Finite Group Databasebeta

← C2×#22626466 #34030673

Subgroups — C2×#22626466 #34030673

Order 18688.

2422 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626466C2116yesmaximal, direct product factor
2C2×#17989536C214yesmaximal
3C2×#17989536C214yesmaximal
4C2×C4⋊C4×#64408C214yesmaximal
5C2×#17989523C212yesmaximal
6C2×#125×D73C211yesmaximal, characteristic
7C22×C4⋊#64408C2214yes—
8C22×C4×#64408C2214yes—
9C22×C4×#64408C2214yes—
10#17989536C22132yes—
11#17989536C22132yes—
12C4⋊C4×#64408C22132yes—
13#17989523C22116yes—
14C2×(C2×#64408)⋊C4C2212yes—
15C2×(C2×#64408)⋊C4C2212yes—
16C2×(C2×#64408)⋊C4C2214yes—
17C2×(C2×#64408)⋊C4C2214yes—
18C2×(C2×#64408)⋊C4C2214yes—
19C2×#14556583C411yescharacteristic, complemented
20C2×#14556582C412yescomplemented
21C2×#14556581C414yescomplemented
22#125×D73C418yescomplemented
23#125×D73C2218yes—
24C2×C4⋊C4×D73C2214yes—
25C2×C4⋊C4×D73C2212yes—
26C2×C42×D73C2211yescharacteristic
27C2×#125×C73C411yescharacteristic, complemented
28C23×#64408C2314yes—
29C23×#64408C2314yes—
30C2×C4⋊#64408—216no—
31C2×C4⋊#64408C23116yes—
32C2×C4×#64408—216no—
33C2×C4×#64408—216no—
34C2×C4×#64408C23116yes—
35C2×C4×#64408C23116yes—
36(C2×#64408)⋊C4C2318yes—
37(C2×#64408)⋊C4C23116yes—
38(C2×#64408)⋊C4C2318yes—
39(C2×#64408)⋊C4C23116yes—
40(C2×#64408)⋊C4C23116yes—
41C22×C4×D73C2312yes—
42C22×C4×D73C2314yes—
43C22×C4×D73C2311yescharacteristic
44#14556583C2×C418yescomplemented
45#14556582C2×C4116yescomplemented
46#14556581C2×C4132yescomplemented
47C2×C4⋊Dic73C2×C414yes—
48C2×C4⋊Dic73C2×C412yes—
49C2×#11924007C2×C414yes—
50C2×#11924007C2×C414yes—
51C2×#11924007C2×C414yes—
52C4⋊C4×D73C2×C4116yes—
53C4⋊C4×D73C2×C418yes—
54C4⋊C4×D73C23116yes—
55C4⋊C4×D73C2318yes—
56C2×C4×Dic73C2×C412yes—
57C2×C4×Dic73C2×C411yescharacteristic
58C42×D73C2×C414yes—
59C42×D73C2314yes—
60#125×C73C2×C418yescomplemented
61C2×C4⋊C4×C73C2×C414yes—
62C2×C4⋊C4×C73C2×C412yes—
63C2×C42×C73C2×C411yescharacteristic
64C22×#64408—28no—
65C22×#64408—28no—
66C22×#64408—28no—
67C22×#64408—28no—
68C22×#64408—28no—
69C22×#64408#41116yes—
70C22×#64408C2418yes—
71C22×#64408C2418yes—
72C4⋊#64408—232no—
73C4×#64408—232no—
74C4×#64408—232no—
75C23×D73C2411yescharacteristic
76C22×Dic73C22×C411yescharacteristic
77C22×Dic73C22×C412yes—
78C22×Dic73C22×C414yes—
79C2×C4×D73—28no—
80C2×C4×D73—28no—
81C2×C4×D73—28no—
82C2×C4×D73—24no—
83C2×C4×D73C22×C414yes—
84C2×C4×D73C22×C412yes—
85C2×C4×D73C22×C418yes—
86C2×C4×D73C2412yes—
87C2×C4×D73C2414yes—
88C2×C4×D73C2418yes—
89C4⋊Dic73C22×C4116yes—
90C4⋊Dic73C22×C418yes—
91#11924007C22×C4116yes—
92#11924007C22×C4116yes—
93#11924007C22×C4116yes—
