Finite Group Databasebeta

← #34010645

Subgroups — #34010645

Order 18688.

332 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 84 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×#17989473C211yesmaximal, characteristic, complemented
2#22625493C211yesmaximal, characteristic, complemented
3#22624887C212yesmaximal, complemented
4#22624886C212yesmaximal, complemented
5#22620502C211yesmaximal, characteristic
6C22×#11924031—21no—
7#17989608—21no—
8#17989473—21no—
9#17989473C2212yes—
10#17989450C2212yes—
11#17989449—21no—
12#17989123—21no—
13#17988988C411yescharacteristic
14#17988972C411yescharacteristic
15#17988943—21no—
16#17988823C411yescharacteristic
17#17988818—21no—
18Q16×Dic73C2211yescharacteristic
19#17988672—21no—
20#17988669C2211yescharacteristic
21C2×C8×Dic73C2211yescharacteristic
22#467×C73C411yescharacteristic
23C2×#11924031—24no—
24C2×#11924031—22no—
25C4⋊#10596833—22no—
26C22⋊#10596833—41no—
27#14556646—41no—
28C2×#11924015—21no—
29C2×D584C2×C411yescharacteristic
30C22×#10596831D411yescharacteristic
31#14556586—41no—
32#14556582—21no—
33#14556580—21no—
34#14556575—41no—
35#14556565—21no—
36Q8×Dic73—21no—
37#14556553—21no—
38#14556551—41no—
39#14556549—41no—
40C2×C4×Dic73D411yescharacteristic
41#14556525C2×C411yescharacteristic
42#14556522—21no—
43#14556515—21no—
44#14556513—21no—
45#14556511—21no—
46C8×Dic73C2×C411yescharacteristic
47C8×Dic73D411yescharacteristic
48C8×Dic73D411yescharacteristic
49C8×Dic73C2311yescharacteristic
50C2×Q16×C73C2×C411yescharacteristic
51C22×C8×C73C2×C411yescharacteristic
52C22⋊Q8×C73—21no—
53#107×C73C2×C411yescharacteristic
54#103×C73—21no—
55C2×#10596833—42no—
56#11924031—24no—
57C2×D292—21no—
58C2×C22⋊D73—41no—
59C22×Dic73D811yescharacteristic
60C22×Dic73SD1611yescharacteristic
61C2×C4×D73—41no—
62#11924015—42no—
63D584—22no—
64C2×#10596831—22no—
65C2×#10596831C22⋊C411yescharacteristic
66C2×#10596831C2×D411yescharacteristic
67C4⋊Dic73—21no—
68C4⋊Dic73—41no—
69C4⋊Dic73—41no—
70#11924007—41no—
71#11924005—41no—
72#11924005—41no—
73#11924005—41no—
74#11924005—41no—
75C4×Dic73—41no—
76C4×Dic73C22⋊C411yescharacteristic
77C4×Dic73D811yescharacteristic
78C4×Dic73SD1611yescharacteristic
79C4×Dic73C2×D411yescharacteristic
80C2×Q8×C73—21no—
81C22×C4×C73C22⋊C411yescharacteristic
82Q16×C73—22no—
83C2×C8×C73—22no—
84C2×C8×C73C22⋊C411yescharacteristic
85C2×C8×C73C22×C411yescharacteristic
86C4⋊C4×C73—21no—
87C4⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C23×C73#10211yescharacteristic
90C22×D73—41no—
91Q8×C73—41no—
92Q8×C73—41no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—41no—
96C2×C4×C73—41no—
97C2×C4×C73#10211yescharacteristic
98C2×C4×C73C2×C22⋊C411yescharacteristic
99C22⋊D73—82no—
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—41no—
103C2×Dic73#10211yescharacteristic
104C2×Dic73#10211yescharacteristic
105C2×Dic73C2×D811yescharacteristic
106C2×Dic73C2×SD1611yescharacteristic
107D292—41no—
108D292—41no—
109C4×D73—82no—
110#10596833—82no—
111C8×C73—21no—
112C8×C73#13112yes—
113#10596831—21no—
114#10596831#10412yes—
115C22×C73—21no—
116C22×C73—21no—
117C22×C73#35912yes—
118C22×C73C2×#10211yescharacteristic
119C2×D73—81no—
120C2×D73—81no—
121C2×D73—81no—
122C4×C73—21no—
123C4×C73—81no—
124C4×C73—81no—
125C4×C73#40811yescharacteristic
