Finite Group Databasebeta

← #33012669

Subgroups — #33012669

Order 18688.

1782 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×#17988450C211yesmaximal, characteristic
2C2×C4×C22⋊Dic73C211yesmaximal, characteristic
3#22608202C212yesmaximal, complemented
4#22608199C212yesmaximal, complemented
5#22608180C214yesmaximal
6#22608163C214yesmaximal, complemented
7C22×C22⋊Q8×C73C211yesmaximal, characteristic
8C22×C22⋊Dic73C2212yes—
9C22×C22⋊Dic73C2212yes—
10C22×C4×Dic73C2212yes—
11#17988450—24no—
12C4×C22⋊Dic73—24no—
13C2×#14556498C2212yes—
14C2×#14556498C2218yes—
15C2×#14556498C2212yes—
16C2×#14556498C2212yes—
17C2×#14556498C2218yes—
18C23×Q8×C73C2211yescharacteristic
19C24×C4×C73C2211yescharacteristic
20C2×C22⋊Q8×C73—24no—
21C2×C22⋊Q8×C73—24no—
22C2×C22⋊Q8×C73C418yescomplemented
23C22×C4⋊C4×C73C2212yes—
24C22×C4⋊C4×C73C2211yescharacteristic
25C22×C22⋊C4×C73C2212yes—
26C23×Dic73C2314yes—
27C23×Dic73C2314yes—
28C2×C22⋊Dic73—28no—
29C2×C22⋊Dic73—24no—
30C2×C22⋊Dic73—24no—
31C2×C22⋊Dic73—44no—
32C2×C4×Dic73—24no—
33C2×C4×Dic73—44no—
34#14556498—216no—
35#14556498—24no—
36#14556498—24no—
37#14556498—24no—
38#14556498—48no—
39C25×C73C2311yescharacteristic
40C22×Q8×C73—22no—
41C22×Q8×C73—22no—
42C22×Q8×C73—44no—
43C22×Q8×C73C2×C414yes—
44C23×C4×C73—44no—
45C23×C4×C73C2×C414yes—
46C23×C4×C73D414yes—
47C23×C4×C73D414yes—
48C23×C4×C73C2312yes—
49C23×C4×C73C2312yes—
50C23×C4×C73C2311yescharacteristic
51C23×C4×C73C2311yescharacteristic
52C22⋊Q8×C73—232no—
53C2×C4⋊C4×C73—22no—
54C2×C4⋊C4×C73—22no—
55C2×C4⋊C4×C73—24no—
56C2×C4⋊C4×C73—24no—
57C2×C4⋊C4×C73C2×C414yes—
58C2×C4⋊C4×C73C2×C418yes—
59C2×C22⋊C4×C73—24no—
60C2×C22⋊C4×C73—24no—
61C2×C22⋊C4×C73C2×C418yes—
62C22×Dic73—24no—
63C22×Dic73—24no—
64C22×Dic73—28no—
65C22×Dic73—28no—
66C22×Dic73—24no—
67C22×Dic73—28no—
68C22×Dic73—28no—
69C22×Dic73—24no—
70C22×Dic73—24no—
71C22×Dic73#4118yes—
72C22⋊Dic73—216no—
73C22⋊Dic73—48no—
74C4×Dic73—48no—
75C24×C73—24no—
76C24×C73—24no—
77C24×C73—24no—
78C24×C73C22×C412yes—
79C24×C73C2×D412yes—
80C24×C73C2×D412yes—
81C24×C73C2411yescharacteristic
82C2×Q8×C73—28no—
83C2×Q8×C73—48no—
84C2×Q8×C73—48no—
85C2×Q8×C73—48no—
86C22×C4×C73—24no—
87C22×C4×C73—24no—
88C22×C4×C73—22no—
89C22×C4×C73—22no—
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—28no—
93C22×C4×C73—28no—
94C22×C4×C73—48no—
95C22×C4×C73—48no—
96C22×C4×C73—48no—
97C22×C4×C73C22⋊C4116yes—
98C22×C4×C73C22×C414yes—
99C22×C4×C73C22×C414yes—
100C22×C4×C73C22×C412yes—
101C22×C4×C73C22×C412yes—
102C22×C4×C73C2×D412yes—
103C22×C4×C73C2×D412yes—
104C22×C4×C73C2×D412yes—
105C22×C4×C73C2×D412yes—
106C4⋊C4×C73—28no—
107C4⋊C4×C73—216no—
108C22⋊C4×C73—216no—
109C23×C73—24no—
110C23×C73—24no—
111C23×C73—24no—
112C23×C73—24no—
113C23×C73—24no—
