Finite Group Databasebeta

← C22×#17988635 #33949676

Subgroups — C22×#17988635 #33949676

Order 18688.

2718 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924012C211yesmaximal, characteristic
2C23×#11924012C211yesmaximal, characteristic
3C2×#17988635C2124yesmaximal, direct product factor
4C22×#14556526C214yesmaximal, complemented
5C23×M16×C73C211yesmaximal, characteristic
6C22×#11924012—21no—
7C22×#11924012—26no—
8C22×#11924012C412yes—
9C22×#11924012C412yes—
10C22×#11924012C4112yes—
11C22×#11924012C2212yes—
12C22×#11924012C22112yes—
13C22×#11924012C22112yes—
14C23×#10596831—21no—
15C23×#10596831C2212yes—
16#17988635C22164yesdirect product factor
17C2×#14556526C22148yescomplemented
18C24×C4×C73C2211yescharacteristic
19C22×M16×C73—21no—
20C22×M16×C73—26no—
21C22×M16×C73C2212yes—
22C22×M16×C73C22112yes—
23C23×C8×C73—21no—
24C2×#11924012—212no—
25C2×#11924012—26no—
26C2×#11924012—224no—
27C2×#11924012C2×C4124yes—
28C2×#11924012C2×C4124yes—
29C2×#11924012C2×C4148yes—
30C2×#11924012C23112yes—
31C2×#11924012C23116yes—
32C2×#11924012C23116yes—
33C22×#10596831—212no—
34C22×#10596831—22no—
35C22×#10596831C2×C4112yes—
36C22×#10596831C2×C414yes—
37C22×#10596831C23112yes—
38#14556526C23164yescomplemented
39C25×C73Q811yescharacteristic
40C23×C4×C73C2×C411yescharacteristic
41C23×C4×C73C2×C411yescharacteristic
42C23×C4×C73C2×C416yes—
43C23×C4×C73D411yescharacteristic
44C23×C4×C73D411yescharacteristic
45C23×C4×C73D416yes—
46C23×C4×C73Q811yescharacteristic
47C23×C4×C73Q816yes—
48C23×C4×C73C2311yescharacteristic
49C23×C4×C73C2316yes—
50C2×M16×C73—212no—
51C2×M16×C73—26no—
52C2×M16×C73—224no—
53C2×M16×C73C23112yes—
54C2×M16×C73C23116yes—
55C22×C8×C73—212no—
56C22×C8×C73—22no—
57#11924012—216no—
58#11924012—212no—
59#11924012C22×C4132yes—
60#11924012C22×C4132yes—
61#11924012C2418yes—
62C2×#10596831—212no—
63C2×#10596831—216no—
64C2×#10596831C22×C4124yes—
65C2×#10596831C22×C4124yes—
66C2×#10596831C2418yes—
67C24×C73—44no—
68C24×C73C4⋊C411yescharacteristic
69C24×C73C4⋊C411yescharacteristic
70C24×C73C4⋊C416yes—
71C24×C73C2×Q811yescharacteristic
72C24×C73C2×Q816yes—
73C22×C4×C73C4⋊C416yes—
74C22×C4×C73C4⋊C416yes—
75C22×C4×C73C4⋊C416yes—
76C22×C4×C73C4⋊C416yes—
77C22×C4×C73C4⋊C416yes—
78C22×C4×C73C4⋊C4124yes—
79C22×C4×C73C4⋊C411yescharacteristic
80C22×C4×C73C4⋊C411yescharacteristic
81C22×C4×C73C22×C413yes—
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C22×C416yes—
84C22×C4×C73C22×C416yes—
85C22×C4×C73C22×C4112yes—
86C22×C4×C73C2×D416yes—
87C22×C4×C73C2×D416yes—
88C22×C4×C73C2×D413yes—
89C22×C4×C73C2×D4112yes—
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73C2×Q813yes—
92C22×C4×C73C2×Q816yes—
93C22×C4×C73C2×Q8112yes—
94C22×C4×C73C2413yes—
95C22×C4×C73C2414yes—
96M16×C73—216no—
97M16×C73—212no—
98M16×C73C2418yes—
99C2×C8×C73—212no—
100C2×C8×C73—216no—
101C23×C73—24no—
102C23×C73—24no—
103C23×C73—24no—
104C23×C73—412no—
105C23×C73—412no—
