Finite Group Databasebeta

← #28308633

Subgroups — #28308633

Order 13568.

382 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 86 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#20291079C211yesmaximal, characteristic, complemented
2#20291078C211yesmaximal, characteristic, complemented
3#20289668C212yesmaximal, complemented
4#20289667C212yesmaximal, complemented
5#436×D53C211yesmaximal, characteristic
6#16811544—21no—
7C4⋊#11719910C2212yescomplemented
8#16811518—21no—
9#16811518—21no—
10#16811512—21no—
11#16811511C2212yescomplemented
12C2×M16×D53—21no—
13C4×D4×D53C411yescharacteristic
14#16810702C411yescharacteristic
15#16810701C2211yescharacteristic
16#16810700C2211yescharacteristic
17#16810699—21no—
18#16810698—21no—
19C4⋊C8×D53C411yescharacteristic
20C22⋊C8×D53—21no—
21C4×C8×D53C2211yescharacteristic
22C53×#436C411yescharacteristic
23C2×#11719910—22no—
24C2×#11719910—22no—
25C2×D4×D53C811yescharacteristic
26C22×C4×D53—21no—
27C2×#11719917—21no—
28C2×#11719912—21no—
29C2×#11719912—21no—
30M16×D53—44no—
31C2×C8×D53—21no—
32C2×C8×D53D411yescharacteristic
33C2×C8×D53D411yescharacteristic
34C4×D212C2×C411yescharacteristic
35D4×Dic53C811yescharacteristic
36#13133179C811yescharacteristic
37C4×C22⋊D53—21no—
38#13133177—21no—
39C4⋊C4×D53C811yescharacteristic
40C22⋊C4×D53—21no—
41C42×D53C2×C411yescharacteristic
42C4⋊#88322C2×C411yescharacteristic
43#13133146—21no—
44C22⋊#88322—21no—
45#13133129—21no—
46#13133129—21no—
47C4×#88322C2311yescharacteristic
48C8×Dic53D411yescharacteristic
49C8×Dic53D411yescharacteristic
50C2×M16×C53—21no—
51C4×D4×C53C2×C411yescharacteristic
52C4⋊C8×C53C2×C411yescharacteristic
53C22⋊C8×C53—21no—
54C4×C8×C53C2×C411yescharacteristic
55C23×D53—21no—
56C2×D4×C53C2×C811yescharacteristic
57C22×C4×C53—21no—
58C2×C22⋊D53—21no—
59C22×Dic53—21no—
60D4×D53—42no—
61C2×D212C2×C811yescharacteristic
62C2×C4×D53—21no—
63C2×C4×D53—21no—
64C2×C4×D53—21no—
65C2×C4×D53—42no—
66C2×C4×D53—42no—
67C2×C4×D53C22⋊C411yescharacteristic
68C2×C4×D53C2×C811yescharacteristic
69C2×C4×D53M1611yescharacteristic
70M16×C53—42no—
71C2×C8×C53—21no—
72C2×C8×C53C22⋊C411yescharacteristic
73C2×C8×C53C22⋊C411yescharacteristic
74C4⋊C4×C53C2×C811yescharacteristic
75C22⋊C4×C53—21no—
76C42×C53C22×C411yescharacteristic
77C22⋊Dic53—21no—
78#11719921—21no—
79#11719921—21no—
80C4⋊Dic53C2×C811yescharacteristic
81#11719919—21no—
82C4×Dic53C22⋊C411yescharacteristic
83C4×Dic53C2×C811yescharacteristic
84C4×Dic53M1611yescharacteristic
85#11719917—42no—
86C2×#88322—21no—
87C2×#88322C2×D411yescharacteristic
88C2×#88322C2×D411yescharacteristic
89#11719912—42no—
90#11719912—42no—
91C8×D53—22no—
92C8×D53—41no—
93C8×D53—42no—
94#11719910—42no—
95#11719910—42no—
96C23×C53—21no—
97C22×D53—21no—
98C22×D53—42no—
99C22×D53—42no—
100C22×D53—42no—
101C22×D53C22⋊C811yescharacteristic
102D4×C53—41no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53C2×C22⋊C411yescharacteristic
107C2×C4×C53C22×C811yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C22⋊D53—42no—
110C2×Dic53—21no—
111C2×Dic53—42no—
112C2×Dic53C22⋊C811yescharacteristic
113C2×Dic53C22⋊C811yescharacteristic
114C2×Dic53C22⋊C811yescharacteristic
115D212—41no—
116C4×D53—22no—
117C4×D53—21no—
118C4×D53—21no—
119C4×D53—41no—
120C4×D53—42no—
121C4×D53—42no—
122C4×D53—41no—
123C4×D53—41no—
124C8×C53—21no—
125C8×C53—21no—
126C8×C53—41no—
127#88322—21no—
128#88322—41no—
129#88322AGL(1,ℤ/8)12yes—
130#11636—531nomaximal, Sylow
131C22×C53—41no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53C2×C22⋊C811yescharacteristic
135C2×D53—21no—
136C2×D53—21no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—84no—
