Finite Group Databasebeta

← #34023828

Subgroups — #34023828

Order 18688.

1270 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989607C211yesmaximal, characteristic
2C2×#17989603C211yesmaximal, characteristic
3#22626104C214yesmaximal, complemented
4#22626102C214yesmaximal, complemented
5#22625822C212yesmaximal
6#22625821C212yesmaximal
7C22×#14556556C211yesmaximal, characteristic
8#17989607—24no—
9#17989603—24no—
10C2×#14556658C2212yes—
11C2×C4⋊#10596833C2212yes—
12C2×C22⋊#10596833C2212yes—
13C2×C22⋊#10596833C2212yes—
14C2×#14556646C2212yes—
15C2×C4×#10596833C2212yes—
16#17989453C2218yes—
17#17989453C2218yes—
18C23×Dic146C411yescharacteristic
19C2×#14556556—24no—
20C2×#14556556—24no—
21C2×#14556556C412yes—
22C2×#14556556C412yes—
23C2×#14556556C2212yes—
24C2×#14556556C2212yes—
25C22×#11924007C411yescharacteristic
26C22×#11924007C411yescharacteristic
27C22×C4×Dic73C2211yescharacteristic
28C22×C4×Dic73C2211yescharacteristic
29C22×C4×Dic73C2211yescharacteristic
30C22×C4⋊C4×C73C411yescharacteristic
31C22×#10596833C2314yes—
32C22×#10596833C2314yes—
33#14556658—24no—
34C4⋊#10596833—24no—
35C22⋊#10596833—24no—
36C22⋊#10596833—24no—
37#14556646—24no—
38C4×#10596833—24no—
39C22×Dic146—28no—
40C22×Dic146—24no—
41C22×Dic146C2×C412yes—
42C22×Dic146C2×C412yes—
43C23×Dic73C2×C411yescharacteristic
44C23×Dic73C2×C411yescharacteristic
45C23×Dic73C2×C411yescharacteristic
46C23×Dic73C2311yescharacteristic
47#14556556—28no—
48#14556556—28no—
49#14556556—24no—
50#14556556—24no—
51#14556556—24no—
52#14556556—24no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007C2×C412yes—
56C2×#11924007C2×C412yes—
57C2×#11924007C2×C412yes—
58C2×#11924007C2×C412yes—
59C2×C4×Dic73—22no—
60C2×C4×Dic73—22no—
61C2×C4×Dic73—22no—
62C2×C4×Dic73—22no—
63C2×C4×Dic73C2×C412yes—
64C2×C4×Dic73C2×C412yes—
65C2×C4×Dic73C2×C411yescharacteristic
66C2×C4×Dic73C2×C411yescharacteristic
67C2×C4×Dic73D414yes—
68C2×C4×Dic73D414yes—
69C2×C4×Dic73C2312yes—
70C2×C4×Dic73C2312yes—
71C2×C4×Dic73C2311yescharacteristic
72C2×C4×Dic73C2311yescharacteristic
73C23×C4×C73C2×C411yescharacteristic
74C23×C4×C73C2×C411yescharacteristic
75C23×C4×C73C2×C411yescharacteristic
76C2×C4⋊C4×C73—24no—
77C2×C4⋊C4×C73C2×C412yes—
78C2×C4⋊C4×C73C2×C412yes—
79C2×#10596833—24no—
80C2×#10596833—24no—
81C2×#10596833—28no—
82C2×#10596833—28no—
83C2×Dic146—232no—
84C2×Dic146—24no—
85C2×Dic146—24no—
86C2×Dic146—24no—
87C2×Dic146—24no—
88C2×Dic146—24no—
89C2×Dic146M1618yes—
90C22×Dic73—22no—
91C22×Dic73—24no—
92C22×Dic73—24no—
93C22×Dic73—24no—
94C22×Dic73—22no—
95C22×Dic73—24no—
96C22×Dic73C22⋊C414yes—
97C22×Dic73C22×C412yes—
98C22×Dic73C22×C411yescharacteristic
99C22×Dic73C22×C411yescharacteristic
100C22×Dic73C22×C412yes—
101C22×Dic73C22×C411yescharacteristic
102C22×Dic73C2×D412yes—
103C22×Dic73C2×D412yes—
104C22×Dic73C2411yescharacteristic
105#11924007—24no—
106#11924007—24no—
107#11924007—24no—
108#11924007—24no—
109C4×Dic73—24no—
110C4×Dic73—24no—
111C4×Dic73—22no—
112C4×Dic73—22no—
113C4×Dic73—22no—
