Finite Group Databasebeta

← #28308604

Subgroups — #28308604

Order 13568.

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

normal subgroups only

This group has 306 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
1#20291078C211yesmaximal, characteristic, complemented
2#20291048C211yesmaximal, characteristic, complemented
3#20289684C212yesmaximal, complemented
4#20289667C212yesmaximal, complemented
5#20283687C211yesmaximal, characteristic
6C2×#13133292—21no—
7#16811544—21no—
8#16811544—21no—
9C4⋊#11719910C2212yes—
10#16811518—21no—
11#16811511C2212yes—
12#16811155C411yescharacteristic
13#16810750C2211yescharacteristic
14C8×Dic106C411yescharacteristic
15#16810701C2211yescharacteristic
16#16810697—21no—
17C8×C22⋊D53—21no—
18#16810681—21no—
19#16810679C411yescharacteristic
20#16810578—21no—
21C4×C8×D53C2211yescharacteristic
22C8×D4×C53C411yescharacteristic
23#13133292—42no—
24C2×#11719910—22no—
25C2×#11719910—22no—
26C2×#11719917—21no—
27C2×C8×D53D411yescharacteristic
28C2×C8×D53D411yescharacteristic
29#13133210—21no—
30#13133208C811yescharacteristic
31#13133204C811yescharacteristic
32#13133199—21no—
33#13133197C811yescharacteristic
34#13133188C811yescharacteristic
35C4×Dic106C2×C411yescharacteristic
36C4×C22⋊D53—21no—
37#13133176—21no—
38C42×D53C2×C411yescharacteristic
39C4⋊#88322C2×C411yescharacteristic
40#13133146—21no—
41#13133146—21no—
42C22⋊#88322—21no—
43#13133129—21no—
44#13133127—21no—
45#13133127—21no—
46C4×#88322C2311yescharacteristic
47C8×Dic53—21no—
48C8×Dic53D411yescharacteristic
49C8×Dic53D411yescharacteristic
50C22×C8×C53—21no—
51C4×D4×C53C2×C411yescharacteristic
52C4⋊C8×C53C2×C411yescharacteristic
53C22⋊C8×C53—21no—
54C4×C8×C53C2×C411yescharacteristic
55C2×D4×C53C2×C811yescharacteristic
56C22×C4×C53—21no—
57C2×C22⋊D53—21no—
58C2×C4×D53C22⋊C411yescharacteristic
59C2×C4×D53C2×C811yescharacteristic
60C2×C4×D53M1611yescharacteristic
61C2×Dic106C2×C811yescharacteristic
62C2×C8×C53—21no—
63C2×C8×C53—42no—
64C2×C8×C53C22⋊C411yescharacteristic
65C2×C8×C53C22⋊C411yescharacteristic
66C4⋊C4×C53C2×C811yescharacteristic
67C22⋊C4×C53—21no—
68C42×C53C22×C411yescharacteristic
69C22⋊Dic53—21no—
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72#11719921—21no—
73C4⋊Dic53—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53C2×C811yescharacteristic
76#11719919—21no—
77#11719919—21no—
78C4×Dic53—21no—
79C4×Dic53C22⋊C411yescharacteristic
80C4×Dic53C2×C811yescharacteristic
81C4×Dic53M1611yescharacteristic
82#11719917—41no—
83#11719917—41no—
84C2×#88322—21no—
85C2×#88322C2×D411yescharacteristic
86C2×#88322C2×D411yescharacteristic
87C8×D53—21no—
88C8×D53—21no—
89C8×D53—41no—
90#11719910—42no—
91#11719910—42no—
92C23×C53—21no—
93C22×D53C22⋊C811yescharacteristic
94D4×C53—41no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53C22×C811yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C22⋊D53—42no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53C22⋊C811yescharacteristic
105C2×Dic53C22⋊C811yescharacteristic
106C2×Dic53C22⋊C811yescharacteristic
107C4×D53—21no—
108C4×D53—21no—
109C4×D53—41no—
110C4×D53—41no—
111Dic106—41no—
112C8×C53—21no—
113C8×C53—21no—
114C8×C53—41no—
115#88322—21no—
116#88322—41no—
117#88322AGL(1,ℤ/8)11yescharacteristic
118#88322#13711yescharacteristic
119#11627—531nomaximal, Sylow
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53C2×C22⋊C811yescharacteristic
124C2×D53—21no—
125C2×D53—21no—
