Finite Group Databasebeta

← #27294825

Subgroups — #27294825

Order 13568.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#16810562C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272741C214yesmaximal, complemented
4C53×#1821C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic
6#16810587—21no—
7#16810562—21no—
8#16810562—21no—
9#16810562C412yes—
10#16810562C2212yes—
11C2×#13133124—21no—
12C2×#13133124—21no—
13C2×#13133124C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#343—21no—
19C53×#343C2212yes—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C2×#11719917—41no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53C2×C412yes—
27C22⋊#88322—22no—
28C22⋊#88322—41no—
29#13133124—21no—
30#13133124—22no—
31#13133124—22no—
32#13133124—22no—
33#13133124—22no—
34#13133124C2×C412yes—
35C23×C4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C2×M16×C53—22no—
39C2×C4⋊C4×C53—41no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53C2311yescharacteristic
45C22⋊C8×C53—41no—
46C22⋊C8×C53—41no—
47C24×C53C22⋊C411yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—41no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53—44no—
69C22×Dic53—42no—
70C22×Dic53—41no—
71C22×Dic53—41no—
72M16×C53—42no—
73M16×C53—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81#11719917—82no—
82C2×#88322—42no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53#9911yescharacteristic
88C23×C53#9911yescharacteristic
89C23×C53C2×C22⋊C411yescharacteristic
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—81no—
109C2×C4×C53#13712yes—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—44no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—84no—
116C2×Dic53—82no—
117C2×Dic53—82no—
118C8×C53—82no—
119#88322—82no—
120#11814—531nomaximal, Sylow
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53#35212yes—
130C22×C53#35512yes—
131C22×C53C2×#9911yescharacteristic
132C22×C53#41911yescharacteristic
133C22×C53#42111yescharacteristic
134C4×C53—22no—
135C4×C53—41no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—41no—
139C4×C53—41no—
140Dic53—82no—
141Dic53—82no—
142C2×#400—531no—
143#1821D5311yescharacteristic
144#1463—531no—
145#1353—534no—
146C2×C53—21no—
147C2×C53—22no—
148C2×C53—22no—
149C2×C53—21no—
150C2×C53#151111yescharacteristic
151C2×C53C2×#35211yescharacteristic
152C2×C53C2×#35511yescharacteristic
153C22×C4⋊C4—531no—
154C22×C4⋊C4C2×D5311yescharacteristic
155#407—21no—
156#407—1061no—
157#400—532no—
158#400—532no—
159#400—1061no—
160#400—1061no—
161C2×#95—531no—
162C2×#95—1061no—
163C2×#95—1061no—
164#343—21no—
165#343C2×D5312yes—
166#324—532no—
167#324—1061no—
168C53#1181411yescharacteristic, complemented, Sylow
169C23×C4—1061no—
170C23×C4—1061no—
171C23×C4C4×D5311yescharacteristic
172C23×C4D21211yescharacteristic
173C23×C4C22⋊D5311yescharacteristic
174C2×M16—22no—
175C2×M16—2121no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—532no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4—2121no—
181C2×C4⋊C4—2121no—
182C2×C4⋊C4C4×D5311yescharacteristic
183C2×C4⋊C4D21211yescharacteristic
184C2×C4⋊C4C22⋊D5311yescharacteristic
185C2×C4⋊C4C22×D5311yescharacteristic
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—1062no—
189C22⋊C8—2121no—
190#95—532no—
191#95—1062no—
192#95—1062no—
193#95—1061no—
194#95—1062no—
195#95—1062no—
196C24#1171992111yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—1062no—
208C22×C4—1062no—
209C22×C4—2122no—
210C22×C4—2122no—
211C22×C4—2122no—
212C22×C4—2124no—
213C22×C4—2121no—
214C22×C4—2121no—
215C22×C4#1171992111yescharacteristic
216C22×C4#1171992111yescharacteristic
217C22×C4#1171992111yescharacteristic
218C22×C4C2×C4×D5311yescharacteristic
219C22×C4C2×D21211yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221M16—42no—
222M16—42no—
223M16—4242no—
224C2×C8—42no—
225C2×C8—2122no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—2122no—
231C4⋊C4—2122no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23#1313313111yescharacteristic
237C23#1313313111yescharacteristic
238C23C2×#1171992111yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4—4244no—
259C2×C4—4242no—
260C2×C4—4242no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2124no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4#1313324712yes—
267C8—82no—
268C8—4242no—
269C22—24no—
270C22—24no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22#1681041412yes—
278C22#1681041912yes—
279C22C2×#1313313111yescharacteristic
280C22#1681063111yescharacteristic
281C22#1681064111yescharacteristic
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—4242no—
289C4—4242no—
290C2—21no—
291C2—21no—
292C2—22no—
293C2—22no—
294C2#2027324711yescharacteristic
295C2C2×#1681041411yescharacteristic
296C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1821C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1821D53
hypocenterC53#11814
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #20272741C2semidirect
168. C53#11814semidirect

Maximal subgroups

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

subgroupindex
C2×#168105622
#202730452
#202727412
C53×#18212
#1181453