Finite Group Databasebeta

← #34011305

Subgroups — #34011305

Order 18688.

358 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 74 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
1#22626294C211yesmaximal, characteristic, complemented
2#22625853C211yesmaximal, characteristic, complemented
3#22624918C214yesmaximal, complemented
4#22620511C211yesmaximal, characteristic
5#17989531—21no—
6#17989508—21no—
7#17989482C2212yes—
8#17989475—21no—
9#17989461—21no—
10#17989461C2212yes—
11#17989138—21no—
12#17988964C411yescharacteristic, complemented
13#17988935—21no—
14#17988824—21no—
15#17988824C2212yes—
16#17988669—21no—
17#17988669C412yescomplemented
18C4×C8×D73C2211yescharacteristic
19#500×C73C411yescharacteristic, complemented
20C2×C4⋊#64408—22no—
21C2×#11924031—22no—
22(C2×#64408)⋊C4—41no—
23(C2×#64408)⋊C4—41no—
24C2×#11924029—41no—
25C2×D584—21no—
26C2×D584C2×C412yes—
27C2×C8×D73D411yescharacteristic
28C2×C8×D73C2311yescharacteristic
29#14556588—21no—
30#14556580—41no—
31#14556566—22no—
32#14556566—41no—
33#14556553—22no—
34#14556553—41no—
35C42×D73D411yescharacteristic
36#14556511—22no—
37#14556511—22no—
38C4×#10596831D411yescharacteristic
39C8×Dic73C2×C411yescharacteristic
40C8×Dic73D411yescharacteristic
41C4⋊Q8×C73—21no—
42#107×C73—21no—
43#107×C73C2×C412yes—
44C4×C8×C73C2×C411yescharacteristic
45C22×#64408—42no—
46C4⋊#64408—42no—
47C4⋊#64408—42no—
48#11924031—42no—
49#11924029—84no—
50C2×D292—22no—
51C2×D292—42no—
52C2×D292—41no—
53C2×D292—41no—
54C2×C4×D73—42no—
55C2×C4×D73D811yescharacteristic
56C2×C4×D73D811yescharacteristic
57C2×C4×D73C2×D411yescharacteristic
58D584—42no—
59D584—42no—
60C2×#10596831C22⋊C411yescharacteristic
61C2×#10596831C2×D411yescharacteristic
62C8×D73—41no—
63C8×D73—41no—
64#11924005—42no—
65#11924005—42no—
66C4×Dic73C22⋊C411yescharacteristic
67C4×Dic73SD1611yescharacteristic
68C4×Dic73SD1611yescharacteristic
69C2×Q8×C73—41no—
70C2×C8×C73C22⋊C411yescharacteristic
71C2×C8×C73C22×C411yescharacteristic
72C4⋊C4×C73—22no—
73C4⋊C4×C73—41no—
74C42×C73C22⋊C411yescharacteristic
75C22×D73—42no—
76C22×D73C2×D811yescharacteristic
77C2×#64408—82no—
78C2×#64408—82no—
79C2×#64408—82no—
80Q8×C73—82no—
81C2×C4×C73—42no—
82C2×C4×C73#10211yescharacteristic
83C2×C4×C73#10211yescharacteristic
84C2×C4×C73C2×C22⋊C411yescharacteristic
85C2×Dic73#10211yescharacteristic
86C2×Dic73#10211yescharacteristic
87C2×Dic73C2×SD1611yescharacteristic
88D292—42no—
89D292—42no—
90D292—82no—
91D292—82no—
92D292—84no—
93C4×D73—22no—
94C4×D73—21no—
95C4×D73—21no—
96C4×D73—41no—
97C4×D73—84no—
98C8×C73—21no—
99C8×C73—21no—
100#10596831—21no—
101#10596831—21no—
102C22×C73C2×#10211yescharacteristic
103C2×D73—21no—
104C2×D73—21no—
105C2×D73—82no—
106C2×D73—82no—
107C2×D73—82no—
108C2×D73#50912yes—
109#64408—162no—
110C4×C73—21no—
111C4×C73—21no—
112C4×C73—82no—
113C4×C73#41011yescharacteristic
114C4×C73#42111yescharacteristic
115Dic73—21no—
116Dic73—21no—
117Dic73—21no—
118#11858—731nomaximal, Sylow
119C2×C73#151711yescharacteristic