114C23×C73—24no—
115C23×C73—24no—
116C23×C73—24no—
117C23×C73—24no—
118C23×C73—48no—
119C23×C73—48no—
120C23×C73C2×C22⋊C414yes—
121C23×C73C23×C411yescharacteristic
122C23×C73C22×D411yescharacteristic
123C23×C73C22×D411yescharacteristic
124C23×C73C2×#4114yes—
125C23×C73H(5,𝔽2)14yes—
126C23×C732−1+414yes—
127Q8×C73—416no—
128C2×C4×C73—24no—
129C2×C4×C73—24no—
130C2×C4×C73—24no—
131C2×C4×C73—24no—
132C2×C4×C73—24no—
133C2×C4×C73—24no—
134C2×C4×C73—24no—
135C2×C4×C73—24no—
136C2×C4×C73—28no—
137C2×C4×C73—28no—
138C2×C4×C73—416no—
139C2×C4×C73—44no—
140C2×C4×C73—44no—
141C2×C4×C73—44no—
142C2×C4×C73—44no—
143C2×C4×C73C2×C22⋊C414yes—
144C2×C4×C73C2×C22⋊C414yes—
145C2×Dic73—28no—
146C2×Dic73—216no—
147C2×Dic73—44no—
148C2×Dic73—44no—
149C2×Dic73—48no—
150C2×Dic73—44no—
151C2×Dic73—44no—
152C2×Dic73—44no—
153C2×Dic73—48no—
154C2×Dic73—48no—
155C22×C73—24no—
156C22×C73—24no—
157C22×C73—24no—
158C22×C73—48no—
159C22×C73—48no—
160C22×C73—48no—
161C22×C73C22×C22⋊C411yescharacteristic
162C22×C73C4×#4112yes—
163C22×C73#51812yes—
164C22×C73#51912yes—
165C22×C73#52512yes—
166C22×C73#53212yes—
167C22×C73#53412yes—
168C22×C73#53412yes—
169C22×C73#53612yes—
170C22×C73#53612yes—
171C22×C73#53918yes—
172C22×C73#54218yes—
173C4×C73—24no—
174C4×C73—24no—
175C4×C73—44no—
176C4×C73—44no—
177Dic73—48no—
178Dic73—84no—
179#14292—731nomaximal, Sylow
180C2×C73—44no—
181C2×C73#229711yescharacteristic
182C2×C73#230811yescharacteristic
183C2×C73#231011yescharacteristic
184C2×C73#233214yes—
185C2×C73#263814yes—
186C2×C73#266314yes—
187C22×C22⋊Q8D7311yescharacteristic
188C2×#381—731no—
189C2×C4×C22⋊C4—731no—
190#1446—732no—
191#1445—732no—
192#1440—734no—
193#1436—734no—
194C73#1429211yescharacteristic, complemented, Sylow
195C23×Q8C2×D7311yescharacteristic
196C24×C4C2×D7311yescharacteristic
197C2×C22⋊Q8—24no—
198C2×C22⋊Q8—24no—
199C2×C22⋊Q8Dic7318yescomplemented
200C22×C4⋊C4C2×D7312yes—
201C22×C4⋊C4C2×D7311yescharacteristic
202C22×C22⋊C4—732no—
203C22×C22⋊C4—732no—
204C22×C22⋊C4C2×D7312yes—
205C22×C42—732no—
206#381—1464no—
207C4×C22⋊C4—1464no—
208C2×#95—738no—
209C2×#95—732no—
210C2×#95—738no—
211C2×#95—732no—
212C2×#95—732no—
213C25C22×D7311yescharacteristic
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—44no—
217C22×Q8C2×Dic7314yes—
218C23×C4—44no—
219C23×C4—734no—
220C23×C4—734no—
221C23×C4C2×Dic7314yes—
222C23×C4C22⋊D7314yes—
223C23×C4C22⋊D7314yes—
224C23×C4C22×D7312yes—
225C23×C4C22×D7312yes—
226C23×C4C22×D7311yescharacteristic
227C23×C4C22×D7311yescharacteristic
228C22⋊Q8—232no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4C2×Dic7314yes—
234C2×C4⋊C4C2×Dic7318yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—2924no—
238C2×C22⋊C4—1464no—
239C2×C22⋊C4—1468no—
