Finite Group Databasebeta

← #28308291

Subgroups — #28308291

Order 13568.

334 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 70 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#20291089C212yesmaximal, complemented
2#20289650C212yesmaximal, complemented
3#20289647C212yesmaximal, complemented
4#20283418C211yesmaximal, characteristic
5#16811545—22no—
6#16811519C2214yes—
7#16811516—22no—
8C2×#13133223C411yescharacteristic
9#16811134C411yescharacteristic
10#16810699—22no—
11#16810692—22no—
12#16810649C411yescharacteristic
13#16810589C2211yescharacteristic
14#16810587C2211yescharacteristic
15#16810578C2211yescharacteristic
16C22×M16×C53C411yescharacteristic
17#13133291—24no—
18C2×#11719945C2×C411yescharacteristic
19#13133223—42no—
20C2×#11719917C2×C411yescharacteristic
21C2×#11719917C2311yescharacteristic
22C2×#11719912—21no—
23C2×C8×D53—21no—
24#13133200—21no—
25#13133196—21no—
26#13133191—21no—
27#13133186—21no—
28C4×C22⋊D53—22no—
29#13133176C2×C411yescharacteristic
30#13133146—22no—
31C22⋊#88322—22no—
32#13133129—22no—
33#13133127D412yes—
34C8×Dic53D412yes—
35C23×C4×C53C2×C411yescharacteristic
36C2×M16×C53—21no—
37C2×M16×C53—21no—
38C2×M16×C53—42no—
39C22×C8×C53C2×C411yescharacteristic
40C22×C8×C53C2×C411yescharacteristic
41C24×C53M1611yescharacteristic
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53—41no—
45C22×C4×C53—41no—
46C22×C4×C53M1611yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C2×C22⋊D53—21no—
49C2×C22⋊D53—22no—
50#11719945—41no—
51#11719945—41no—
52C2×D212M1611yescharacteristic
53C2×C4×D53—21no—
54C2×Dic106M1611yescharacteristic
55M16×C53—42no—
56M16×C53—42no—
57C2×C8×C53—22no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53C22⋊C411yescharacteristic
61C2×C8×C53C22⋊C411yescharacteristic
62C22⋊Dic53—22no—
63C22⋊Dic53—21no—
64#11719921—22no—
65C4⋊Dic53—21no—
66#11719919—22no—
67C4×Dic53C22⋊C412yes—
68#11719917—41no—
69#11719917—41no—
70C2×#88322—21no—
71C2×#88322C2×D412yes—
72#11719912—42no—
73C8×D53—42no—
74#11719910—44no—
75C23×C53—21no—
76C23×C53—41no—
77C23×C53—81no—
78C23×C53C2×M1611yescharacteristic
79C22×D53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—42no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53C2×M1611yescharacteristic
89C22⋊D53—42no—
90C22⋊D53—82no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93D212—41no—
94C4×D53—41no—
95C4×D53—41no—
96Dic106—41no—
97C8×C53—21no—
98C8×C53—41no—
99C8×C53#10012yes—
100#88322—22no—
101#88322—41no—
102#11538—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—82no—
109C22×C53#40711yescharacteristic
110C2×D53—42no—
111C2×D53—41no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53#41011yescharacteristic
115C4×C53C2×#10011yescharacteristic
116Dic53—42no—
117Dic53—41no—
118#2930—531no—
119#2166—532no—
120#1321—532no—
121#1320—532no—
122C2×C53—21no—
123C2×C53—41no—
124C2×C53—81no—
125C2×C53#146611yescharacteristic
126D53—81no—
127C2×#131—531no—
128C22×M16#615711yescharacteristic
129#525—531no—
130#436—1062no—
131#435—1062no—
132#424—531no—
133#408—531no—
134#407—531no—
135#405—531no—
136#364—1062no—
137#349—534no—
138#347—1062no—
139C53#1153811yescharacteristic, complemented, Sylow
140C2×#41—531no—
141C23×C4C2×#615711yescharacteristic
142#131—2122no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—42no—
146C2×M16—531no—
147C2×M16—531no—
148C2×M16—1061no—
149C22×C8—1061no—
150C22×C8C2×#615711yescharacteristic
151C22×C8C2×#615711yescharacteristic
152#123—1061no—
153C22⋊Q8—1061no—
154#121—1061no—
155#120—1061no—
156C4×D4—1062no—
157#117—531no—
158M32—1064no—
159C4⋊C8—1062no—
160C22⋊C8—1062no—
161C22⋊C8—1062no—
162#97—532no—
163C4×C8—532no—
164C24#1171993611yescharacteristic
165#41—2121no—
166#41—2121no—
167C2×Q8—531no—
168C2×D4—531no—
169C2×D4—1062no—
170C2×D4—1061no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—1061no—
176C22×C4#1171993611yescharacteristic
177C22×C4C22×#615711yescharacteristic
178M16—42no—
179M16—42no—
180M16—2121no—
181M16—2122no—
182M16—2121no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—532no—
187C2×C8—1061no—
188C2×C8—2122no—
189C2×C8C22⋊#615711yescharacteristic
190C2×C8C22⋊#615711yescharacteristic
191C4⋊C4—1062no—
192C4⋊C4—1061no—
193C22⋊C4—1062no—
194C22⋊C4—1062no—
195C22⋊C4—1061no—
196C42—532no—
197C16—2124no—
198C23—21no—
199C23—41no—
200C23—81no—
201C23—1061no—
202C23C2×#1171993611yescharacteristic
203Q8—2121no—
204D4—4242no—
205D4—2121no—
206D4—2122no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1062no—
215C2×C4—1061no—
216C2×C4—2121no—
217C2×C4—2121no—
218C2×C4C2×C22⋊#615711yescharacteristic
219C2×C4D53⋊M1611yescharacteristic
220C8—21no—
221C8—41no—
222C8—1062no—
223C8—2121no—
224C8#1313327712yes—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—82no—
231C22—2122no—
232C22—2121no—
233C22#1681159411yescharacteristic
234C4—21no—
235C4—41no—
236C4—2122no—
237C4—2121no—
238C4#1681160211yescharacteristic
239C4C2×#1313327711yescharacteristic
240C2—21no—
241C2—41no—
242C2—81no—
243C2—4241no—
244C2#2028997411yescharacteristic
subgroupG/H
centerC4#16811602
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×M16×C53C4
socleC2×C53#1466
prime radical#16811134C4
cosocleC2×#11719917C23
hypercenterC22×M16#6157
hypocenterC53#11538
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #20291089C2semidirect
2. #20289650C2semidirect
3. #20289647C2semidirect
139. C53#11538semidirect

Maximal subgroups

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

subgroupindex
#202910892
#202896502
#202896472
#202834182
#1153853