120C2×C73#214011yescharacteristic
121C2×C73#214511yescharacteristic
122D73—22no—
123D73—162no—
124#3169—731no—
125#2163—731no—
126#1835—731no—
127#1347—734no—
128C73#1185811yescharacteristic, complemented, Sylow
129#532—1461no—
130#500#6440811yescharacteristic, complemented
131#493—731no—
132#486—1461no—
133#466—732no—
134#466—1461no—
135C4×D8—732no—
136C4×D8—1461no—
137C2×C4×C8—731no—
138#382—1461no—
139C4⋊C16—1461no—
140C22⋊C16—732no—
141#346—1461no—
142#340—732no—
143#340—1461no—
144C2×#41—2921no—
145C2×D8—732no—
146C2×D8—1461no—
147C22×C8—731no—
148C22×C8—731no—
149C4⋊Q8—21no—
150#127—1461no—
151#124—2921no—
152#121—2921no—
153#121—1462no—
154C4×D4—2921no—
155C4×D4—1462no—
156C2×C4⋊C4—1462no—
157C2×C42—731no—
158C2×C16—1462no—
159#107—21no—
160#107C2×#6440812yes—
161#102—1462no—
162#102—1462no—
163C4×C8—731no—
164C4×C8—731no—
165C4×C8—731no—
166C4×C8C2×#6440811yescharacteristic
167#95—2921no—
168#95—2921no—
169#41—5844no—
170C2×Q8—41no—
171C2×D4—2922no—
172C2×D4—2921no—
173C2×D4—2921no—
174C2×D4—1462no—
175C22×C4—2922no—
176C22×C4—2922no—
177C22×C4—731no—
178C22×C4—731no—
179C22×C4—731no—
180D8—2922no—
181D8—2922no—
182C2×C8—2921no—
183C2×C8—2921no—
184C2×C8—731no—
185C2×C8—731no—
186C2×C8C22⋊#6440811yescharacteristic
187C2×C8C22×#6440811yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—2922no—
191C4⋊C4—2922no—
192C22⋊C4—2922no—
193C22⋊C4—2922no—
194C42—731no—
195C42—731no—
196C42—731no—
197C42C22⋊#6440811yescharacteristic
198C16—2922no—
199C23—2922no—
200C23—731no—
201Q8—82no—
202D4—2922no—
203D4—2922no—
204D4—5842no—
205D4—5842no—
206D4—5844no—
207C2×C4—42no—
208C2×C4—2921no—
209C2×C4—5844no—
210C2×C4—5842no—
211C2×C4—5842no—
212C2×C4—5842no—
213C2×C4—731no—
214C2×C4—731no—
215C2×C4—731no—
216C2×C4—1462no—
217C2×C4—1461no—
218C2×C4—1461no—
219C2×C4#1455665311yescharacteristic
220C2×C4#1455665311yescharacteristic
221C2×C4C2×C22⋊#6440811yescharacteristic
222C8—21no—
223C8—21no—
224C8—1461no—
225C8—1461no—
226C22—5842no—
227C22—5842no—
228C22—5842no—
229C22—732no—
230C22—1461no—
231C22—1461no—
232C22C2×#1455665311yescharacteristic
233C4—21no—
234C4—21no—
235C4—82no—
236C4—11682no—
237C4—1461no—
238C4—1461no—
239C4—1461no—
240C4#1798956811yescharacteristic
241C4#1798958611yescharacteristic
242C2—11682no—
243C2—1462no—
244C2#2262536211yescharacteristic
245C2#2262624011yescharacteristic
246C2#2262625311yescharacteristic
subgroupG/H
centerC22C2×#14556653
commutatorC2×C8×C73C22×C4
FrattiniC2×C8C22×#64408
Fitting#500×C73C4
socleC22×C73C2×#102
prime radical#22620511C2
cosocleC2×C8×D73C23
hypercenter#500#64408
hypocenterC73#11858
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #22626294C2semidirect
2. #22625853C2semidirect
3. #22624918C2semidirect
12. #17988964C4semidirect
17. #17988669C4semidirect
19. #500×C73C4semidirect
128. C73#11858semidirect
130. #500#64408semidirect

Maximal subgroups

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

subgroupindex
#226262942
#226258532
#226249182
#226205112
#1185873