106C23×C73C2×C4⋊C413yes—
107C23×C73C2×C4⋊C416yes—
108C23×C73C2×C4⋊C416yes—
109C23×C73C2×C4⋊C4112yes—
110C23×C73C2×C4⋊C411yescharacteristic
111C23×C73C22×Q813yes—
112C23×C73C22×Q814yes—
113C2×C4×C73C2×C4⋊C416yes—
114C2×C4×C73C2×C4⋊C4112yes—
115C2×C4×C73C2×C4⋊C4112yes—
116C2×C4×C73C2×C4⋊C4112yes—
117C2×C4×C73C2×C4⋊C4112yes—
118C2×C4×C73C2×C4⋊C4112yes—
119C2×C4×C73C2×C4⋊C413yes—
120C2×C4×C73C2×C4⋊C416yes—
121C2×C4×C73C2×C4⋊C416yes—
122C2×C4×C73C2×C4⋊C413yes—
123C2×C4×C73C23×C413yes—
124C2×C4×C73C23×C413yes—
125C2×C4×C73C23×C414yes—
126C2×C4×C73C23×C414yes—
127C2×C4×C73C22×D413yes—
128C2×C4×C73C22×D414yes—
129C2×C4×C73C22×D414yes—
130C2×C4×C73C22×D413yes—
131C2×C4×C73C22×Q813yes—
132C2×C4×C73C22×Q814yes—
133C2×C4×C73C2511yescharacteristic
134C8×C73—28no—
135#10596831—28no—
136#10596831C23×C4116yes—
137C22×C73—212no—
138C22×C73—212no—
139C22×C73—26no—
140C22×C73—26no—
141C22×C73—412no—
142C22×C73—44no—
143C22×C73#43014yescomplemented
144C22×C73C22×C4⋊C413yes—
145C22×C73C22×C4⋊C414yes—
146C22×C73C22×C4⋊C414yes—
147C22×C73C22×C4⋊C413yes—
148C22×C73C23×Q811yescharacteristic
149C4×C73C22×C4⋊C413yes—
150C4×C73C22×C4⋊C413yes—
151C4×C73C22×C4⋊C414yes—
152C4×C73C22×C4⋊C414yes—
153C4×C73C24×C411yescharacteristic
154C4×C73C23×D411yescharacteristic
155C22×#430—731nomaximal, Sylow
156C2×C73—24no—
157C2×C73—24no—
158C2×C73—23no—
159C2×C73—21no—
160C2×C73C2×#43016yescomplemented
161C2×C73C23×C4⋊C411yescharacteristic
162C23×M16—731no—
163C23×M16—731no—
164C23×M16D7311yescharacteristic
165C2×#430—7324no—
166C22×#108—734no—
167C73C22×#43011yescharacteristic, complemented, Sylow
168C24×C4C2×D7311yescharacteristic
169C22×M16—26no—
170C22×M16—21no—
171C22×M16—732no—
172C22×M16—732no—
173C22×M16—7312no—
174C22×M16—7312no—
175C22×M16—732no—
176C22×M16—7312no—
177C22×M16—1466no—
178C22×M16—1461no—
179C22×M16C2×D7312yes—
180C22×M16C2×D73112yes—
181C23×C8—21no—
182C23×C8—732no—
183C23×C8—1461no—
184#430—7364no—
185C2×#108—7348no—
186C25Dic14611yescharacteristic
187C23×C4Dic14611yescharacteristic
188C23×C4Dic14616yes—
189C23×C4C4×D7311yescharacteristic
190C23×C4C4×D7311yescharacteristic
191C23×C4C4×D7316yes—
192C23×C4C22⋊D7316yes—
193C23×C4C22⋊D7311yescharacteristic
194C23×C4C22⋊D7311yescharacteristic
195C23×C4C22×D7316yes—
196C23×C4C22×D7311yescharacteristic
197C2×M16—212no—
198C2×M16—26no—
199C2×M16—224no—
200C2×M16—7324no—
201C2×M16—7324no—
202C2×M16—7348no—
203C2×M16—7316no—
204C2×M16—7312no—
205C2×M16—7316no—
206C2×M16—14612no—
207C2×M16—1466no—
208C2×M16—14624no—
209C2×M16C22×D73112yes—
210C2×M16C22×D73116yes—
211C22×C8—212no—
212C22×C8—22no—
213C22×C8—734no—
214C22×C8—7312no—
215C22×C8—7312no—
216C22×C8—14612no—
217C22×C8—1462no—
218#108—7364no—
219C24—44no—
220C24#1192400711yescharacteristic
221C24#1192400711yescharacteristic
222C24#1192400716yes—
223C24C2×Dic14611yescharacteristic
224C24C2×Dic14616yes—
225C22×C4#1192400716yes—
226C22×C4#1192400716yes—
227C22×C4#1192400716yes—