114C4×Dic73—22no—
115C4×Dic73C22⋊C414yes—
116C4×Dic73C22⋊C414yes—
117C4×Dic73C2×D412yes—
118C4×Dic73C2×D412yes—
119C4×Dic73C2×D412yes—
120C4×Dic73C2×D412yes—
121C24×C73C22×C411yescharacteristic
122C22×C4×C73—22no—
123C22×C4×C73—24no—
124C22×C4×C73—24no—
125C22×C4×C73—22no—
126C22×C4×C73—24no—
127C22×C4×C73C22⋊C414yes—
128C22×C4×C73C22×C412yes—
129C22×C4×C73C22×C411yescharacteristic
130C22×C4×C73C22×C411yescharacteristic
131C22×C4×C73C22×C412yes—
132C4⋊C4×C73—24no—
133C4⋊C4×C73—24no—
134C23×C73—22no—
135C23×C73—22no—
136C23×C73C2×C22⋊C412yes—
137C23×C73#13114yes—
138C23×C73C23×C411yescharacteristic
139C2×C4×C73—28no—
140C2×C4×C73—22no—
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—28no—
145C2×C4×C73—22no—
146C2×C4×C73—24no—
147C2×C4×C73—44no—
148C2×C4×C73C2×C22⋊C412yes—
149C2×C4×C73C2×C22⋊C412yes—
150C2×C4×C73C2×M1614yes—
151C2×Dic73—28no—
152C2×Dic73—22no—
153C2×Dic73—22no—
154C2×Dic73—22no—
155C2×Dic73—22no—
156C2×Dic73—24no—
157C2×Dic73—24no—
158C2×Dic73—28no—
159C2×Dic73—22no—
160C2×Dic73—24no—
161C2×Dic73—44no—
162C2×Dic73C2×C22⋊C412yes—
163C2×Dic73C2×C22⋊C412yes—
164C2×Dic73C2×C22⋊C412yes—
165C2×Dic73C2×M1614yes—
166C2×Dic73C2×M1614yes—
167C2×Dic73C22×D411yescharacteristic
168C2×Dic73C22×D411yescharacteristic
169C2×Dic732−1+414yes—
170Dic146—216no—
171Dic146—216no—
172#10596833—44no—
173#10596833—44no—
174C22×C73—22no—
175C22×C73—22no—
176C22×C73—28no—
177C22×C73#40814yes—
178C22×C73C22×C22⋊C411yescharacteristic
179C22×C73#51912yes—
180C22×C73C22×M1612yes—
181C22×C73C2×#13112yes—
182C4×C73—24no—
183C4×C73—22no—
184C4×C73—22no—
185C4×C73—42no—
186Dic73—22no—
187Dic73—22no—
188Dic73—24no—
189Dic73—42no—
190Dic73#40714yes—
191Dic73#53612yes—
192Dic73#53612yes—
193#21820—731nomaximal, Sylow
194C2×C73—24no—
195C2×C73#231011yescharacteristic
196C2×C73C2×#40711yescharacteristic
197C2×C73C2×#40811yescharacteristic
198C2×C73#298414yes—
199C22×C4×Q8—731no—
200C2×#432—731no—
201C2×#431—731no—
202#1985—734no—
203#1984—734no—
204#1817—732no—
205#1817—732no—
206C73#2182011yescharacteristic, complemented, Sylow
207C23×Q8—731no—
208C2×C4×Q8—732no—
209C2×C4×Q8—732no—
210C2×C4×Q8—732no—
211C2×C4×Q8—732no—
212C2×C4×Q8—1464no—
213C2×C4×Q8—1464no—
214C22×C4⋊C4—731no—
215C22×C4⋊C4—731no—
216C22×C4⋊C4#6440811yescharacteristic
217C22×C42—731no—
218C22×C42—731no—
219C22×C42—731no—
220#432—1464no—
221#431—1464no—
222C2×C4⋊C8—732no—
223C2×C4⋊C8—732no—
224C2×C22⋊C8—732no—
225C2×C22⋊C8—732no—
226C2×#97—732no—
227C2×C4×C8—732no—
228#336—738no—
229#336—738no—
230C22×Q8—732no—
231C22×Q8—732no—
232C22×Q8—1464no—
233C22×Q8—1468no—
234C23×C4—731no—
235C23×C4—731no—
236C23×C4—731no—
237C23×C4—731no—
238C23×C4C2×#6440811yescharacteristic
239C23×C4C2×#6440811yescharacteristic
240C23×C4C2×#6440811yescharacteristic
241C22×C8—734no—
242C22×C8—734no—
243C4×Q8—1468no—
244C4×Q8—1468no—
245C4×Q8—1464no—
246C4×Q8—1464no—
247C4×Q8—1464no—
248C4×Q8—1464no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—732no—
251C2×C4⋊C4—732no—
252C2×C4⋊C4—732no—
