Finite Group Databasebeta

← #28309416

Subgroups — #28309416

Order 13568.

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

normal subgroups only

This group has 314 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#20291078C211yesmaximal, characteristic
2#20290637C211yesmaximal, characteristic
3#20289704C214yesmaximal
4#20285302C211yesmaximal, characteristic
5#16811567—21no—
6#16811544—21no—
7#16811518C2212yes—
8#16811511—21no—
9#16811497—21no—
10#16811497C2212yes—
11#16811178—21no—
12#16811003C411yescharacteristic, complemented
13#16810975—21no—
14#16810865—21no—
15#16810865C2212yes—
16#16810717—21no—
17#16810717C412yescomplemented
18C4×C8×D53C2211yescharacteristic
19C53×#500C411yescharacteristic, complemented
20C2×C4⋊#6157—22no—
21C2×#11719910—22no—
22(C2×#6157)⋊C4—41no—
23(C2×#6157)⋊C4—41no—
24C2×Dic212—21no—
25C2×Dic212C2×C412yes—
26C2×C8×D53D411yescharacteristic
27C2×C8×D53C2311yescharacteristic
28#13133217—21no—
29#13133208—41no—
30#13133197—22no—
31#13133193—41no—
32Q8×Dic53—41no—
33#13133182—22no—
34#13133175—41no—
35C42×D53D411yescharacteristic
36#13133140—22no—
37#13133140—22no—
38C4×#88322D411yescharacteristic
39C8×Dic53C2×C411yescharacteristic
40C8×Dic53D411yescharacteristic
41C4⋊Q8×C53—21no—
42#107×C53—21no—
43#107×C53C2×C412yes—
44C4×C8×C53C2×C411yescharacteristic
45C22×#6157—42no—
46C2×Q8×C53—41no—
47C2×C4×D53D811yescharacteristic
48C2×C4×D53D811yescharacteristic
49C2×C4×D53C2×D411yescharacteristic
50C2×Dic106—22no—
51C4⋊#6157—42no—
52C4⋊#6157—42no—
53C2×C8×C53C22⋊C411yescharacteristic
54C2×C8×C53C22×C411yescharacteristic
55C4⋊C4×C53—22no—
56C4⋊C4×C53—41no—
57C42×C53C22⋊C411yescharacteristic
58#11719921—42no—
59C4⋊Dic53—41no—
60C4⋊Dic53—41no—
61C4⋊Dic53—42no—
62#11719919—42no—
63C4×Dic53—42no—
64C4×Dic53C22⋊C411yescharacteristic
65C4×Dic53SD1611yescharacteristic
66C4×Dic53SD1611yescharacteristic
67C2×#88322C22⋊C411yescharacteristic
68C2×#88322C2×D411yescharacteristic
69Dic212—42no—
70Dic212—42no—
71C8×D53—41no—
72C8×D53—41no—
73#11719910—42no—
74C22×D53C2×D811yescharacteristic
75C2×#6157—82no—
76C2×#6157—82no—
77C2×#6157—82no—
78Q8×C53—82no—
79C2×C4×C53—42no—
80C2×C4×C53#10211yescharacteristic
81C2×C4×C53#10211yescharacteristic
82C2×C4×C53C2×C22⋊C411yescharacteristic
83C2×Dic53—42no—
84C2×Dic53#10211yescharacteristic
85C2×Dic53#10211yescharacteristic
86C2×Dic53C2×SD1611yescharacteristic
87C4×D53—22no—
88C4×D53—21no—
89C4×D53—21no—
90C4×D53—41no—
91Dic106—42no—
92Dic106—42no—
93C8×C53—21no—
94C8×C53—21no—
95#88322—21no—
96#88322—21no—
97#11859—531nomaximal, Sylow
98C22×C53C2×#10211yescharacteristic
99C2×D53—21no—
100C2×D53—21no—
101C2×D53#51012yes—
102#6157—162no—
103C4×C53—21no—
104C4×C53—21no—
105C4×C53—82no—
106C4×C53#41011yescharacteristic
107C4×C53#42111yescharacteristic
108Dic53—21no—
109Dic53—21no—
110Dic53—21no—
111Dic53—82no—
112#3170—531no—
