Finite Group Databasebeta

← C22×#17988863 #33953372

Subgroups — C22×#17988863 #33953372

Order 18688.

2630 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556624C212yesmaximal
2C23×Dic292C211yesmaximal, characteristic, complemented
3C22×#14556597C211yesmaximal, characteristic
4C2×#17988863C2124yesmaximal, direct product factor
5C22×#14556509C212yesmaximal, complemented
6C22×#108×C73C211yesmaximal, characteristic, complemented
7C22×#11924028C2211yescharacteristic
8C23×Dic146C2212yes—
9C2×#14556624—212no—
10C2×#14556624C22124yes—
11C22×Dic292—22no—
12C22×Dic292—21no—
13C22×Dic292—21no—
14C22×Dic292—26no—
15C22×Dic292C22112yescomplemented
16C22×#11924017—22no—
17C2×#14556597—26no—
18C2×#14556597C22112yes—
19C22×#11924012C2211yescharacteristic
20C22×#11924011—21no—
21C22×C8×D73—21no—
22#17988863C22164yesdirect product factor
23C2×#14556509C22124yescomplemented
24C22×M16×C73C2212yes—
25C23×C8×C73C2211yescharacteristic
26C2×#108×C73C22112yescomplemented
27C2×#11924028—23no—
28C2×#11924028—23no—
29C2×#11924028D416yes—
30C2×#11924028C2316yes—
31C22×Dic146—26no—
32C22×Dic146—22no—
33C22×Dic146—22no—
34C22×Dic146—26no—
35C22×Dic146—26no—
36C22×Dic146D411yescharacteristic
37C22×Dic146C23112yes—
38C22×D292D411yescharacteristic
39C22×C22⋊D73—21no—
40C23×Dic73—22no—
41C22×C4×D73—21no—
42#14556624—248no—
43#14556624C23132yes—
44C2×Dic292—224no—
45C2×Dic292—212no—
46C2×Dic292—212no—
47C2×Dic292—224no—
48C2×Dic292C23116yescomplemented
49C2×#11924017—224no—
50#14556597—224no—
51#14556597C23116yes—
52C2×#11924012—23no—
53C2×#11924012C2316yes—
54C2×#11924011—212no—
55C22×#10596831—21no—
56C2×C8×D73—212no—
57#14556509C23132yescomplemented
58C23×C4×C73C2311yescharacteristic
59C2×M16×C73—26no—
60C2×M16×C73C23112yes—
61C22×C8×C73—22no—
62C22×C8×C73D411yescharacteristic
63C22×C8×C73D411yescharacteristic
64C22×C8×C73D416yes—
65C22×C8×C73C2316yes—
66#108×C73C23116yescomplemented
67C23×D73—21no—
68#11924028—212no—
69#11924028—26no—
70#11924028—26no—
71#11924028C2×D4112yes—
72#11924028C2414yes—
73C2×Dic146—23no—
74C2×Dic146—212no—
75C2×Dic146—212no—
76C2×Dic146—212no—
77C2×Dic146—212no—
78C2×Dic146—212no—
79C2×Dic146—46no—
80C2×Dic146—46no—
81C2×Dic146—412no—
82C2×Dic146C2×D416yes—
83C2×Dic146C2418yes—
84C2×D292—23no—
85C2×D292C2×D416yes—
86C2×C22⋊D73—26no—
87C2×C22⋊D73—26no—
88C22×Dic73—21no—
89C22×Dic73—212no—
90C22×Dic73—46no—
91C22×Dic73—42no—
92C2×C4×D73—26no—
93C2×C4×D73—26no—
94Dic292—232no—
95Dic292—216no—
96Dic292—216no—
97#11924017—232no—
98#11924012—26no—
99#11924012C2414yes—
100#11924011—216no—
101C2×#10596831—26no—
102C8×D73—216no—
103C24×C73#4111yescharacteristic
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D413yes—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D413yes—
108C22×C4×C73#4113yes—
109C22×C4×C73C2413yes—
110M16×C73—212no—
111M16×C73C2418yes—
112C2×C8×C73—212no—