228C22×C4#11924007124yes—
229C22×C4#1192400711yescharacteristic
230C22×C4#1192400711yescharacteristic
231C22×C4#1192400716yes—
232C22×C4#1192400716yes—
233C22×C4C2×C4×D7311yescharacteristic
234C22×C4C2×C4×D7313yes—
235C22×C4C2×C4×D7316yes—
236C22×C4C2×C4×D7316yes—
237C22×C4C2×C4×D73112yes—
238C22×C4C2×C22⋊D7316yes—
239C22×C4C2×C22⋊D7316yes—
240C22×C4C2×C22⋊D7313yes—
241C22×C4C2×C22⋊D73112yes—
242C22×C4C2×C22⋊D7311yescharacteristic
243C22×C4C2×Dic14616yes—
244C22×C4C2×Dic146112yes—
245C22×C4C2×Dic14613yes—
246C22×C4C23×D7313yes—
247C22×C4C23×D7314yes—
248M16—216no—
249M16—212no—
250M16—7332no—
251M16—7332no—
252M16—738no—
253M16—14616no—
254M16—14612no—
255M16C23×D7318yes—
256C2×C8—212no—
257C2×C8—216no—
258C2×C8—7324no—
259C2×C8—7324no—
260C2×C8—738no—
261C2×C8—14612no—
262C2×C8—14616no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—412no—
267C23—412no—
268C23C2×#1192400713yes—
269C23C2×#1192400716yes—
270C23C2×#1192400716yes—
271C23C2×#11924007112yes—
272C23C2×#1192400711yescharacteristic
273C23C22×Dic14613yes—
274C23C22×Dic14614yes—
275C2×C4C2×#1192400716yes—
276C2×C4C2×#11924007112yes—
277C2×C4C2×#11924007112yes—
278C2×C4C2×#11924007112yes—
279C2×C4C2×#11924007112yes—
280C2×C4C2×#11924007112yes—
281C2×C4C2×#1192400713yes—
282C2×C4C2×#1192400716yes—
283C2×C4C2×#1192400716yes—
284C2×C4C2×#1192400713yes—
285C2×C4C22×C4×D7313yes—
286C2×C4C22×C4×D7313yes—
287C2×C4C22×C4×D7314yes—
288C2×C4C22×C4×D7314yes—
289C2×C4C22×C22⋊D7313yes—
290C2×C4C22×C22⋊D7314yes—
291C2×C4C22×C22⋊D7314yes—
292C2×C4C22×C22⋊D7313yes—
293C2×C4C22×Dic14613yes—
294C2×C4C22×Dic14614yes—
295C2×C4C24×D7311yescharacteristic
296C8—28no—
297C8—7316no—
298C8—1468no—
299C22—212no—
300C22—212no—
301C22—26no—
302C22—26no—
303C22—412no—
304C22—44no—
305C22#1798863514yesdirect product factor
306C22C22×#1192400713yes—
307C22C22×#1192400714yes—
308C22C22×#1192400714yes—
309C22C22×#1192400713yes—
310C22C23×Dic14611yescharacteristic
311C4C22×#1192400713yes—
312C4C22×#1192400714yes—
313C4C22×#1192400714yes—
314C4C22×#1192400713yes—
315C4C23×C4×D7311yescharacteristic
316C4C23×C22⋊D7311yescharacteristic
317C2—21no—
318C2—24no—
319C2—24no—
320C2—23no—
321C2C2×#1798863516yesdirect product factor
322C2C23×#1192400711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D73
commutatorC4×C73C24×C4
FrattiniC2×C4C24×D73
FittingC23×M16×C73C2
socleC23×C73C2×C4⋊C4
prime radicalC25×C73Q8
cosocleC2×C4×C73C25
hypercenterC23×M16D73
hypocenterC73C22×#430
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
3. C2×#17988635C2direct (N × H)
4. C22×#14556526C2semidirect
16. #17988635C22direct (N × H)
17. C2×#14556526C22semidirect
38. #14556526C23semidirect
143. C22×C73#430semidirect
160. C2×C73C2×#430semidirect
167. C73C22×#430semidirect
305. C22#17988635direct (N × H)
321. C2C2×#17988635direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240122
C2×#179886352
C22×#145565262
C23×M16×C732
C22×#43073