Finite Group Databasebeta

← #33008685

Subgroups — #33008685

Order 18688.

2078 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556550C211yesmaximal, characteristic
2C2×#17988450C211yesmaximal, characteristic
3#22608200C212yesmaximal, complemented
4#22608198C212yesmaximal, complemented
5#22608155C214yesmaximal, complemented
6#22608154C214yesmaximal
7C22×#95×C73C211yesmaximal, characteristic
8C2×#14556550—24no—
9C2×#14556550—24no—
10C2×#14556550C418yescomplemented
11C22×C4⋊Dic73C2211yescharacteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C22⋊Dic73C2212yes—
14C22×C22⋊Dic73C2212yes—
15C22×C4×Dic73C2212yes—
16#17988450—24no—
17C2×#14556498C2218yes—
18C2×#14556498C2212yes—
19C2×#14556498C2212yes—
20C2×#14556498C2218yes—
21C24×C4×C73C2212yes—
22C24×C4×C73C2211yescharacteristic
23C2×#95×C73—24no—
24C2×#95×C73C2212yes—
25C2×#95×C73C2212yes—
26C23×Dic73C2314yes—
27C23×Dic73C2312yes—
28C23×Dic73C2312yes—
29#14556550—232no—
30C2×C4⋊Dic73—22no—
31C2×C4⋊Dic73—22no—
32C2×C4⋊Dic73—22no—
33C2×C4⋊Dic73—22no—
34C2×C4⋊Dic73C2×C414yescomplemented
35C2×C4⋊Dic73C2×C414yescomplemented
36C2×C22⋊Dic73—24no—
37C2×C22⋊Dic73—24no—
38C2×C22⋊Dic73—24no—
39C2×C22⋊Dic73—28no—
40C2×C22⋊Dic73C2×C418yes—
41C2×C4×Dic73—24no—
42C2×C4×Dic73—24no—
43C2×C4×Dic73C2×C418yescomplemented
44#14556498—24no—
45#14556498—24no—
46#14556498—216no—
47#14556498—48no—
48C25×C73C2311yescharacteristic
49C23×C4×C73—28no—
50C23×C4×C73—24no—
51C23×C4×C73—28no—
52C23×C4×C73—28no—
53C23×C4×C73—44no—
54C23×C4×C73C2×C414yes—
55C23×C4×C73D414yes—
56C23×C4×C73D414yes—
57C23×C4×C73C2314yes—
58C23×C4×C73C2311yescharacteristic
59C23×C4×C73C2311yescharacteristic
60#95×C73—24no—
61#95×C73—24no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—22no—
65C22×Dic73—22no—
66C22×Dic73—28no—
67C22×Dic73—24no—
68C22×Dic73—24no—
69C22×Dic73—24no—
70C22×Dic73—28no—
71C22×Dic73—28no—
72C22×Dic73—24no—
73C22×Dic73—24no—
74C22×Dic73C22×C414yes—
75C22×Dic73C22×C414yes—
76C4⋊Dic73—28no—
77C4⋊Dic73—28no—
78C22⋊Dic73—216no—
79C22⋊Dic73—48no—
80C4×Dic73—216no—
81C24×C73—24no—
82C24×C73—22no—
83C24×C73—22no—
84C24×C73C22×C412yes—
85C24×C73C2×D412yes—
86C24×C73C2×D412yes—
87C24×C73#4118yes—
88C24×C73C2411yescharacteristic
89C22×C4×C73—216no—
90C22×C4×C73—216no—
91C22×C4×C73—216no—
92C22×C4×C73—24no—
93C22×C4×C73—24no—
94C22×C4×C73—216no—
95C22×C4×C73—24no—
96C22×C4×C73—28no—
97C22×C4×C73—24no—
98C22×C4×C73—28no—
99C22×C4×C73—432no—
100C22×C4×C73—48no—
101C22×C4×C73—48no—
102C22×C4×C73C22⋊C4116yes—
103C22×C4×C73C22×C412yes—
104C22×C4×C73C22×C412yes—
105C22×C4×C73C2×D412yes—
106C22×C4×C73C2×D412yes—
107C22×C4×C73C2×D412yes—
108C22×C4×C73C2×D412yes—
109C22×C4×C73#4118yes—
110C23×C73—24no—
111C23×C73—24no—
112C23×C73—24no—
113C23×C73—24no—
114C23×C73—22no—
115C23×C73—22no—
116C23×C73—24no—
117C23×C73—216no—
118C23×C73—216no—
119C23×C73C2×C22⋊C414yes—
120C23×C73#11718yes—
121C23×C73#12318yes—
122C23×C73#12318yes—
