Finite Group Databasebeta

← #28308618

Subgroups — #28308618

Order 13568.

422 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 96 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#20291074C211yesmaximal, characteristic, complemented
3#20289666C212yesmaximal, complemented
4D4⋊#11719910C212yesmaximal, complemented
5C8×D4×D53C211yesmaximal, characteristic
6#16811544—21no—
7C4⋊#11719910C2212yescomplemented
8#16811518—21no—
9#16811518—21no—
10#16811512—21no—
11C4×#11719910C2212yescomplemented
12C22×C8×D53—21no—
13C4×D4×D53C411yescharacteristic
14C8×D212C411yescharacteristic
15D4×#88322C2211yescharacteristic
16#16810688C2211yescharacteristic
17C8×C22⋊D53—21no—
18#16810686—21no—
19C4⋊C8×D53C411yescharacteristic
20C22⋊C8×D53—21no—
21C4×C8×D53C2211yescharacteristic
22C8×D4×C53C411yescharacteristic
23C2×#11719910—22no—
24C2×#11719910—22no—
25C2×D4×D53C811yescharacteristic
26C22×C4×D53—21no—
27C22×#88322—21no—
28C2×C8×D53—21no—
29C2×C8×D53—21no—
30C2×C8×D53—21no—
31C2×C8×D53—44no—
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
50C22×C8×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—22no—
61D4×D53—22no—
62C2×D212C2×C811yescharacteristic
63C2×C4×D53—21no—
64C2×C4×D53—21no—
65C2×C4×D53—21no—
66C2×C4×D53—42no—
67C2×C4×D53—42no—
68C2×C4×D53C22⋊C411yescharacteristic
69C2×C4×D53C2×C811yescharacteristic
70C2×C4×D53M1611yescharacteristic
71C2×C8×C53—21no—
72C2×C8×C53—42no—
73C2×C8×C53C22⋊C411yescharacteristic
74C2×C8×C53C22⋊C411yescharacteristic
75C4⋊C4×C53C2×C811yescharacteristic
76C22⋊C4×C53—21no—
77C42×C53C22×C411yescharacteristic
78C22⋊Dic53—21no—
79#11719921—21no—
80#11719921—21no—
81C4⋊Dic53C2×C811yescharacteristic
82#11719919—21no—
83C4×Dic53C22⋊C411yescharacteristic
84C4×Dic53C2×C811yescharacteristic
85C4×Dic53M1611yescharacteristic
86C2×#88322—21no—
87C2×#88322—42no—
88C2×#88322C2×D411yescharacteristic
89C2×#88322C2×D411yescharacteristic
90C8×D53—22no—
91C8×D53—41no—
92C8×D53—42no—
93C8×D53—42no—
94C8×D53—42no—
95#11719910—42no—
96#11719910—42no—
97C23×C53—21no—
98C22×D53—24no—
99C22×D53—21no—
100C22×D53—42no—
101C22×D53—42no—
102C22×D53C22⋊C811yescharacteristic
103D4×C53—21no—
104D4×C53M3212yes—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53C2×C22⋊C411yescharacteristic
109C2×C4×C53C22×C811yescharacteristic
110C2×C4×C53C2×M1611yescharacteristic
111C22⋊D53—24no—
112C2×Dic53—21no—
113C2×Dic53—42no—
114C2×Dic53C22⋊C811yescharacteristic
115C2×Dic53C22⋊C811yescharacteristic
116C2×Dic53C22⋊C811yescharacteristic
117D212—21no—
118D212M3212yes—
119C4×D53—22no—
120C4×D53—21no—
121C4×D53—21no—
122C4×D53—21no—
123C4×D53—21no—
124C4×D53—42no—
125C4×D53—42no—
126C4×D53—41no—
127C4×D53—41no—
128C8×C53—21no—
129C8×C53—21no—
130C8×C53—41no—
131#88322—21no—
132#88322—41no—
133#88322AGL(1,ℤ/8)12yes—
134#11631—531nomaximal, Sylow
135C22×C53—22no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53C2×C22⋊C811yescharacteristic
139C2×D53—21no—
140C2×D53—21no—
141C2×D53—22no—
142C2×D53—48no—
143C2×D53—41no—
144C2×D53—41no—
145C2×D53#35012yes—
146C4×C53—21no—
147C4×C53—41no—
148C4×C53#41811yescharacteristic
149C4×C53#42111yescharacteristic
150C4×C53C2×M3211yescharacteristic
151C4×C53C2×M3211yescharacteristic
152Dic53—21no—
153Dic53—21no—
154Dic53—21no—
155Dic53—41no—
156C2×C8×D4—531no—
157#2163—531no—
158#2162—531no—
159D4⋊C16—532no—
160D4⋊C16—532no—
161C2×C53—42no—
162C2×C53#147811yescharacteristic
163C2×C53#211211yescharacteristic
