Finite Group Databasebeta

← #33071269

Subgroups — #33071269

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620858C211yesmaximal, characteristic
2C2×#17988872C211yesmaximal, characteristic, complemented
3#22619403C211yesmaximal, characteristic
4#22609361C214yesmaximal, complemented
5#22609279C214yesmaximal, complemented
6#22608702C211yesmaximal, characteristic, complemented
7#22608615C211yesmaximal, characteristic, complemented
8#22608578C211yesmaximal, characteristic
9C73×#2982C211yesmaximal, characteristic, complemented
10C2×#14556622—21no—
11#17989137C2211yescharacteristic
12C2×#14556591C2211yescharacteristic
13#17988960C2212yes—
14#17988938C2212yes—
15#17988872—22no—
16#17988872C2214yes—
17#17988809—22no—
18#17988797C2212yes—
19#17988793—22no—
20#17988787C2212yes—
21#17988728—22no—
22C2×#14556522—21no—
23#17988611C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25#17988586—21no—
26C2×#14556515—21no—
27#17988338C2214yes—
28#17988334C2214yes—
29#17988332C2212yes—
30#17988330C2212yes—
31C2×C4×Q8×C73C2211yescharacteristic
32C8×Q8×C73—22no—
33C8×Q8×C73C2214yes—
34#431×C73—22no—
35#431×C73C2211yescharacteristic
36#431×C73C2211yescharacteristic
37C2×#11924028—21no—
38C22×Dic146—21no—
39#14556622—44no—
40C2×#11924018—22no—
41C2×#11924017—24no—
42C2×#11924015—22no—
43C2×#11924012—21no—
44C22×#10596831—21no—
45#14556591—21no—
46#14556591—21no—
47#14556591—21no—
48#14556591C2312yes—
49#14556591C2311yescharacteristic
50#14556588C2311yescharacteristic
51#14556579—21no—
52#14556565—21no—
53#14556565—21no—
54C4×D292—21no—
55C4×D292—21no—
56C2×C4⋊Dic73—21no—
57C2×C4⋊Dic73—21no—
58#14556522—24no—
59C4⋊#10596831—21no—
60C4⋊#10596831—21no—
61C4⋊#10596831C2312yes—
62C4⋊#10596831C2312yes—
63#14556516—24no—
64#14556515—24no—
65#14556514—22no—
66#14556511—22no—
67C22×Q8×C73D411yescharacteristic
68C4×Q8×C73—21no—
69C4×Q8×C73—21no—
70C4×Q8×C73D412yes—
71C4×Q8×C73C2312yes—
72C2×C4⋊C4×C73—21no—
73C2×C4⋊C4×C73D411yescharacteristic
74C2×C42×C73—21no—
75C2×C42×C73C2311yescharacteristic
76C4⋊C8×C73—24no—
77C4⋊C8×C73D412yes—
78C4⋊C8×C73C2312yes—
79C22⋊C8×C73—21no—
80C22⋊C8×C73D412yes—
81C4×C8×C73—24no—
82C4×C8×C73—21no—
83C4×C8×C73C2312yes—
84#11924028—42no—
85#11924028—42no—
86C2×Dic146—21no—
87C2×Dic146—22no—
88C2×Dic146—41no—
89C2×Dic146—82no—
90C2×D292—21no—
91C2×D292—21no—
92C2×D292—21no—
93C2×C22⋊D73—41no—
94C22×Dic73—41no—
95C2×C4×D73—41no—
96#11924018—44no—
97#11924017—84no—
98#11924015—44no—
99#11924012—42no—
100C2×#10596831—22no—
101C2×#10596831—22no—
102C2×#10596831—42no—
103C4⋊Dic73—22no—
104C4⋊Dic73—21no—
105C4⋊Dic73—22no—
106C4⋊Dic73—21no—
107C4⋊Dic73—41no—
108C4⋊Dic73—41no—
109#11924005—41no—
110#11924005—41no—
111C2×Q8×C73—21no—
112C2×Q8×C73—22no—
113C2×Q8×C73SD1614yescomplemented
114C2×Q8×C73C2×D412yes—
115C22×C4×C73—21no—
116C22×C4×C73—21no—
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73#4111yescharacteristic
120C2×C8×C73—22no—
121C2×C8×C73C2×D414yes—