253C2×C4⋊C4—732no—
254C2×C4⋊C4—1464no—
255C2×C4⋊C4—1464no—
256C2×C4⋊C4C2×#6440812yes—
257C2×C4⋊C4C2×#6440812yes—
258C2×C42—734no—
259C2×C42—734no—
260C2×C42—731no—
261C2×C42—731no—
262C2×C42—731no—
263C2×C42—731no—
264C2×C42—732no—
265C2×C42—732no—
266C2×C42—732no—
267C2×C42—732no—
268C2×C42—1462no—
269C2×C42—1462no—
270C2×C42—1462no—
271C2×C42—1462no—
272C4⋊C8—1464no—
273C4⋊C8—1464no—
274C22⋊C8—1464no—
275C22⋊C8—1464no—
276#97—1464no—
277C4×C8—1464no—
278C24C22×#6440811yescharacteristic
279C2×Q8—738no—
280C2×Q8—1464no—
281C2×Q8—1464no—
282C2×Q8—1464no—
283C2×Q8—1464no—
284C2×Q8—14632no—
285C2×Q8—1464no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—731no—
292C22×C4—731no—
293C22×C4—731no—
294C22×C4—731no—
295C22×C4—732no—
296C22×C4—734no—
297C22×C4—732no—
298C22×C4—732no—
299C22×C4—732no—
300C22×C4—1464no—
301C22×C4—1462no—
302C22×C4—1464no—
303C22×C4—1464no—
304C22×C4—1464no—
305C22×C4—1462no—
306C22×C4C22⋊#6440814yes—
307C22×C4C22×#6440812yes—
308C22×C4C22×#6440812yes—
309C22×C4C22×#6440811yescharacteristic
310C22×C4C22×#6440811yescharacteristic
311C2×C8—1464no—
312C2×C8—1464no—
313C2×C8—1468no—
314C2×C8—1468no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—1464no—
318C4⋊C4—1464no—
319C4⋊C4—1464no—
320C4⋊C4—1464no—
321C42—734no—
322C42—734no—
323C42—732no—
324C42—732no—
325C42—732no—
326C42—732no—
327C42—1462no—
328C42—1462no—
329C42—1464no—
330C42—1462no—
331C42—1462no—
332C42—1464no—
333C23—22no—
334C23—22no—
335C23C2×C22⋊#6440812yes—
336C23#1455667814yes—
337C23C23×#6440811yescharacteristic
338Q8—14616no—
339Q8—14616no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—44no—
349C2×C4—2924no—
350C2×C4—731no—
351C2×C4—731no—
352C2×C4—732no—
353C2×C4—734no—
354C2×C4—734no—
355C2×C4—732no—
356C2×C4—732no—
357C2×C4—734no—
358C2×C4—1462no—
359C2×C4—1464no—
360C2×C4—1468no—
361C2×C4—1464no—
362C2×C4—1464no—
363C2×C4—1468no—
364C2×C4—1462no—
365C2×C4—1462no—
366C2×C4—1462no—
367C2×C4—1462no—
368C2×C4C2×C22⋊#6440812yes—
369C2×C4C2×C22⋊#6440812yes—
370C2×C4D73⋊M1614yes—
371C8—2924no—
372C8—2924no—
373C22—28no—
374C22—22no—
375C22—22no—
376C22#1798956014yes—
377C22C22×C22⋊#6440811yescharacteristic
378C22#1798965512yes—
379C22C2×D73⋊M1612yes—
380C22C2×#1455667812yes—
381C4—24no—
382C4—22no—
383C4—22no—
384C4—42no—
385C4—2922no—
386C4—732no—
387C4—732no—
388C4—734no—
389C4—1462no—
390C4—1462no—
391C4—1464no—
392C2—24no—
393C2#2262640511yescharacteristic
394C2D73⋊#40711yescharacteristic
395C2C2×#1798956011yescharacteristic
396C2#2262683114yes—
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC22×C4⋊C4×C73C4
prime radicalC24×C73C22×C4
cosocleC22×Dic73C24
hypercenterC22×C4⋊C4#64408
hypocenterC73#21820
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22626104C2semidirect
4. #22626102C2semidirect
206. C73#21820semidirect

Maximal subgroups

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

subgroupindex
C2×#179896072
C2×#179896032
#226261042
#226261022
#226258222
#226258212
C22×#145565562
#2182073