Finite Group Databasebeta

← C22×#17988636 #33949677

Subgroups — C22×#17988636 #33949677

Order 18688.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924012C212yesmaximal
2C2×#17988636C2124yesmaximal, direct product factor
3C22×#14556527C214yesmaximal, complemented
4C23×M16×C73C211yesmaximal, characteristic
5C22×#11924012—22no—
6C22×#11924012—212no—
7C22×#11924012C2214yes—
8C22×#11924012C22124yes—
9C23×#10596831—22no—
10#17988636C22164yesdirect product factor
11C2×#14556527C22148yescomplemented
12C24×C4×C73C2211yescharacteristic
13C22×M16×C73C414yes—
14C22×M16×C73C4112yes—
15C22×M16×C73C22112yes—
16C23×C8×C73C2212yes—
17C2×#11924012—224no—
18C2×#11924012—212no—
19C2×#11924012—248no—
20C2×#11924012C23124yes—
21C2×#11924012C23132yes—
22C22×#10596831—224no—
23C22×#10596831—24no—
24#14556527C23164yescomplemented
25C25×C73Q811yescharacteristic
26C23×C4×C73C2×C412yes—
27C23×C4×C73C2×C416yes—
28C23×C4×C73D412yes—
29C23×C4×C73D416yes—
30C23×C4×C73Q811yescharacteristic
31C23×C4×C73Q816yes—
32C23×C4×C73C2311yescharacteristic
33C23×C4×C73C2316yes—
34C2×M16×C73C2×C4148yes—
35C2×M16×C73C2×C4148yes—
36C2×M16×C73C23116yes—
37C22×C8×C73C2×C4112yes—
38C22×C8×C73C2×C414yes—
39C22×C8×C73C23112yes—
40#11924012—232no—
41#11924012—224no—
42#11924012C24116yes—
43C2×#10596831—224no—
44C2×#10596831—232no—
45C24×C73—44no—
46C24×C73C4⋊C412yes—
47C24×C73C4⋊C416yes—
48C24×C73C2×Q811yescharacteristic
49C24×C73C2×Q816yes—
50C22×C4×C73C4⋊C4112yes—
51C22×C4×C73C4⋊C4112yes—
52C22×C4×C73C4⋊C416yes—
53C22×C4×C73C4⋊C4124yes—
54C22×C4×C73C4⋊C412yes—
55C22×C4×C73C22×C413yes—
56C22×C4×C73C22×C411yescharacteristic
57C22×C4×C73C22×C4112yes—
58C22×C4×C73C22×C4112yes—
59C22×C4×C73C2×D4112yes—
60C22×C4×C73C2×D413yes—
61C22×C4×C73C2×D4112yes—
62C22×C4×C73C2×D411yescharacteristic
63C22×C4×C73C2×Q813yes—
64C22×C4×C73C2×Q816yes—
65C22×C4×C73C2×Q8112yes—
66C22×C4×C73C2413yes—
67C22×C4×C73C2414yes—
68M16×C73C22×C4164yes—
69C2×C8×C73C22×C4124yes—
70C2×C8×C73C22×C4124yes—
71C2×C8×C73C2418yes—
72C23×C73—28no—
73C23×C73—24no—
74C23×C73—412no—
75C23×C73—412no—
76C23×C73C2×C4⋊C413yes—
77C23×C73C2×C4⋊C4112yes—
78C23×C73C2×C4⋊C4112yes—
79C23×C73C2×C4⋊C411yescharacteristic
80C23×C73C22×Q813yes—
81C23×C73C22×Q814yes—
82C2×C4×C73C2×C4⋊C416yes—
83C2×C4×C73C2×C4⋊C4124yes—
84C2×C4×C73C2×C4⋊C4124yes—
85C2×C4×C73C2×C4⋊C4112yes—
86C2×C4×C73C2×C4⋊C413yes—
87C2×C4×C73C2×C4⋊C4112yes—
88C2×C4×C73C2×C4⋊C413yes—
89C2×C4×C73C23×C413yes—
90C2×C4×C73C23×C413yes—
91C2×C4×C73C23×C418yes—
92C2×C4×C73C22×D413yes—
93C2×C4×C73C22×D418yes—
94C2×C4×C73C22×D413yes—
95C2×C4×C73C22×Q813yes—
96C2×C4×C73C22×Q814yes—
97C2×C4×C73C2511yescharacteristic
98C8×C73C23×C4116yes—
99#10596831—216no—
100C22×C73—224no—
101C22×C73—26no—
102C22×C73—26no—
103C22×C73—412no—
104C22×C73—44no—
105C22×C73#43014yescomplemented
106C22×C73C22×C4⋊C413yes—
107C22×C73C22×C4⋊C418yes—
108C22×C73C22×C4⋊C413yes—
109C22×C73C23×Q811yescharacteristic
110C4×C73C22×C4⋊C413yes—
111C4×C73C22×C4⋊C413yes—
112C4×C73C22×C4⋊C418yes—
113C4×C73C24×C411yescharacteristic
114C4×C73C23×D411yescharacteristic
115C22×#430—731nomaximal, Sylow
116C2×C73—28no—
117C2×C73—23no—
118C2×C73—21no—
119C2×C73C2×#43016yescomplemented
120C2×C73C23×C4⋊C411yescharacteristic
121C23×M16—732no—
122C23×M16D7311yescharacteristic