123C23×C73C23×C411yescharacteristic
124C23×C73C22×D411yescharacteristic
125C23×C73C22×D411yescharacteristic
126C23×C73C2×#4114yes—
127C23×C73C2×#4114yes—
128C23×C73H(5,𝔽2)12yes—
129C23×C732−1+412yes—
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—232no—
137C2×C4×C73—28no—
138C2×C4×C73—216no—
139C2×C4×C73—432no—
140C2×C4×C73—44no—
141C2×C4×C73—44no—
142C2×C4×C73—48no—
143C2×C4×C73C2×C22⋊C414yes—
144C2×C4×C73C2×C22⋊C414yes—
145C2×Dic73—24no—
146C2×Dic73—24no—
147C2×Dic73—28no—
148C2×Dic73—28no—
149C2×Dic73—44no—
150C2×Dic73—44no—
151C2×Dic73—42no—
152C2×Dic73—42no—
153C2×Dic73—44no—
154C2×Dic73—44no—
155C2×Dic73—44no—
156C2×Dic73—44no—
157C2×Dic73—48no—
158C2×Dic73—48no—
159C22×C73—24no—
160C22×C73—216no—
161C22×C73—28no—
162C22×C73—28no—
163C22×C73—216no—
164C22×C73—24no—
165C22×C73#38118yes—
166C22×C73C22×C22⋊C411yescharacteristic
167C22×C73C2×#11712yes—
168C22×C73C4×#4112yes—
169C22×C73#52012yes—
170C22×C73C2×#12312yes—
171C22×C73C2×#12312yes—
172C22×C73#52512yes—
173C22×C73#53212yes—
174C22×C73#53511yescharacteristic
175C22×C73#53611yescharacteristic
176C22×C73#53712yes—
177C22×C73#55118yes—
178C22×C73#55218yes—
179C4×C73—24no—
180C4×C73—24no—
181C4×C73—48no—
182Dic73—44no—
183Dic73—44no—
184Dic73—84no—
185#13958—731nomaximal, Sylow
186C2×C73—28no—
187C2×C73C2×#38111yescharacteristic
188C2×C73#229711yescharacteristic
189C2×C73#231111yescharacteristic
190C2×C73#234314yes—
191C2×C73#248212yes—
192C2×C73#248412yes—
193C2×C73#249814yes—
194C22×#117—731no—
195C2×#381—731no—
196C22×#95D7311yescharacteristic
197#1445—732no—
198#1445—732no—
199#1434—734no—
200C22⋊#95—734no—
201C73#1395811yescharacteristic, complemented, Sylow
202C24×C4C2×D7312yes—
203C24×C4C2×D7311yescharacteristic
204C2×#117—738no—
205C2×#117—1464no—
206C2×#117—1464no—
207C22×C4⋊C4—731no—
208C22×C4⋊C4—731no—
209C22×C22⋊C4—732no—
210C22×C22⋊C4—732no—
211C22×C42—732no—
212#381—1464no—
213C2×#95—24no—
214C2×#95—738no—
215C2×#95—738no—
216C2×#95—732no—
217C2×#95—732no—
218C2×#95C2×D7312yes—
219C2×#95C2×D7312yes—
220C25C22×D7311yescharacteristic
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—44no—
226C23×C4—732no—
227C23×C4—732no—
228C23×C4—734no—
229C23×C4C4×D7314yes—
230C23×C4D29214yes—
231C23×C4C22⋊D7314yes—
232C23×C4C22×D7314yes—
233C23×C4C22×D7311yescharacteristic
234C23×C4C22×D7311yescharacteristic
235#117—14632no—
236C2×C4⋊C4—734no—
237C2×C4⋊C4—734no—
238C2×C4⋊C4—1462no—
239C2×C4⋊C4—1462no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1462no—
242C2×C22⋊C4—738no—
243C2×C22⋊C4—1464no—
244C2×C22⋊C4—1468no—
245C2×C22⋊C4—1464no—
246C2×C22⋊C4—1464no—
247C2×C42—738no—
248C2×C42—1464no—
249C2×C42—1464no—
250#95—24no—
251#95—24no—
252#95—2928no—
253#95—1464no—
254#95—1464no—
255#95—14616no—
256C24—24no—
257C24—22no—
258C24—22no—
259C24C2×C4×D7312yes—
260C24C2×C22⋊D7312yes—
261C24C2×D29212yes—
262C24#1192402718yes—
263C24C23×D7311yescharacteristic
264C22×C4—216no—
265C22×C4—216no—
