Finite Group Databasebeta

← C2×#22624920 #34010742

Subgroups — C2×#22624920 #34010742

Order 18688.

506 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989482C211yesmaximal, characteristic
2C2×#17989461C211yesmaximal, characteristic
3#22624920C214yesmaximal, direct product factor
4C2×#17988829C211yesmaximal, characteristic
5C22×C4⋊#64408—21no—
6C22×#11924031—21no—
7#17989482—21no—
8#17989482C2212yes—
9#17989461—21no—
10#17989461C2212yes—
11C22×Dic292C411yescharacteristic, complemented
12C22×C8×D73C2211yescharacteristic
13C2×#14556571—21no—
14#17988829—21no—
15#17988829—21no—
16#17988829C412yescomplemented
17#17988829C2212yes—
18C2×#14556514—21no—
19C2×#107×C73C411yescharacteristic, complemented
20C23×#64408—41no—
21C2×C4⋊#64408—24no—
22C2×C4⋊#64408—22no—
23C2×C4⋊#64408—42no—
24C2×C4⋊#64408—41no—
25C2×#11924031—22no—
26C2×#11924031—42no—
27C22×Dic146—21no—
28C22×C4×D73D411yescharacteristic
29C2×Dic292—21no—
30C2×Dic292—42no—
31C2×Dic292C2×C412yescomplemented
32C22×#10596831D411yescharacteristic
33C2×C8×D73—42no—
34C2×C8×D73C2×C411yescharacteristic
35C2×C8×D73D411yescharacteristic
36C2×C8×D73D411yescharacteristic
37C2×C8×D73C2311yescharacteristic
38#14556571—22no—
39#14556571—22no—
40#14556571—41no—
41#14556571—41no—
42C2×C4⋊Dic73—41no—
43C2×#11924005—41no—
44#14556514—22no—
45#14556514—22no—
46C22×C8×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73—21no—
48#107×C73—21no—
49#107×C73C2×C412yescomplemented
50C22×#64408—44no—
51C22×#64408—42no—
52C22×#64408—82no—
53C22×#64408—82no—
54C4⋊#64408—24no—
55C4⋊#64408—44no—
56C4⋊#64408—42no—
57#11924031—42no—
58C23×D73D811yescharacteristic
59C2×Dic146—22no—
60C2×Dic146—42no—
61C2×Dic146—42no—
62C2×Dic146—41no—
63C22×Dic73—41no—
64C22×Dic73SD1611yescharacteristic
65C2×C4×D73—22no—
66C2×C4×D73—22no—
67C2×C4×D73C22⋊C411yescharacteristic
68C2×C4×D73D811yescharacteristic
69C2×C4×D73SD1611yescharacteristic
70C2×C4×D73C2×D411yescharacteristic
71Dic292—42no—
72Dic292—42no—
73C2×#10596831—22no—
74C2×#10596831C22⋊C411yescharacteristic
75C2×#10596831C2×D411yescharacteristic
76C8×D73—41no—
77C8×D73—41no—
78C8×D73—41no—
79C8×D73—41no—
80C4⋊Dic73—42no—
81C4⋊Dic73—42no—
82#11924005—42no—
83#11924005—42no—
84C22×C4×C73—41no—
85C22×C4×C73C22⋊C411yescharacteristic
86C2×C8×C73—22no—
87C2×C8×C73C22⋊C411yescharacteristic
88C2×C8×C73C22×C411yescharacteristic
89C4⋊C4×C73—22no—
90C4⋊C4×C73—41no—
91C23×C73#10211yescharacteristic
92C22×D73—22no—
93C22×D73—22no—
94C22×D73#10211yescharacteristic
95C22×D73SD3212yes—
96C22×D73Q3212yes—
97C22×D73C2×D811yescharacteristic
98C2×#64408—44no—
99C2×#64408—88no—
100C2×#64408—82no—
101C2×#64408—82no—
102C2×C4×C73—42no—
103C2×C4×C73—82no—
104C2×C4×C73#9912yes—
105C2×C4×C73#10211yescharacteristic
106C2×C4×C73#10412yes—
107C2×C4×C73C2×C22⋊C411yescharacteristic
108C2×Dic73—22no—
109C2×Dic73—42no—
110C2×Dic73—82no—
111C2×Dic73#10211yescharacteristic
112C2×Dic73C2×SD1611yescharacteristic
113C4×D73—22no—
114C4×D73—22no—
115C4×D73—21no—
116C4×D73—21no—
117C4×D73—21no—
118C4×D73—21no—
119Dic146—42no—
120Dic146—42no—
121Dic146—82no—
122C8×C73—21no—
123C8×C73—21no—
124#10596831—21no—
125#10596831—21no—
126C22×C73#32712yes—
127C22×C73#35812yes—
128C22×C73#36012yes—
129C22×C73C2×#10211yescharacteristic
130C2×D73—24no—
131C2×D73—24no—
132C2×D73—21no—
133C2×D73—21no—
134C2×D73—21no—
135C2×D73—21no—
136C2×D73#35812yes—
137C2×D73C2×SD3211yescharacteristic
138C2×D73C2×Q3211yescharacteristic
139#64408—84no—
140C4×C73—82no—
141C4×C73#32712yes—
142C4×C73C2×#9911yescharacteristic
143C4×C73C2×#10411yescharacteristic
144Dic73—21no—
145Dic73—21no—
146Dic73—82no—
147C2×#1348—731nomaximal, Sylow
148C2×C73#134814yescomplemented
149C2×C73C2×#32711yescharacteristic
150C2×C73C2×#35811yescharacteristic
151C2×C73C2×#36011yescharacteristic