240C2×C22⋊C4—1464no—
241C2×C22⋊C4C2×Dic7318yes—
242C2×C42—2924no—
243C2×C42—1464no—
244#95—2928no—
245#95—14616no—
246#95—1464no—
247#95—1464no—
248#95—1464no—
249C24—24no—
250C24—24no—
251C24—24no—
252C24C22×Dic7312yes—
253C24C2×C22⋊D7312yes—
254C24C2×C22⋊D7312yes—
255C24C23×D7311yescharacteristic
256C2×Q8—28no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—48no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—738no—
272C22×C4—1464no—
273C22×C4—1468no—
274C22×C4—1468no—
275C22×C4—1464no—
276C22×C4—1464no—
277C22×C4—1464no—
278C22×C4—1464no—
279C22×C4—1468no—
280C22×C4—1468no—
281C22×C4C22⋊Dic73116yes—
282C22×C4C22×Dic7314yes—
283C22×C4C22×Dic7314yes—
284C22×C4C22×Dic7312yes—
285C22×C4C22×Dic7312yes—
286C22×C4C2×C22⋊D7312yes—
287C22×C4C2×C22⋊D7312yes—
288C22×C4C2×C22⋊D7312yes—
289C22×C4C2×C22⋊D7312yes—
290C4⋊C4—28no—
291C4⋊C4—216no—
292C22⋊C4—216no—
293C22⋊C4—2928no—
294C22⋊C4—14616no—
295C42—2928no—
296C23—24no—
297C23—24no—
298C23—24no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—24no—
303C23—24no—
304C23—24no—
305C23—48no—
306C23—48no—
307C23C2×C22⋊Dic7314yes—
308C23C23×Dic7311yescharacteristic
309C23C22×C22⋊D7311yescharacteristic
310C23C22×C22⋊D7311yescharacteristic
311C23#41×D7314yes—
312C23#1455663614yes—
313C23#1455663914yes—
314Q8—416no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—416no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—2924no—
331C2×C4—2924no—
332C2×C4—2928no—
333C2×C4—2924no—
334C2×C4—2924no—
335C2×C4—2924no—
336C2×C4—2928no—
337C2×C4—2928no—
338C2×C4—14616no—
339C2×C4—1468no—
340C2×C4C2×C22⋊Dic7314yes—
341C2×C4C2×C22⋊Dic7314yes—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—48no—
346C22—48no—
347C22—48no—
348C22C22×C22⋊Dic7311yescharacteristic
349C22#41×Dic7312yes—
350C22#1798906212yes—
351C22#1798906512yes—
352C22#1798909412yes—
353C22#1798913312yes—
354C22#1798914612yes—
355C22#1798914612yes—
356C22#1798916112yes—
357C22#1798916112yes—
358C22#1798918318yes—
359C22#1798920718yes—
360C4—24no—
361C4—24no—
362C4—44no—
363C4—44no—
364C4—2928no—
365C4—5844no—
366C2—44no—
367C2#2261242211yescharacteristic
368C2#2261250911yescharacteristic
369C2#2261251711yescharacteristic
370C2#2261268914yes—
371C2#2261584914yes—
372C2#2261608214yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC22×C22⋊Q8×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22×C22⋊Q8D73
hypocenterC73#14292
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22608202C2semidirect
4. #22608199C2semidirect
6. #22608163C2semidirect
22. C2×C22⋊Q8×C73C4semidirect
194. C73#14292semidirect
199. C2×C22⋊Q8Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884502
C2×C4×C22⋊Dic732
#226082022
#226081992
#226081802
#226081632
C22×C22⋊Q8×C732
#1429273