266C22×C4—216no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—216no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—432no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—734no—
278C22×C4—734no—
279C22×C4—1464no—
280C22×C4—1468no—
281C22×C4—1468no—
282C22×C4—1464no—
283C22×C4—1464no—
284C22×C4—1462no—
285C22×C4—1462no—
286C22×C4—1462no—
287C22×C4—1462no—
288C22×C4—1468no—
289C22×C4—1464no—
290C22×C4—1464no—
291C22×C4#11924005116yes—
292C22×C4C2×C4×D7312yes—
293C22×C4C2×C4×D7312yes—
294C22×C4C2×C22⋊D7312yes—
295C22×C4C2×C22⋊D7312yes—
296C22×C4C2×D29212yes—
297C22×C4C2×D29212yes—
298C22×C4#1192402818yes—
299C4⋊C4—1468no—
300C4⋊C4—1468no—
301C22⋊C4—2928no—
302C22⋊C4—14616no—
303C42—14616no—
304C23—24no—
305C23—22no—
306C23—22no—
307C23—24no—
308C23—216no—
309C23—216no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23C2×#1192400514yes—
314C23#1455654718yes—
315C23#1455657218yes—
316C23#1455657518yes—
317C23C22×C4×D7311yescharacteristic
318C23C22×C22⋊D7311yescharacteristic
319C23C22×D29211yescharacteristic
320C23C2×#1192402714yes—
321C23C2×#1192402814yes—
322C23#1455663612yes—
323C23#1455663912yes—
324C2×C4—232no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—216no—
332C2×C4—28no—
333C2×C4—432no—
334C2×C4—44no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—2924no—
338C2×C4—2922no—
339C2×C4—2922no—
340C2×C4—2924no—
341C2×C4—2924no—
342C2×C4—2924no—
343C2×C4—2924no—
344C2×C4—2928no—
345C2×C4—2928no—
346C2×C4—2924no—
347C2×C4—1464no—
348C2×C4—1464no—
349C2×C4—1468no—
350C2×C4—1468no—
351C2×C4C2×#1192400514yes—
352C2×C4C2×#1192400514yes—
353C22—24no—
354C22—216no—
355C22—216no—
356C22—28no—
357C22—24no—
358C22—28no—
359C22#1798844818yes—
360C22C22×#1192400511yescharacteristic
361C22C2×#1455654712yes—
362C22C4×#1192402812yes—
363C22#1798906812yes—
364C22C2×#1455657212yes—
365C22C2×#1455657512yes—
366C22#1798909812yes—
367C22#1798913712yes—
368C22#1798915411yescharacteristic
369C22#1798916111yescharacteristic
370C22#1798916812yes—
371C22#1798926318yes—
372C22#1798927218yes—
373C4—24no—
374C4—24no—
375C4—48no—
376C4—2924no—
377C4—2924no—
378C4—5844no—
379C2—28no—
380C2C2×#1798844811yescharacteristic
381C2#2261242111yescharacteristic
382C2#2261252411yescharacteristic
383C2#2261278714yes—
384C2#2261420512yes—
385C2#2261421812yes—
386C2#2261437514yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC22×#95×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC22×#95D73
hypocenterC73#13958
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
3. #22608200C2semidirect
4. #22608198C2semidirect
5. #22608155C2semidirect
10. C2×#14556550C4semidirect
34. C2×C4⋊Dic73C2×C4semidirect
35. C2×C4⋊Dic73C2×C4semidirect
43. C2×C4×Dic73C2×C4semidirect
201. C73#13958semidirect

Maximal subgroups

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

subgroupindex
C22×#145565502
C2×#179884502
#226082002
#226081982
#226081552
#226081542
C22×#95×C732
#1395873