94C4×Dic73C22×C418yes—
95C4×Dic73C22×C414yes—
96C22×C4×C73C22×C412yes—
97C22×C4×C73C22×C411yescharacteristic
98C22×C4×C73C22×C414yes—
99C4⋊C4×C73C22×C4116yes—
100C4⋊C4×C73C22×C418yes—
101C42×C73C22×C414yes—
102C23×C73C23×C411yescharacteristic
103C22×D73C23×C411yescharacteristic
104C22×D73C2×#4118yes—
105C22×D732−1+414yes—
106C22×D73C2511yescharacteristic
107C2×#64408—232no—
108C2×#64408—28no—
109C2×#64408—28no—
110C2×#64408—28no—
111C2×#64408—28no—
112C2×#64408—28no—
113C2×#64408—416no—
114C2×#64408C2×#41116yes—
115C2×C4×C73—24no—
116C2×C4×C73—28no—
117C2×C4×C73—22no—
118C2×C4×C73C23×C414yes—
119C2×C4×C73C23×C412yes—
120C2×C4×C73C23×C418yes—
121C2×Dic73—24no—
122C2×Dic73—28no—
123C2×Dic73—22no—
124C2×Dic73C23×C412yes—
125C2×Dic73C23×C414yes—
126C2×Dic73C23×C418yes—
127C4×D73—28no—
128C4×D73—216no—
129C4×D73—24no—
130C4×D73—28no—
131C4×D73—28no—
132C4×D73—28no—
133C4×D73—24no—
134C22×C73C4×#4114yes—
135C22×C73#51912yes—
136C22×C73C24×C411yescharacteristic
137C2×D73C4×#4114yes—
138C2×D73#51912yes—
139C2×D73#552116yes—
140C2×D73C22×#4114yes—
141C2×D73C2×2−1+412yes—
142#64408—216no—
143#64408—48no—
144C4×C73—24no—
145C4×C73—28no—
146C4×C73—22no—
147Dic73—24no—
148Dic73—28no—
149Dic73—22no—
150C2×#2371—731nomaximal, Sylow
151C2×C73#237114yescomplemented
152C2×C73C2×C4×#4112yes—
153C2×C73C2×#51911yescharacteristic
154D73#237114yescomplemented
155D73C2×#55214yes—
156C22×#125—731no—
157#2371—7316no—
158C2×#387—734no—
159C2×#387—734no—
160C2×C4×C4⋊C4—734no—
161C2×#376—732no—
162C73C2×#237111yescharacteristic, complemented, Sylow
163C2×#125—738no—
164C2×#125—738no—
165C2×#125—734no—
166C2×#125—732no—
167C2×#125—731no—
168C2×#125#6440811yescharacteristic, complemented
169C22×C4⋊C4—734no—
170C22×C4⋊C4—732no—
171C22×C4⋊C4—734no—
172C22×C42—731no—
173C22×C42—734no—
174C22×C42—734no—
175#387—7332no—
176#387—7332no—
177C4×C4⋊C4—7332no—
178#376—7316no—
179C2×#95—732no—
180C2×#95—734no—
181C2×#95—732no—
182C2×#95—734no—
183C2×#95—734no—
184C23×C4—734no—
185C23×C4—734no—
186C23×C4—731no—
187C23×C4—732no—
188C23×C4—734no—
189#125—7332no—
190#125—7316no—
191#125—738no—
192#125C2×#6440818yescomplemented
193C2×C4⋊C4—734no—
194C2×C4⋊C4—734no—
195C2×C4⋊C4—734no—
196C2×C4⋊C4—734no—
197C2×C4⋊C4—732no—
198C2×C4⋊C4—7316no—
199C2×C4⋊C4—7316no—
200C2×C4⋊C4—738no—
201C2×C4⋊C4—738no—
202C2×C4⋊C4—7316no—
203C2×C4⋊C4—14616no—
204C2×C4⋊C4C2×#6440814yes—
205C2×C4⋊C4C2×#6440812yes—
206C2×C42—732no—
207C2×C42—731no—
208C2×C42—734no—
209C2×C42—734no—
210C2×C42—7316no—
211C2×C42—7316no—
212C2×C42—14616no—
213C2×C42—14616no—
214C2×C42C2×#6440811yescharacteristic
215#95—738no—
216#95—7316no—