122C4⋊C4×C73—22no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73C2×D412yes—
126C42×C73—22no—
127C42×C73—22no—
128C42×C73C2×D411yescharacteristic
129C42×C73C2×D411yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73C2411yescharacteristic
132C23×C73#12111yescharacteristic
133C22×D73—41no—
134Q8×C73—22no—
135Q8×C73—24no—
136Q8×C73C2×SD1614yes—
137C2×C4×C73—22no—
138C2×C4×C73—24no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—41no—
142C2×C4×C73—81no—
143C2×C4×C73#12111yescharacteristic
144C2×C4×C73#12111yescharacteristic
145C2×C4×C73#12111yescharacteristic
146C2×C4×C73C2×SD1612yes—
147C2×C4×C73#13512yes—
148C2×C4×C73C22×D411yescharacteristic
149C2×C4×C73C22×D411yescharacteristic
150C2×C4×C73C2×#4111yescharacteristic
151C22⋊D73—82no—
152C2×Dic73—41no—
153C2×Dic73—42no—
154C2×Dic73—82no—
155D292—41no—
156D292—41no—
157D292—42no—
158D292—81no—
159C4×D73—82no—
160Dic146—41no—
161Dic146—41no—
162Dic146—44no—
163Dic146—82no—
164C8×C73—24no—
165C8×C73—42no—
166#10596831—42no—
167#10596831—42no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73#46512yes—
171C22×C73C2×#12111yescharacteristic
172C2×D73—81no—
173C2×D73—81no—
174C2×D73—81no—
175C4×C73—21no—
176C4×C73—24no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—42no—
180C4×C73#46512yes—
181C4×C73C22×SD1611yescharacteristic
182C4×C73C2×#13511yescharacteristic
183C4×C73#57712yes—
184Dic73—82no—
185Dic73—81no—
186#19397—731nomaximal, Sylow
187C2×C73—22no—
188C2×C73#304311yescharacteristic
189C2×C73C2×#46511yescharacteristic
190C2×C73#306911yescharacteristic
191D73—161no—
192#3206—731no—
193C2×#475—731no—
194#3037—731no—
195#2982D7311yescharacteristic, complemented
196#1680—734no—
197#1666—734no—
198#1570—731no—
199#1549—731no—
200#1540—731no—
201C73#1939711yescharacteristic, complemented, Sylow
202C2×#137—1461no—
203#532—731no—
204C2×C4⋊Q8—731no—
205C2×C4×Q8C2×D7311yescharacteristic
206#492—732no—
207#487—732no—
208#475—734no—
209#475—1462no—
210#464—1462no—
211#462—732no—
212#461—1462no—
213#460—732no—
214#448—1462no—
215C8×Q8—22no—
216C8×Q8C2×D7314yes—
217#431—22no—
218#431C2×D7311yescharacteristic
219#431C2×D7311yescharacteristic
220C2×#106—1461no—
221#423—731no—
222C2×C4⋊C8—731no—
223#417—1461no—
224C2×#103—1461no—
225#334—734no—
226#332—732no—
227#332—734no—
228#331—732no—
229C2×#41—1461no—
230C22×Q8—1461no—
231C22×Q8D29211yescharacteristic
232#137—2924no—
233C2×Q16—1462no—
234C2×SD16—1462no—
235C2×SD16—1464no—
236C2×M16—1461no—
237C22×C8—1461no—
238C4⋊Q8—732no—
239C4⋊Q8—731no—
240C4⋊Q8—1461no—
241C4⋊Q8—1461no—
242C4⋊Q8—1461no—
243#127—731no—
244#124—1461no—
245#121—1461no—
246#121—1461no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D29212yes—
250C4×Q8C22×D7312yes—
251C4×D4—1461no—
252C4×D4—1461no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1461no—
255C2×C4⋊C4—1461no—
256C2×C4⋊C4D29211yescharacteristic
257C2×C42—21no—
258C2×C42C22×D7311yescharacteristic
259#106—1464no—
260C4⋊C8—24no—
261C4⋊C8—732no—
262C4⋊C8—732no—
263C4⋊C8—1461no—
