Finite Group Databasebeta

← #27276634

Subgroups — #27276634

Order 13568.

656 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 131 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#20273441C212yesmaximal, complemented
2#20273342C212yesmaximal, complemented
3#20273049C212yesmaximal
4C2×C53×#541C211yesmaximal, characteristic
5#16810680C2214yes—
6#16810372—24no—
7#16810365—24no—
8#16810349—22no—
9#16810347—22no—
10C53×#541—24no—
11C53×#541—24no—
12C2×C4⋊Q8×C53C2211yescharacteristic
13C2×#126×C53—22no—
14C2×#125×C53—22no—
15C2×#125×C53C2211yescharacteristic
16C2×C22⋊Q8×C53C412yes—
17C2×C4×Q8×C53C412yes—
18C2×#117×C53C2211yescharacteristic
19C4⋊#88322—24no—
20C22⋊#88322—24no—
21#13133128—24no—
22C22×Q8×C53C2×C412yes—
23C23×C4×C53C2×C411yescharacteristic
24C4⋊Q8×C53—22no—
25C4⋊Q8×C53D414yes—
26#126×C53—216no—
27#125×C53—216no—
28#125×C53—22no—
29#125×C53D414yes—
30C22⋊Q8×C53—24no—
31C22⋊Q8×C53—24no—
32C4×Q8×C53—24no—
33C4×Q8×C53—24no—
34#117×C53—22no—
35#117×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53D412yes—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53D412yes—
44C2×C42×C53—22no—
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53C2311yescharacteristic
47C24×C53C22⋊C411yescharacteristic
48C2×Q8×C53—22no—
49C2×Q8×C53—44no—
50C2×Q8×C53C22⋊C414yes—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—42no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D412yes—
60C4⋊C4×C53—28no—
61C4⋊C4×C53—28no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—28no—
64C4⋊C4×C53—28no—
65C4⋊C4×C53—24no—
66C4⋊C4×C53C22⋊C414yes—
67C22⋊C4×C53—28no—
68C22⋊C4×C53—24no—
69C42×C53—28no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53C2×D412yes—
73C42×C53C2×D412yes—
74C2×#88322—44no—
75C23×C53—21no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53C2×C22⋊C411yescharacteristic
79Q8×C53—44no—
80Q8×C53—44no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—24no—
84C2×C4×C53—24no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—44no—
91C2×C4×C53—44no—
92C2×C4×C53—42no—
93C2×C4×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53C2×C22⋊C411yescharacteristic
95C2×C4×C53#12014yes—
96#88322—84no—
97#8527—531nomaximal, Sylow
98C22×C53—22no—
99C22×C53—42no—
100C22×C53—41no—
101C22×C53—82no—
102C22×C53#37911yescharacteristic
103C22×C53#41312yes—
104C4×C53—24no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—42no—
108C4×C53—44no—
109C4×C53—41no—
110C4×C53—44no—
111C2×#541D5311yescharacteristic
112#1558—532no—
113#1534—532no—
114#1464—532no—
115C2×C53—22no—
116C2×C53—81no—
117C2×C53#178511yescharacteristic
118C2×C53#179612yes—
119#541—24no—
120#541—24no—
121C2×C4⋊Q8C2×D5311yescharacteristic
122C2×#126—22no—
123C2×#125—22no—
124C2×#125C2×D5311yescharacteristic
125C2×C22⋊Q8Dic5312yes—
126C2×C4×Q8Dic5312yes—
127C2×#117C2×D5311yescharacteristic
128#432—534no—
129#333—1064no—
130#330—1064no—
131#324—1062no—
132#323—1062no—
133C53#852711yescharacteristic, complemented, Sylow
134C22×Q8C2×Dic5312yes—
135C23×C4C2×Dic5311yescharacteristic
136C4⋊Q8—22no—
137C4⋊Q8C22⋊D5314yes—
138#126—216no—
139#125—216no—
140#125—22no—
141#125C22⋊D5314yes—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C2×Dic5312yes—
153C2×C4⋊C4C22⋊D5312yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C22⋊D5312yes—
156C2×C42—22no—
157C2×C42C2×Dic5311yescharacteristic
158C2×C42C22×D5311yescharacteristic
159C4⋊C8—1064no—
160C22⋊C8—1064no—
161#97—1064no—
162C24C22⋊Dic5311yescharacteristic
163C2×Q8—22no—
164C2×Q8—44no—
165C2×Q8C22⋊Dic5314yes—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4C22⋊Dic5311yescharacteristic
172C22×C4C22⋊Dic5312yes—
173C22×C4C22×Dic5311yescharacteristic
174C22×C4C2×C22⋊D5312yes—
175C2×C8—2124no—
176C4⋊C4—28no—
177C4⋊C4—28no—
178C4⋊C4—22no—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4C22⋊Dic5314yes—
183C22⋊C4—28no—
184C22⋊C4—24no—
185C42—28no—
186C42—21no—
187C42—21no—
188C42C2×C22⋊D5312yes—
189C42C2×C22⋊D5312yes—
190C23—21no—
191C23—22no—
192C23—42no—
193C23C2×C22⋊Dic5311yescharacteristic
194Q8—44no—
195Q8—44no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—21no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4C2×C22⋊Dic5311yescharacteristic
209C2×C4C2×C22⋊Dic5311yescharacteristic
210C2×C4#1313318614yes—
211C8—4244no—
212C22—22no—
213C22—42no—
214C22—41no—
215C22—82no—
216C22#1681047911yescharacteristic
217C22#1681060712yes—
218C4—24no—
219C4—21no—
220C4—21no—
221C4—44no—
222C4—42no—
223C4—41no—
224C4—44no—
225C2—22no—
226C2—81no—
227C2#2027464311yescharacteristic
228C2#2027469012yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#541C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#541D53
hypocenterC53#8527
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. #20273441C2semidirect
2. #20273342C2semidirect
133. C53#8527semidirect

Maximal subgroups

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

subgroupindex
#202734412
#202733422
#202730492
C2×C53×#5412
#852753