126C2×D53—21no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—41no—
130C4×C53#41711yescharacteristic
131C4×C53#42111yescharacteristic
132Dic53—21no—
133Dic53—21no—
134Dic53—21no—
135Dic53—41no—
136Dic53—41no—
137#2966—531no—
138#2163—531no—
139#2150—531no—
140#1338—532no—
141#1330—532no—
142C2×C53—81no—
143C2×C53#147711yescharacteristic
144C2×C53#211511yescharacteristic
145C2×C53#211711yescharacteristic
146D53—41no—
147#529—531no—
148C2×M32—1061no—
149#446—531no—
150C8×Q8—531no—
151#436—531no—
152#435—1061no—
153C8×D4—1061no—
154C8×D4#615711yescharacteristic
155#433—1061no—
156#432—531no—
157#405—1061no—
158C2×C4×C8—531no—
159C4⋊C16—532no—
160C4⋊C16—1061no—
161C4⋊C16—1061no—
162C22⋊C16—1061no—
163#346—532no—
164C53#1162711yescharacteristic, complemented, Sylow
165C2×M16—1061no—
166C22×C8—21no—
167C22×C8—531no—
168C22×C8—531no—
169#126—1061no—
170#125—531no—
171#124—531no—
172#123—1061no—
173C22⋊Q8—531no—
174#121—531no—
175C4×Q8—531no—
176C4×D4—1061no—
177C4×D4C2×#615711yescharacteristic
178#117—1061no—
179C2×C42—531no—
180M32—2122no—
181C2×C16—1062no—
182C2×C16—1062no—
183C4⋊C8—531no—
184C4⋊C8—1061no—
185C4⋊C8—1061no—
186C4⋊C8C2×#615711yescharacteristic
187C22⋊C8—21no—
188C22⋊C8—1061no—
189C22⋊C8—1061no—
190#97—1061no—
191#97—1061no—
192C4×C8—531no—
193C4×C8—531no—
194C4×C8—531no—
195C4×C8—1061no—
196C4×C8C2×#615711yescharacteristic
197C2×Q8—531no—
198C2×D4—1061no—
199C2×D4D53⋊C811yescharacteristic
200C22×C4—21no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—531no—
204M16—2121no—
205M16—2121no—
206C2×C8—21no—
207C2×C8—42no—
208C2×C8—531no—
209C2×C8—531no—
210C2×C8—1061no—
211C2×C8—1061no—
212C2×C8—1061no—
213C2×C8—2121no—
214C2×C8C22⋊#615711yescharacteristic
215C2×C8C22⋊#615711yescharacteristic
216C4⋊C4—531no—
217C4⋊C4—1061no—
218C4⋊C4—1061no—
219C4⋊C4—1061no—
220C4⋊C4—1061no—
221C4⋊C4D53⋊C811yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—1061no—
224C22⋊C4—1061no—
225C22⋊C4—1061no—
226C22⋊C4—1061no—
227C42—531no—
228C42—531no—
229C42—531no—
230C42—1061no—
231C42C22×#615711yescharacteristic
232C16—2122no—
233C16—2122no—
234C23—21no—
235C23—531no—
236Q8—2121no—
237D4—41no—
238D4—2122no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—531no—
243C2×C4—531no—
244C2×C4—531no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4C2×C22⋊#615711yescharacteristic
252C2×C4C2×D53⋊C811yescharacteristic
253C2×C4D53⋊M1611yescharacteristic
254C8—21no—
255C8—21no—
256C8—41no—
257C8—531no—
258C8—531no—
259C8—1061no—
260C8—2121no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—1061no—
265C22—1061no—
266C22—1061no—
267C22D53⋊(C22⋊C8)11yescharacteristic
268C4—21no—
269C4—21no—
270C4—41no—
271C4—1061no—
272C4—1061no—
273C4—1061no—
274C4—2121no—
275C4—2121no—
276C4#1681161411yescharacteristic
277C4#1681162111yescharacteristic
278C2—81no—
279C2—2121no—
280C2#2029001111yescharacteristic
281C2#2029095211yescharacteristic
282C2#2029095811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC42C22×#6157
FittingC8×D4×C53C4
socleC22×C53C2×C22⋊C8
prime radical#13133188C8
cosocleC4×#88322C23
hypercenterC8×D4#6157
hypocenterC53#11627
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20291078C2semidirect
2. #20291048C2semidirect
3. #20289684C2semidirect
4. #20289667C2semidirect
164. C53#11627semidirect

Maximal subgroups

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

subgroupindex
#202910782
#202910482
#202896842
#202896672
#202836872
#1162753