264C4⋊C8—1461no—
265C4⋊C8C22⋊D7312yes—
266C4⋊C8C22×D7312yes—
267#103—1464no—
268#103—1464no—
269#103—1462no—
270#102—1462no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D7312yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D7312yes—
276#41—2922no—
277#41—2922no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—2921no—
281C2×Q8—5842no—
282C2×Q8—1462no—
283C2×Q8—1461no—
284C2×Q8#1192401714yescomplemented
285C2×Q8C2×D29212yes—
286C2×D4—2921no—
287C2×D4—1461no—
288C2×D4—1461no—
289C2×D4—1461no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4C2×C22⋊D7311yescharacteristic
295C22×C4C2×D29211yescharacteristic
296C22×C4#1192402811yescharacteristic
297Q16—2924no—
298SD16—2924no—
299SD16—5844no—
300M16—2922no—
301C2×C8—22no—
302C2×C8—2922no—
303C2×C8—1462no—
304C2×C8—1462no—
305C2×C8C2×C22⋊D7314yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—2921no—
310C4⋊C4—2921no—
311C4⋊C4—1461no—
312C4⋊C4—1462no—
313C4⋊C4—1462no—
314C4⋊C4—1461no—
315C4⋊C4C2×D29212yes—
316C22⋊C4—2921no—
317C22⋊C4—2921no—
318C42—22no—
319C42—22no—
320C42C2×C22⋊D7311yescharacteristic
321C42C2×D29211yescharacteristic
322C42#1192402811yescharacteristic
323C42C23×D7311yescharacteristic
324C23—2921no—
325C23#1455656511yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—2924no—
329Q8—2921no—
330Q8—2921no—
331Q8—5842no—
332Q8C2×#1192401714yes—
333D4—2922no—
334D4—2921no—
335D4—2921no—
336D4—5841no—
337D4—5842no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—2921no—
345C2×C4—2922no—
346C2×C4—5842no—
347C2×C4—5842no—
348C2×C4#1455656511yescharacteristic
349C2×C4#1455656511yescharacteristic
350C2×C4#1455656511yescharacteristic
351C2×C4C2×#1192401712yes—
352C2×C4#1455661312yes—
353C2×C4C22×C22⋊D7311yescharacteristic
354C2×C4C22×D29211yescharacteristic
355C2×C4C2×#1192402811yescharacteristic
356C8—24no—
357C8—42no—
358C8—2922no—
359C8—2922no—
360C22—21no—
361C22—21no—
362C22—5841no—
363C22—5841no—
364C22—5841no—
365C22#1798881612yes—
366C22C2×#1455656511yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—24no—
371C4—42no—
372C4—5842no—
373C4—5841no—
374C4#1798881612yes—
375C4C22×#1192401711yescharacteristic
376C4C2×#1455661311yescharacteristic
377C4#1798939512yes—
378C2—22no—
379C2—11681no—
380C2#2261945211yescharacteristic
381C2C2×#1798881611yescharacteristic
382C2#2261966811yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2982C2
socleC22×C73C2×#121
prime radical#17989137C22
hypercenter#2982D73
hypocenterC73#19397
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#17988872C2semidirect
4. #22609361C2semidirect
5. #22609279C2semidirect
6. #22608702C2semidirect
7. #22608615C2semidirect
9. C73×#2982C2semidirect
113. C2×Q8×C73SD16semidirect
195. #2982D73semidirect
201. C73#19397semidirect
284. C2×Q8#11924017semidirect

Maximal subgroups

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

subgroupindex
#226208582
C2×#179888722
#226194032
#226093612
#226092792
#226087022
#226086152
#226085782
C73×#29822
#1939773