2semidirect
10. #17989144C4semidirect
138. C2×C4×C73#103semidirect
194. C73#21400semidirect
332. C2×C4#14556514semidirect

Maximal subgroups

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

subgroupindex
C2×#179891442
C22×C4×#105968312
#226108912
#226107322
#226103522
C4×#145565142
C73×#22852
#2140073