Finite Group Databasebeta

← #28299732

Subgroups — #28299732

Order 13568.

624 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 118 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#20290631C212yesmaximal, complemented
2#20289531C214yesmaximal, complemented
3#127×Dic53C211yesmaximal, characteristic
4C2×#13133284—22no—
5#16811489—22no—
6#16811489C2214yes—
7C2×D4×Dic53—22no—
8C2×D4×Dic53C412yes—
9#16810526—22no—
10#16810526C2212yes—
11#16810493C411yescharacteristic
12C42×Dic53C2211yescharacteristic
13C2×#127×C53C411yescharacteristic
14C22×#88324—24no—
15#13133284—44no—
16C23×Dic53—22no—
17D4×Dic53—44no—
18D4×Dic53—44no—
19D4×Dic53—44no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53C2×C412yes—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—24no—
24C2×C4×Dic53C2×C411yescharacteristic
25C2×C4×Dic53D412yes—
26C2×C4×Dic53D412yes—
27C2×C4×Dic53C2311yescharacteristic
28C22×D4×C53—22no—
29C22×D4×C53C2×C412yes—
30#127×C53—24no—
31C2×C42×C53C2×C411yescharacteristic
32C24×C53—22no—
33C2×D4×C53—28no—
34C2×D4×C53—22no—
35C2×D4×C53—44no—
36C2×D4×C53—42no—
37C2×D4×C53—42no—
38C2×D4×C53—44no—
39C2×D4×C53M1614yes—
40C22×C4×C53C22⋊C412yes—
41C22×C4×C53C22×C411yescharacteristic
42C22×Dic53—22no—
43C22×Dic53—44no—
44C22×Dic53—48no—
45C22×Dic53C22⋊C411yescharacteristic
46C22×Dic53C22⋊C411yescharacteristic
47C22×Dic53C2×D412yes—
48C2×#88324—44no—
49C2×#88324—44no—
50C2×#88324—44no—
51C42×C53—22no—
52C22⋊Dic53—44no—
53C22⋊Dic53—44no—
54C22⋊Dic53—44no—
55C4⋊Dic53—44no—
56C4⋊Dic53—42no—
57C4×Dic53—22no—
58C4×Dic53—42no—
59C4×Dic53—41no—
60C4×Dic53D814yes—
61C4×Dic53SD1614yes—
62C23×C53—24no—
63C23×C53—42no—
64C23×C53—42no—
65C23×C53—44no—
66C23×C53—44no—
67C23×C53C2×C22⋊C411yescharacteristic
68D4×C53—44no—
69D4×C53—44no—
70D4×C53—416no—
71C2×C4×C53—22no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53#10214yes—
76C2×C4×C53C2×M1612yes—
77C2×Dic53—22no—
78C2×Dic53—22no—
79C2×Dic53—21no—
80C2×Dic53—22no—
81C2×Dic53—42no—
82C2×Dic53—44no—
83C2×Dic53—88no—
84C2×Dic53#10012yes—
85C2×Dic53#10214yes—
86C2×Dic53C2×D812yes—
87C2×Dic53C2×SD1612yes—
88#88324—84no—
89#7651—531nomaximal, Sylow
90C22×C53—44no—
91C22×C53—48no—
92C22×C53—82no—
93C22×C53—82no—
94C22×C53—84no—
95C22×C53#40711yescharacteristic
96C22×C53#41011yescharacteristic
97C22×C53C2×#10011yescharacteristic
98C22×C53C2×#10212yes—
99C22×C53#42112yes—
100C4×C53—22no—
101C4×C53—24no—
102Dic53—22no—
103Dic53—22no—
104Dic53—41no—
105Dic53—82no—
106Dic53#33114yes—
107C4×#127—531no—
108#1831—532no—
109#1270—534no—
110C2×C53—84no—
111C2×C53#146611yescharacteristic
112C2×C53#148612yes—
113C2×C53#151712yes—
114C2×C53#153611yescharacteristic
115C2×C53C2×#33111yescharacteristic
116C2×#127#615711yescharacteristic
117C2×C4×D4—532no—
118C2×C4×D4—1062no—
119C2×C4⋊C8—1062no—
120#390—532no—
121#390—1062no—
122#384—531no—
123C43—531no—
124#336—534no—
125#336—1062no—
126C53#765111yescharacteristic, complemented, Sylow
127C22×D4—22no—
128C22×D4C2×#615712yes—
129C23×C4—1062no—
130C22×C8—1064no—
131#127—24no—
132C4×D4—2124no—
133C4×D4—2124no—
134C4×D4—2124no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—1062no—
137C2×C22⋊C4—1062no—
138C2×C22⋊C4—1064no—
139C2×C42—532no—
140C2×C42—531no—
141C2×C42—532no—
142C2×C42—531no—
143C2×C42C2×#615711yescharacteristic
144C4⋊C8—2124no—
145C24—22no—
146C2×D4—28no—
147C2×D4—22no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4#1171994014yes—
153C22×C4—531no—
154C22×C4—532no—
155C22×C4—531no—
156C22×C4—1062no—
157C22×C4—2128no—
158C22×C4—2124no—
159C22×C4C22⋊#615712yes—
160C22×C4C22×#615711yescharacteristic
161C2×C8—2124no—
162C2×C8—2124no—
163C2×C8—2124no—
164C4⋊C4—2122no—
165C4⋊C4—2124no—
166C22⋊C4—2124no—
167C22⋊C4—2124no—
168C22⋊C4—2124no—
169C42—22no—
170C42—534no—
171C42—534no—
172C42—1062no—
173C42—2121no—
174C42—2122no—
175C23—24no—
176C23—42no—
177C23—42no—
178C23—44no—
179C23—44no—
180C23C2×C22⋊#615711yescharacteristic
181D4—44no—
182D4—44no—
183D4—416no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—532no—
189C2×C4—534no—
190C2×C4—532no—
191C2×C4—532no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1061no—
195C2×C4—1062no—
196C2×C4—4248no—
197C2×C4—2122no—
198C2×C4—2124no—
199C2×C4#1313327914yes—
200C2×C4C2×#1171994012yes—
201C8—4244no—
202C22—44no—
203C22—48no—
204C22—84no—
205C22—82no—
206C22—82no—
207C22#1681159311yescharacteristic
208C22#1681160311yescharacteristic
209C22D53⋊#10011yescharacteristic
210C22C2×#1313327912yes—
211C22#1681162112yes—
212C4—22no—
213C4—24no—
214C4—534no—
215C4—1062no—
216C4—1062no—
217C4—4242no—
218C4—2121no—
219C2—84no—
220C2#2028997511yescharacteristic
221C2#2029004012yes—
222C2#2029014412yes—
223C2#2029020411yescharacteristic
224C2D53⋊#33111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#127#6157
hypocenterC53#7651
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20290631C2semidirect
2. #20289531C2semidirect
126. C53#7651semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202895312
#127×Dic532
#765153