126C4×C73C2×#10411yescharacteristic
127Dic73—21no—
128Dic73—21no—
129Dic73—21no—
130Dic73—81no—
131Dic73#50812yes—
132#11673—731nomaximal, Sylow
133C2×C73—22no—
134C2×C73#149211yescharacteristic
135C2×C73#213811yescharacteristic
136C2×C73C2×#35911yescharacteristic
137D73—161no—
138#3168—731no—
139C2×C4×C16—731no—
140#1568—731no—
141#1332—732no—
142#1331—732no—
143C73#1167311yescharacteristic, complemented, Sylow
144#529—1461no—
145C22×C16—1461no—
146#499—731no—
147#495—731no—
148#488—1461no—
149#467#6440811yescharacteristic
150#466—731no—
151#465—1461no—
152C4×Q16—731no—
153C4×SD16—1461no—
154C4×D8—731no—
155#432—1461no—
156C2×C4×C8—731no—
157C4×C16—732no—
158C4×C16—1461no—
159#335—732no—
160#334—1461no—
161C2×Q16C2×#6440811yescharacteristic
162C2×SD16—1461no—
163C2×D8—731no—
164C22×C8—731no—
165C22×C8C2×#6440811yescharacteristic
166#126—2921no—
167#125—1461no—
168#124—1461no—
169#123—2921no—
170C22⋊Q8—21no—
171#121—1461no—
172C4×Q8—1461no—
173C4×D4—2921no—
174C4×D4—1461no—
175#117—2921no—
176C2×C42—731no—
177C2×C16—1462no—
178C2×C16—1464no—
179#107—731no—
180#107C2×#6440811yescharacteristic
181#106—1461no—
182C4⋊C8—1462no—
183#103—21no—
184#103—1461no—
185#102—1461no—
186#102—1461no—
187C22⋊C8—2921no—
188#97—2921no—
189C4×C8—731no—
190C4×C8—731no—
191C4×C8—731no—
192C4×C8—731no—
193C2×Q8—21no—
194C2×D4—2921no—
195C2×D4—1461no—
196C22×C4—2921no—
197C22×C4—731no—
198C22×C4—731no—
199C22×C4C22⋊#6440811yescharacteristic
200Q16—22no—
201SD16—2922no—
202D8—1462no—
203C2×C8—22no—
204C2×C8—2922no—
205C2×C8—731no—
206C2×C8—731no—
207C2×C8—1462no—
208C2×C8C22⋊#6440811yescharacteristic
209C2×C8C22×#6440811yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—2921no—
213C4⋊C4—2921no—
214C4⋊C4—2921no—
215C4⋊C4—1461no—
216C22⋊C4—41no—
217C22⋊C4—2921no—
218C22⋊C4—2921no—
219C22⋊C4—2921no—
220C22⋊C4—2921no—
221C42—2921no—
222C42—731no—
223C42—731no—
224C42—731no—
225C42—731no—
226C16—1464no—
227C23—2921no—
228C23#1455665311yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—2921no—
232D4—2921no—
233D4—5842no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—2921no—
239C2×C4—5842no—
240C2×C4—731no—
241C2×C4—731no—
242C2×C4—731no—
243C2×C4—731no—
244C2×C4—1462no—
245C2×C4—1462no—
246C2×C4#1455665311yescharacteristic
247C2×C4C2×C22⋊#6440811yescharacteristic
248C8—21no—
249C8—5842no—
250C8—732no—
251C8—1461no—
252C8#1455667812yes—
253C22—21no—
254C22—21no—
255C22—5841no—
256C22—5841no—
257C22—5841no—
258C22#1798950212yes—
259C22C2×#1455665311yescharacteristic
260C4—21no—
261C4—81no—
262C4—81no—
263C4—5841no—
264C4—732no—
265C4—1461no—
266C4—1461no—
267C4—1461no—
268C4#1798956011yescharacteristic
269C4D73⋊#10411yescharacteristic
270C2—22no—
271C2—11681no—
272C2#2262527611yescharacteristic
273C2#2262623811yescharacteristic
274C2C2×#1798950211yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#467×C73C4
socleC22×C73C2×#102
prime radical#17988823C4
cosocleC8×Dic73C23
hypercenter#467#64408
hypocenterC73#11673
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#17989473C2semidirect
2. #22625493C2semidirect
3. #22624887C2semidirect
4. #22624886C2semidirect
143. C73#11673semidirect

Maximal subgroups

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

subgroupindex
C2×#179894732
#226254932
#226248872
#226248862
#226205022
#1167373