141C2×D53#34912yes—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—41no—
145C4×C53#41811yescharacteristic
146C4×C53#42111yescharacteristic
147Dic53—21no—
148Dic53—21no—
149Dic53—21no—
150Dic53—41no—
151C2×#436—531no—
152#2163—531no—
153#2163—531no—
154#1330—532no—
155#1330—532no—
156C2×C53—81no—
157C2×C53#147811yescharacteristic
158C2×C53C2×#34911yescharacteristic
159C2×C53#211711yescharacteristic
160D53—22no—
161D53—81no—
162C22×M16—1061no—
163C2×C4×D4—531no—
164#436—531no—
165#436—531no—
166#436—531no—
167#436—1061no—
168#436—1061no—
169#436#615711yescharacteristic
170C2×C4⋊C8—531no—
171C2×C22⋊C8—1061no—
172C2×C4×C8—531no—
173C4⋊C16—532no—
174C4⋊C16—1061no—
175C22⋊C16—1061no—
176C22⋊C16—1061no—
177#346—532no—
178#346—1061no—
179C53#1163611yescharacteristic, complemented, Sylow
180C22×D4—531no—
181C23×C4—1061no—
182C2×M16—21no—
183C2×M16—1061no—
184C2×M16—1061no—
185C2×M16—1061no—
186C2×M16—2124no—
187C22×C8—531no—
188C22×C8—531no—
189C22×C8—1061no—
190C4×D4—531no—
191C4×D4—531no—
192C4×D4—531no—
193C4×D4—1061no—
194C4×D4—1061no—
195C4×D4C2×#615711yescharacteristic
196C2×C4⋊C4—531no—
197C2×C22⋊C4—1061no—
198C2×C42—531no—
199C2×C16—1062no—
200C2×C16—1062no—
201C4⋊C8—531no—
202C4⋊C8—1061no—
203C4⋊C8C2×#615711yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—1061no—
206C22⋊C8—1061no—
207C22⋊C8—1061no—
208C4×C8—531no—
209C4×C8—531no—
210C4×C8—531no—
211C4×C8C2×#615711yescharacteristic
212C24—1061no—
213C2×D4—531no—
214C2×D4—1061no—
215C2×D4—2122no—
216C2×D4D53⋊C811yescharacteristic
217C22×C4—21no—
218C22×C4—531no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4—2122no—
226C22×C4—2122no—
227M16—42no—
228M16—2122no—
229M16—2122no—
230M16—2122no—
231C2×C8—21no—
232C2×C8—531no—
233C2×C8—531no—
234C2×C8—1062no—
235C2×C8—1061no—
236C2×C8—2121no—
237C2×C8—2122no—
238C2×C8C22⋊#615711yescharacteristic
239C2×C8C22⋊#615711yescharacteristic
240C4⋊C4—531no—
241C4⋊C4—1061no—
242C4⋊C4D53⋊C811yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—1061no—
245C22⋊C4—1061no—
246C22⋊C4—1061no—
247C42—531no—
248C42—531no—
249C42—531no—
250C42C22×#615711yescharacteristic
251C16—2122no—
252C16—2122no—
253C23—21no—
254C23—531no—
255C23—1061no—
256C23—2122no—
257C23—2122no—
258C23—2122no—
259D4—41no—
260D4—2121no—
261D4—2122no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—531no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—2121no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4C2×C22⋊#615711yescharacteristic
279C2×C4C2×D53⋊C811yescharacteristic
280C2×C4D53⋊M1611yescharacteristic
281C8—21no—
282C8—21no—
283C8—41no—
284C8—532no—
285C8—1061no—
286C8—2121no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—532no—
291C22—1061no—
292C22—1061no—
293C22—4244no—
294C22—2121no—
295C22—2121no—
296C22—2121no—
297C22D53⋊(C22⋊C8)11yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—1061no—
302C4—1061no—
303C4—1061no—
304C4—2121no—
305C4#1681161511yescharacteristic
306C4#1681162011yescharacteristic
307C2—81no—
308C2—1062no—
309C2—4241no—
310C2#2029001511yescharacteristic
311C2D53⋊#34911yescharacteristic
312C2#2029095811yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC42C22×#6157
FittingC53×#436C4
socleC22×C53C2×C22⋊C8
prime radicalC2×D4×D53C8
cosocleC4×#88322C23
hypercenter#436#6157
hypocenterC53#11636
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #20291079C2semidirect
2. #20291078C2semidirect
3. #20289668C2semidirect
4. #20289667C2semidirect
7. C4⋊#11719910C22semidirect
11. #16811511C22semidirect
179. C53#11636semidirect

Maximal subgroups

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

subgroupindex
#202910792
#202910782
#202896682
#202896672
#436×D532
#1163653