113#2163—531no—
114#1835—531no—
115#1348—534no—
116C2×C53#151711yescharacteristic
117C2×C53#214111yescharacteristic
118C2×C53#214511yescharacteristic
119D53—22no—
120#533—1061no—
121#500#615711yescharacteristic, complemented
122#494—531no—
123#487—1061no—
124#467—532no—
125#467—1061no—
126C4×Q16—532no—
127C4×Q16—1061no—
128C2×C4×C8—531no—
129#382—1061no—
130C4⋊C16—1061no—
131C22⋊C16—532no—
132#346—1061no—
133#340—532no—
134#340—1061no—
135C53#1185911yescharacteristic, complemented, Sylow
136C2×Q16—532no—
137C2×Q16—1061no—
138C22×C8—531no—
139C22×C8—531no—
140C4⋊Q8—21no—
141C4⋊Q8—1061no—
142#125—2121no—
143C22⋊Q8—1062no—
144C22⋊Q8—2121no—
145C4×Q8—1062no—
146C4×Q8—2121no—
147#117—2121no—
148C2×C4⋊C4—1062no—
149C2×C42—531no—
150C2×C16—1062no—
151#107—21no—
152#107C2×#615712yes—
153#103—1062no—
154#103—1062no—
155C4×C8—531no—
156C4×C8—531no—
157C4×C8—531no—
158C4×C8C2×#615711yescharacteristic
159#95—2121no—
160#95—2121no—
161C2×Q8—41no—
162C2×Q8—1062no—
163C22×C4—531no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—2122no—
167Q16—2122no—
168Q16—2122no—
169C2×C8—531no—
170C2×C8—531no—
171C2×C8—2121no—
172C2×C8—2121no—
173C2×C8C22⋊#615711yescharacteristic
174C2×C8C22×#615711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—2122no—
178C4⋊C4—2122no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C4⋊C4—2121no—
182C4⋊C4—2121no—
183C22⋊C4—2122no—
184C42—531no—
185C42—531no—
186C42—531no—
187C42—2122no—
188C42C22⋊#615711yescharacteristic
189C16—2122no—
190C23—531no—
191Q8—82no—
192Q8—2122no—
193Q8—2122no—
194C2×C4—42no—
195C2×C4—531no—
196C2×C4—531no—
197C2×C4—531no—
198C2×C4—1062no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—4242no—
202C2×C4—4242no—
203C2×C4—4242no—
204C2×C4—2121no—
205C2×C4—2122no—
206C2×C4#1313327911yescharacteristic
207C2×C4#1313327911yescharacteristic
208C2×C4C2×C22⋊#615711yescharacteristic
209C8—21no—
210C8—21no—
211C8—1061no—
212C8—1061no—
213C22—532no—
214C22—1061no—
215C22—1061no—
216C22C2×#1313327911yescharacteristic
217C4—21no—
218C4—21no—
219C4—82no—
220C4—8482no—
221C4—1061no—
222C4—1061no—
223C4—1061no—
224C4—4242no—
225C4#1681160411yescharacteristic
226C4#1681162211yescharacteristic
227C2—1062no—
228C2#2029014611yescharacteristic
229C2#2029102511yescharacteristic
230C2#2029103611yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#500C4
socleC22×C53C2×#102
prime radicalC22×D53C2×D8
cosocleC2×C8×D53C23
hypercenter#500#6157
hypocenterC53#11859
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #500

Complements

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

NH ≅ G/Ntype
12. #16811003C4semidirect
17. #16810717C4semidirect
19. C53×#500C4semidirect
121. #500#6157semidirect
135. C53#11859semidirect

Maximal subgroups

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

subgroupindex
#202910782
#202906372
#202897042
#202853022
#1185953