217#95—738no—
218#95—7316no—
219#95—7316no—
220C24—731no—
221C22×C4—7316no—
222C22×C4—738no—
223C22×C4—732no—
224C22×C4—731no—
225C22×C4—734no—
226C22×C4—732no—
227C22×C4—734no—
228C22×C4—738no—
229C22×C4—732no—
230C22×C4—734no—
231C22×C4—738no—
232C22×C4—738no—
233C22×C4—1468no—
234C22×C4—1468no—
235C22×C4—1468no—
236C22×C4—1464no—
237C22×C4—1468no—
238C22×C4—1468no—
239C22×C4—1468no—
240C22×C4—1468no—
241C22×C4—1468no—
242C22×C4C22×#6440812yes—
243C22×C4C22×#6440811yescharacteristic
244C22×C4C22×#6440814yes—
245C4⋊C4—7316no—
246C4⋊C4—7316no—
247C4⋊C4—7316no—
248C4⋊C4—7316no—
249C4⋊C4—738no—
250C4⋊C4—14632no—
251C4⋊C4C22×#64408116yes—
252C4⋊C4C22×#6440818yes—
253C42—738no—
254C42—734no—
255C42—14632no—
256C42—14632no—
257C42C22×#6440814yes—
258C23—738no—
259C23—734no—
260C23—731no—
261C23—731no—
262C23C23×#6440811yescharacteristic
263C2×C4—24no—
264C2×C4—28no—
265C2×C4—22no—
266C2×C4—29216no—
267C2×C4—7316no—
268C2×C4—734no—
269C2×C4—732no—
270C2×C4—738no—
271C2×C4—1468no—
272C2×C4—1468no—
273C2×C4—1468no—
274C2×C4—1468no—
275C2×C4—14616no—
276C2×C4—1464no—
277C2×C4—1464no—
278C2×C4—1468no—
279C2×C4—1468no—
280C2×C4—1468no—
281C2×C4—1468no—
282C2×C4—1468no—
283C2×C4—14632no—
284C2×C4—1464no—
285C2×C4—1468no—
286C2×C4—1462no—
287C2×C4C23×#6440814yes—
288C2×C4C23×#6440812yes—
289C2×C4C23×#6440818yes—
290C22—734no—
291C22—734no—
292C22—732no—
293C22—732no—
294C22—7316no—
295C22#41×#6440814yes—
296C22#1798965512yes—
297C22C24×#6440811yescharacteristic
298C4—24no—
299C4—28no—
300C4—22no—
301C4—2928no—
302C4—14616no—
303C4—1464no—
304C4—1468no—
305C4—1462no—
306C2—734no—
307C2—734no—
308C2#2262646614yesdirect product factor
309C2C2×#41×#6440812yes—
310C2C2×#1798965511yescharacteristic
subgroupG/H
centerC23C23×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
FittingC2×#125×C73C4
socleC23×C73C23×C4
prime radicalC23×D73C24
cosocleC22×D73C25
hypercenterC2×#125#64408
hypocenterC73C2×#2371
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #22626466C2direct (N × H)
19. C2×#14556583C4semidirect
20. C2×#14556582C4semidirect
21. C2×#14556581C4semidirect
22. #125×D73C4semidirect
27. C2×#125×C73C4semidirect
44. #14556583C2×C4semidirect
45. #14556582C2×C4semidirect
46. #14556581C2×C4semidirect
60. #125×C73C2×C4semidirect
151. C2×C73#2371semidirect
154. D73#2371semidirect
162. C73C2×#2371semidirect
168. C2×#125#64408semidirect
192. #125C2×#64408semidirect
308. C2#22626466direct (N × H)

Maximal subgroups

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

subgroupindex
#226264662
C2×#179895362
C2×C4⋊C4×#644082
C2×#179895232
C2×#125×D732
C2×#237173