113C2×C8×C73C2×D416yes—
114C2×C8×C73C2×D416yes—
115C2×C8×C73C2×D4112yes—
116C2×C8×C73C2414yes—
117C23×C73—24no—
118C23×C73#12113yes—
119C23×C73#12111yescharacteristic
120C23×C73C2×#4113yes—
121C22×D73—26no—
122C22×D73—44no—
123C2×C4×C73#12113yes—
124C2×C4×C73#12113yes—
125C2×C4×C73#12116yes—
126C2×C4×C73C22×D413yes—
127C2×C4×C73C22×D413yes—
128C2×C4×C73C22×D413yes—
129C2×C4×C73C22×D413yes—
130C2×C4×C73C2×#4113yes—
131C2×C4×C73C2511yescharacteristic
132C22⋊D73—212no—
133C22⋊D73—24no—
134C2×Dic73—26no—
135C2×Dic73—28no—
136C2×Dic73—412no—
137C2×Dic73—412no—
138D292—26no—
139D292C22×D414yes—
140C4×D73—212no—
141C4×D73—24no—
142Dic146—26no—
143Dic146—28no—
144Dic146—28no—
145Dic146—412no—
146Dic146—412no—
147Dic146C22×D414yes—
148C8×C73—28no—
149C8×C73C22×D414yes—
150C8×C73C22×D414yes—
151#10596831—24no—
152C22×C73—26no—
153C22×C73—26no—
154C22×C73#47314yescomplemented
155C22×C73C2×#12113yes—
156C22×C73C2×#12113yes—
157C22×C73C22×#4111yescharacteristic
158C2×D73—24no—
159C2×D73—412no—
160C4×C73C2×#12113yes—
161C4×C73C2×#12113yes—
162C4×C73C23×D411yescharacteristic
163C4×C73C23×D411yescharacteristic
164Dic73—24no—
165Dic73—48no—
166C22×#473—731nomaximal, Sylow
167C2×C73—23no—
168C2×C73—21no—
169C2×C73C2×#47316yescomplemented
170C2×C73C22×#12111yescharacteristic
171D73—44no—
172C22×#137—732no—
173C23×Q16—731no—
174C22×#131—731no—
175C2×#473—7324no—
176C22×#108D7311yescharacteristic, complemented
177C22×#101—732no—
178C73C22×#47311yescharacteristic, complemented, Sylow
179C22×#41—731no—
180C23×Q8—732no—
181C2×#137—7324no—
182C2×#137—14612no—
183C22×Q16—7312no—
184C22×Q16—1462no—
185C22×Q16—1461no—
186C22×Q16—1461no—
187C22×Q16—1466no—
188C22×SD16—1462no—
189C2×#131—7312no—
190C2×#131—1466no—
191C22×M16—731no—
192C22×M16—1461no—
193C22×M16C2×D7312yes—
194C23×C8—1461no—
195C23×C8C2×D7311yescharacteristic
196#473—7364no—
197C2×#108C2×D73112yescomplemented
198C2×#101—7324no—
199C2×#41—736no—
200C2×#41—736no—
201C2×#41—1463no—
202C2×#41—1463no—
203C22×Q8—731no—
204C22×Q8—7312no—
205C22×Q8—1462no—
206C22×Q8—1466no—
207C22×Q8—1466no—
208C22×Q8—1462no—
209C22×Q8—1466no—
210C22×D4—731no—
211C22×D4—1461no—
212C23×C4—1462no—
213C23×C4—1461no—
214C23×C4C22×D7311yescharacteristic
215#137—7332no—
216#137—14648no—
217C2×Q16—7316no—
218C2×Q16—14624no—
219C2×Q16—14612no—
220C2×Q16—14612no—
221C2×Q16—14624no—
222C2×SD16—14624no—
223#131—7316no—
224#131—14624no—
225C2×M16—26no—
226C2×M16—736no—
227C2×M16—1463no—
228C2×M16—14612no—
229C2×M16C22×D73112yes—
230C22×C8—22no—
231C22×C8—1461no—
232C22×C8—14612no—
233C22×C8D29211yescharacteristic
234C22×C8D29211yescharacteristic
235C22×C8D29216yes—
236C22×C8C22×D7316yes—
237#108C22×D73116yescomplemented
238#101—7332no—
239C24—1461no—
240C24#1192402911yescharacteristic
241#41—7312no—
242#41—734no—
243#41—14612no—
244#41—1466no—
245#41—1466no—
246C2×Q8—2926no—