164C2×C53C2×#35011yescharacteristic
165D53—22no—
166D53—42no—
167C23×C8—1061no—
168C2×C4×D4—531no—
169C8×D4—531no—
170C8×D4—531no—
171C8×D4—531no—
172C8×D4—1061no—
173C8×D4—1061no—
174C8×D4#615711yescharacteristic
175C2×C4⋊C8—531no—
176C2×C22⋊C8—1061no—
177C2×C4×C8—531no—
178C4⋊C16—532no—
179C4⋊C16—1061no—
180C22⋊C16—1061no—
181C22⋊C16—1061no—
182#346—1061no—
183C4×C16—532no—
184C53#1163111yescharacteristic, complemented, Sylow
185C22×D4—531no—
186C23×C4—1061no—
187C22×C8—21no—
188C22×C8—531no—
189C22×C8—531no—
190C22×C8—1061no—
191C22×C8—1061no—
192C22×C8—1061no—
193C22×C8—1061no—
194C22×C8—2124no—
195C4×D4—531no—
196C4×D4—531no—
197C4×D4—531no—
198C4×D4—1061no—
199C4×D4—1061no—
200C4×D4C2×#615711yescharacteristic
201C2×C4⋊C4—531no—
202C2×C22⋊C4—1061no—
203C2×C42—531no—
204C2×C16—1062no—
205C2×C16—1062no—
206C4⋊C8—531no—
207C4⋊C8—1061no—
208C4⋊C8C2×#615711yescharacteristic
209C22⋊C8—21no—
210C22⋊C8—1061no—
211C22⋊C8—1061no—
212C22⋊C8—1061no—
213C4×C8—531no—
214C4×C8—531no—
215C4×C8—531no—
216C4×C8C2×#615711yescharacteristic
217C24—1061no—
218C2×D4—531no—
219C2×D4—1061no—
220C2×D4—1062no—
221C2×D4—1062no—
222C2×D4D53⋊C811yescharacteristic
223C22×C4—21no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4—1061no—
231C22×C4—2122no—
232C22×C4—2122no—
233C2×C8—21no—
234C2×C8—42no—
235C2×C8—531no—
236C2×C8—531no—
237C2×C8—1062no—
238C2×C8—1061no—
239C2×C8—2121no—
240C2×C8—2122no—
241C2×C8—2122no—
242C2×C8—2122no—
243C2×C8—2122no—
244C2×C8C22⋊#615711yescharacteristic
245C2×C8C22⋊#615711yescharacteristic
246C4⋊C4—531no—
247C4⋊C4—1061no—
248C4⋊C4D53⋊C811yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—1061no—
251C22⋊C4—1061no—
252C22⋊C4—1061no—
253C42—531no—
254C42—531no—
255C42—531no—
256C42C22×#615711yescharacteristic
257C16—2122no—
258C16—2122no—
259C23—21no—
260C23—531no—
261C23—1064no—
262C23—1061no—
263C23—2122no—
264C23—2122no—
265D4—21no—
266D4—532no—
267D4—1064no—
268D4—1061no—
269D4#1313329212yes—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—531no—
274C2×C4—531no—
275C2×C4—531no—
276C2×C4—1062no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4C2×C22⋊#615711yescharacteristic
288C2×C4C2×D53⋊C811yescharacteristic
289C2×C4D53⋊M1611yescharacteristic
290C8—21no—
291C8—21no—
292C8—41no—
293C8—532no—
294C8—1061no—
295C8—2121no—
296C22—22no—
297C22—41no—
298C22—41no—
299C22—532no—
300C22—1061no—
301C22—1061no—
302C22—1062no—
303C22—2128no—
304C22—2121no—
305C22—2121no—
306C22D53⋊(C22⋊C8)11yescharacteristic
307C4—21no—
308C4—41no—
309C4—1061no—
310C4—1061no—
311C4—1061no—
312C4—2121no—
313C4#1681161511yescharacteristic
314C4#1681162011yescharacteristic
315C4D53⋊M3211yescharacteristic
316C4C2×#1313329211yescharacteristic
317C2—42no—
318C2—1062no—
319C2—2122no—
320C2#2029001511yescharacteristic
321C2#2029094311yescharacteristic
322C2D53⋊#35011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC42C22×#6157
FittingC8×D4×C53C4
socleC22×C53C2×C22⋊C8
prime radicalC2×D4×D53C8
cosocleC4×#88322C23
hypercenterC8×D4#6157
hypocenterC53#11631
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20291079C2semidirect
2. #20291074C2semidirect
3. #20289666C2semidirect
4. D4⋊#11719910C2semidirect
7. C4⋊#11719910C22semidirect
11. C4×#11719910C22semidirect
184. C53#11631semidirect

Maximal subgroups

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

subgroupindex
#202910792
#202910742
#202896662
D4⋊#117199102
C8×D4×D532
#1163153