152D73—21no—
153D73—22no—
154D73—21no—
155C2×#467—731no—
156C2×C22⋊C16—731no—
157C2×#340—731no—
158#1348—734no—
159C73C2×#134811yescharacteristic, complemented, Sylow
160C22×Q16—731no—
161C23×C8—731no—
162C2×C22⋊Q8—1461no—
163C22×C4⋊C4—1461no—
164C22×C16—1461no—
165#467—732no—
166#467—732no—
167#467—1461no—
168#467—1461no—
169C2×#107#6440811yescharacteristic, complemented
170C2×#103—1461no—
171C22⋊C16—732no—
172C22⋊C16—1461no—
173#340—732no—
174#340—1461no—
175C22×Q8—1461no—
176C23×C4—2921no—
177C23×C4—731no—
178C2×Q16—2922no—
179C2×Q16—732no—
180C2×Q16—1461no—
181C22×C8—2922no—
182C22×C8—731no—
183C22×C8—731no—
184C22×C8—731no—
185C22×C8—731no—
186C22×C8—731no—
187C22×C8C2×#6440811yescharacteristic
188C22⋊Q8—2921no—
189C22⋊Q8—2921no—
190C22⋊Q8—1462no—
191C22⋊Q8—1462no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—2921no—
194C2×C4⋊C4—2921no—
195C2×C4⋊C4—2922no—
196C2×C4⋊C4—1462no—
197C2×C4⋊C4—1464no—
198C2×C22⋊C4—2921no—
199C2×C16—2922no—
200C2×C16—1462no—
201#107—21no—
202#107C2×#6440812yescomplemented
203#103—1462no—
204#103—1462no—
205C24—731no—
206C2×Q8—2921no—
207C2×Q8—2922no—
208C2×Q8—2922no—
209C2×Q8—1462no—
210C22×C4—41no—
211C22×C4—2922no—
212C22×C4—2924no—
213C22×C4—2921no—
214C22×C4—5842no—
215C22×C4—5842no—
216C22×C4—731no—
217C22×C4—731no—
218C22×C4—731no—
219C22×C4—731no—
220C22×C4—731no—
221C22×C4—1462no—
222C22×C4—1462no—
223C22×C4C22⋊#6440811yescharacteristic
224Q16—2922no—
225Q16—2922no—
226C2×C8—22no—
227C2×C8—2921no—
228C2×C8—2921no—
229C2×C8—2921no—
230C2×C8—2921no—
231C2×C8—731no—
232C2×C8—731no—
233C2×C8—1462no—
234C2×C8C22⋊#6440811yescharacteristic
235C2×C8C22×#6440811yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—41no—
238C4⋊C4—2922no—
239C4⋊C4—2922no—
240C4⋊C4—2924no—
241C4⋊C4—2922no—
242C4⋊C4—1464no—
243C22⋊C4—2922no—
244C22⋊C4—2922no—
245C16—2922no—
246C23—732no—
247C23—732no—
248C23—731no—
249C23—731no—
250C23—1462no—
251C23—1462no—
252C23#1455665311yescharacteristic
253Q8—2922no—
254Q8—2922no—
255Q8—5842no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—2924no—
259C2×C4—2922no—
260C2×C4—5842no—
261C2×C4—5848no—
262C2×C4—5842no—
263C2×C4—5842no—
264C2×C4—731no—
265C2×C4—731no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4—1461no—
270C2×C4—1461no—
271C2×C4—1461no—
272C2×C4—1461no—
273C2×C4#1455665012yes—
274C2×C4#1455665311yescharacteristic
275C2×C4#1455665612yes—
276C2×C4C2×C22⋊#6440811yescharacteristic
277C8—21no—
278C8—21no—
279C8—1461no—
280C8—1461no—
281C22—731no—
282C22—732no—
283C22—731no—
284C22—1461no—
285C22—1461no—
286C22—1464no—
287C22—1464no—
288C22—1461no—
289C22—1461no—
290C22#1798943912yes—
291C22#1798950012yes—
292C22#1798950312yes—
293C22C2×#1455665311yescharacteristic
294C4—82no—
295C4—5844no—
296C4—5842no—
297C4—1461no—
298C4—1461no—
299C4#1798943912yes—
300C4C2×#1455665011yescharacteristic
301C4C2×#1455665611yescharacteristic
302C2—1461no—
303C2—1462no—
304C2—1461no—
305C2#2262492014yesdirect product factor
306C2C2×#1798943911yescharacteristic
307C2C2×#1798950011yescharacteristic
308C2C2×#1798950311yescharacteristic
subgroupG/H
centerC23#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
FittingC2×#107×C73C4
socleC23×C73#102
prime radicalC23×D73D8
cosocleC2×C8×D73C23
hypercenterC2×#107#64408
hypocenterC73C2×#1348
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
3. #22624920C2direct (N × H)
11. C22×Dic292C4semidirect
16. #17988829C4semidirect
19. C2×#107×C73C4semidirect
31. C2×Dic292C2×C4semidirect
49. #107×C73C2×C4semidirect
148. C2×C73#1348semidirect
159. C73C2×#1348semidirect
169. C2×#107#64408semidirect
202. #107C2×#64408semidirect
305. C2#22624920direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179894822
C2×#179894612
#226249202
C2×#179888292
C2×#134873