123C2×#430—7324no—
124C22×#108—734no—
125C73C22×#43011yescharacteristic, complemented, Sylow
126C24×C4C2×D7311yescharacteristic
127C22×M16—734no—
128C22×M16—7324no—
129C22×M16—14612no—
130C22×M16—1462no—
131C22×M16Dic7314yes—
132C22×M16Dic73112yes—
133C22×M16C2×D73112yes—
134C23×C8—1462no—
135C23×C8C2×D7312yes—
136#430—7364no—
137C2×#108—7348no—
138C25Dic14611yescharacteristic
139C23×C4Dic14611yescharacteristic
140C23×C4Dic14616yes—
141C23×C4D29216yes—
142C23×C4D29212yes—
143C23×C4C2×Dic7312yes—
144C23×C4C2×Dic7316yes—
145C23×C4C22×D7316yes—
146C23×C4C22×D7311yescharacteristic
147C2×M16—7324no—
148C2×M16—7332no—
149C2×M16—14624no—
150C2×M16—14612no—
151C2×M16—14648no—
152C2×M16C2×Dic73148yes—
153C2×M16C2×Dic73148yes—
154C2×M16C22×D73116yes—
155C22×C8—14624no—
156C22×C8—1464no—
157C22×C8C2×Dic7314yes—
158C22×C8C2×Dic73112yes—
159C22×C8C22×D73112yes—
160#108—7364no—
161C24—44no—
162C24C4⋊Dic7312yes—
163C24C4⋊Dic7316yes—
164C24C2×Dic14611yescharacteristic
165C24C2×Dic14616yes—
166C22×C4C4⋊Dic73112yes—
167C22×C4C4⋊Dic7316yes—
168C22×C4C4⋊Dic73124yes—
169C22×C4C4⋊Dic7312yes—
170C22×C4C4⋊Dic73112yes—
171C22×C4C22×Dic7311yescharacteristic
172C22×C4C22×Dic7313yes—
173C22×C4C22×Dic73112yes—
174C22×C4C22×Dic73112yes—
175C22×C4C2×D292112yes—
176C22×C4C2×D29213yes—
177C22×C4C2×D292112yes—
178C22×C4C2×D29211yescharacteristic
179C22×C4C2×Dic14616yes—
180C22×C4C2×Dic146112yes—
181C22×C4C2×Dic14613yes—
182C22×C4C23×D7313yes—
183C22×C4C23×D7314yes—
184M16—7316no—
185M16—14632no—
186M16—14624no—
187M16C22×Dic73164yes—
188C2×C8—14624no—
189C2×C8—14632no—
190C2×C8C22×Dic73124yes—
191C2×C8C22×Dic73124yes—
192C2×C8C23×D7318yes—
193C23—28no—
194C23—24no—
195C23—412no—
196C23—412no—
197C23C2×C4⋊Dic7313yes—
198C23C2×C4⋊Dic73112yes—
199C23C2×C4⋊Dic73112yes—
200C23C2×C4⋊Dic7311yescharacteristic
201C23C22×Dic14613yes—
202C23C22×Dic14614yes—
203C2×C4C2×C4⋊Dic7316yes—
204C2×C4C2×C4⋊Dic73124yes—
205C2×C4C2×C4⋊Dic73124yes—
206C2×C4C2×C4⋊Dic73112yes—
207C2×C4C2×C4⋊Dic7313yes—
208C2×C4C2×C4⋊Dic73112yes—
209C2×C4C2×C4⋊Dic7313yes—
210C2×C4C23×Dic7313yes—
211C2×C4C23×Dic7313yes—
212C2×C4C23×Dic7318yes—
213C2×C4C22×D29213yes—
214C2×C4C22×D29218yes—
215C2×C4C22×D29213yes—
216C2×C4C22×Dic14613yes—
217C2×C4C22×Dic14614yes—
218C2×C4C24×D7311yescharacteristic
219C8—14616no—
220C8C23×Dic73116yes—
221C22—224no—
222C22—26no—
223C22—26no—
224C22—412no—
225C22—44no—
226C22#1798863614yesdirect product factor
227C22C22×C4⋊Dic7313yes—
228C22C22×C4⋊Dic7318yes—
229C22C22×C4⋊Dic7313yes—
230C22C23×Dic14611yescharacteristic
231C4C22×C4⋊Dic7313yes—
232C4C22×C4⋊Dic7318yes—
233C4C22×C4⋊Dic7313yes—
234C4C24×Dic7311yescharacteristic
235C4C23×D29211yescharacteristic
236C2—21no—
237C2—28no—
238C2—23no—
239C2C2×#1798863616yesdirect product factor
240C2C23×C4⋊Dic7311yescharacteristic
subgroupG/H
centerC22×C4C2×D292
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
2. C2×#17988636C2direct (N × H)
3. C22×#14556527C2semidirect
10. #17988636C22direct (N × H)
11. C2×#14556527C22semidirect
24. #14556527C23semidirect
105. C22×C73#430semidirect
119. C2×C73C2×#430semidirect
125. C73C22×#430semidirect
226. C22#17988636direct (N × H)
239. C2C2×#17988636direct (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×#179886362
C22×#145565272
C23×M16×C732
C22×#43073