Finite Group Databasebeta

← #27294848

Subgroups — #27294848

Order 13568.

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

normal subgroups only

This group has 368 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
1#20274798C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272745C214yesmaximal, complemented
4C53×#2628C211yesmaximal, characteristic
5#16810680—21no—
6#16810587—21no—
7#16810392—21no—
8#16810392C2212yes—
9#16810349C2212yes—
10#16810347—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×#117×C53C2211yescharacteristic
13C53×#400—21no—
14C53×#399—21no—
15C53×#395C2212yes—
16C53×#393C412yes—
17C53×#389—21no—
18C53×#382C411yescharacteristic
19C53×#380—21no—
20C2×#11719917—22no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—41no—
24C22⋊#88322—41no—
25#13133128—41no—
26C22×Q8×C53C2×C412yes—
27C23×C4×C53D411yescharacteristic
28C4⋊Q8×C53—41no—
29C4⋊Q8×C53—41no—
30#117×C53—41no—
31#117×C53—41no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C4⋊C4×C53C2311yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53D411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43#95×C53—21no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53C2×C412yes—
47C24×C53C22⋊C411yescharacteristic
48C2×Q8×C53—42no—
49C2×Q8×C53—82no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—41no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—82no—
69C22⋊C4×C53—41no—
70C22⋊C4×C53—41no—
71C22⋊C4×C53—82no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76#11719917—42no—
77#11719917—42no—
78C2×#88322—42no—
79C2×#88322—42no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53C2×C22⋊C411yescharacteristic
86Q8×C53—84no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#13712yes—
110#88322—82no—
111#88322—82no—
112#11819—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—42no—
116C22×C53—41no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—81no—
120C22×C53C2×#9911yescharacteristic
121C22×C53#41911yescharacteristic
122C22×C53#42111yescharacteristic
123C4×C53—22no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—41no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—41no—
130C4×C53—82no—
131C4×C53—82no—
132#2628D5311yescharacteristic
133#1822—531no—
134#1464—531no—
135#1354—534no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53—81no—
139C2×C53#151111yescharacteristic
140C2×C53#212211yescharacteristic
141C2×C53#213511yescharacteristic
142C2×C4⋊Q8Dic5311yescharacteristic
143C2×#117C2×D5311yescharacteristic
144#432—1061no—
145#407—1061no—
146#400—21no—
147#399—21no—
148#395C2×D5312yes—
149#393Dic5312yes—
150#389—21no—
151#382Dic5311yescharacteristic
152#380—21no—
153#343—532no—
154#343—1061no—
155#324—532no—
156#323—1061no—
157C53#1181911yescharacteristic, complemented, Sylow
158C22×Q8C2×Dic5312yes—
159C23×C4C22⋊D5311yescharacteristic
160C2×M16—1062no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163#117—41no—
164#117—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊D5311yescharacteristic
170C2×C4⋊C4C22×D5311yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4C2×Dic5311yescharacteristic
173C2×C22⋊C4C22⋊D5311yescharacteristic
174C2×C42C2×Dic5311yescharacteristic
175C2×C42C22⋊D5311yescharacteristic
176C4⋊C8—2121no—
177C22⋊C8—1062no—
178C22⋊C8—2121no—
179C22⋊C8—2121no—
180#97—2121no—
181#95—22no—
182#95—21no—
183#95—22no—
184#95C2×Dic5312yes—
185C24C22⋊Dic5311yescharacteristic
186C2×Q8—42no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C22⋊Dic5311yescharacteristic
195C22×C4C22⋊Dic5311yescharacteristic
196C22×C4C22⋊Dic5311yescharacteristic
197C22×C4C22×Dic5311yescharacteristic
198C22×C4C2×C22⋊D5311yescharacteristic
199C22×C4C2×C22⋊D5311yescharacteristic
200M16—2122no—
201M16—2122no—
202C2×C8—2122no—
203C2×C8—2122no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—82no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23C2×C22⋊Dic5311yescharacteristic
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4#1313313211yescharacteristic
246C2×C4#1313313211yescharacteristic
247C2×C4#1313324812yes—
248C8—4242no—
249C8—4242no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—81no—
257C22C2×#1313313211yescharacteristic
258C22#1681063211yescharacteristic
259C22#1681064011yescharacteristic
260C4—22no—
261C4—21no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4—82no—
269C2—21no—
270C2—21no—
271C2—81no—
272C2#2027324811yescharacteristic
273C2#2027636511yescharacteristic
274C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2628C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2628D53
hypocenterC53#11819
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #20272745C2semidirect
157. C53#11819semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730492
#202727452
C53×#26282
#1181953