247C2×Q8—2926no—
248C2×Q8—29212no—
249C2×Q8—736no—
250C2×Q8—738no—
251C2×Q8—14612no—
252C2×Q8—14612no—
253C2×Q8—14612no—
254C2×Q8—14612no—
255C2×Q8—1463no—
256C2×Q8—14612no—
257C2×D4—736no—
258C2×D4—1466no—
259C2×D4—1466no—
260C2×D4—1463no—
261C22×C4—2926no—
262C22×C4—2922no—
263C22×C4—14612no—
264C22×C4—1461no—
265C22×C4—1466no—
266C22×C4—1466no—
267C22×C4D4×D7313yes—
268C22×C4D4×D7311yescharacteristic
269C22×C4C2×D29213yes—
270C22×C4C2×D29211yescharacteristic
271C22×C4#1192402913yes—
272C22×C4C23×D7313yes—
273Q16—14632no—
274Q16—14616no—
275Q16—14616no—
276SD16—14632no—
277M16—212no—
278M16—734no—
279M16—1466no—
280M16—14616no—
281M16C23×D7318yes—
282C2×C8—212no—
283C2×C8—1466no—
284C2×C8—14616no—
285C2×C8C2×D29216yes—
286C2×C8C2×D29216yes—
287C2×C8C2×D292112yes—
288C2×C8C23×D7314yes—
289C23—24no—
290C23—2924no—
291C23—1466no—
292C23#1455656613yes—
293C23#1455656611yescharacteristic
294C23C2×#1192402913yes—
295Q8—29212no—
296Q8—29212no—
297Q8—734no—
298Q8—1468no—
299Q8—1468no—
300Q8—1466no—
301D4—734no—
302D4—1464no—
303D4—14612no—
304D4—1466no—
305C2×C4—29212no—
306C2×C4—29212no—
307C2×C4—1468no—
308C2×C4—1466no—
309C2×C4—1464no—
310C2×C4—14612no—
311C2×C4#1455656613yes—
312C2×C4#1455656613yes—
313C2×C4#1455656616yes—
314C2×C4C2×D4×D7313yes—
315C2×C4C2×D4×D7313yes—
316C2×C4C22×D29213yes—
317C2×C4C22×D29213yes—
318C2×C4C2×#1192402913yes—
319C2×C4C24×D7311yescharacteristic
320C8—28no—
321C8—1464no—
322C8C22×D29214yes—
323C8C22×D29214yes—
324C22—26no—
325C22—26no—
326C22—29212no—
327C22—1464no—
328C22#1798886314yesdirect product factor
329C22C2×#1455656613yes—
330C22C2×#1455656613yes—
331C22C22×#1192402911yescharacteristic
332C4—2928no—
333C4—1464no—
334C4C2×#1455656613yes—
335C4C2×#1455656613yes—
336C4C22×D4×D7311yescharacteristic
337C4C23×D29211yescharacteristic
338C2—21no—
339C2—23no—
340C2—2924no—
341C2C2×#1798886316yesdirect product factor
342C2C22×#1455656611yescharacteristic
subgroupG/H
centerC23#14556566
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC22×#108×C73C2
socleC23×C73#121
prime radicalC22×#11924028C22
hypercenterC22×#108D73
hypocenterC73C22×#473
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
2. C23×Dic292C2semidirect
4. C2×#17988863C2direct (N × H)
5. C22×#14556509C2semidirect
6. C22×#108×C73C2semidirect
15. C22×Dic292C22semidirect
22. #17988863C22direct (N × H)
23. C2×#14556509C22semidirect
26. C2×#108×C73C22semidirect
48. C2×Dic292C23semidirect
57. #14556509C23semidirect
66. #108×C73C23semidirect
154. C22×C73#473semidirect
169. C2×C73C2×#473semidirect
176. C22×#108D73semidirect
178. C73C22×#473semidirect
197. C2×#108C2×D73semidirect
237. #108C22×D73semidirect
328. C22#17988863direct (N × H)
341. C2C2×#17988863direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145566242
C23×Dic2922
C22×#145565972
C2×#179888632
C22×#145565092
C22×#108×C732
C22×#47373