Finite Group Databasebeta

← #28308292

Subgroups — #28308292

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#20291088C212yesmaximal, complemented
2#20289649C212yesmaximal, complemented
3#20289647C212yesmaximal, complemented
4#20283419C211yesmaximal, characteristic
5#16811546—22no—
6#16811519C2214yes—
7#16811515—22no—
8C2×#13133224C411yescharacteristic
9C22×#11719917C411yescharacteristic
10#16810701—22no—
11#16810695—22no—
12#16810650C411yescharacteristic
13#16810590C2211yescharacteristic
14#16810587C2211yescharacteristic
15#16810579C2211yescharacteristic
16C53×#525C411yescharacteristic
17#13133291—24no—
18#13133224—42no—
19C2×#11719917—21no—
20C2×#11719917—21no—
21C2×#11719917—21no—
22C2×#11719917—42no—
23C2×#11719917C2×C411yescharacteristic
24C2×#11719917C2311yescharacteristic
25C22×#88322—21no—
26C22×#88322C2×C411yescharacteristic
27C22×#88322C2×C411yescharacteristic
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31#13133128D412yes—
32C4×#88322D412yes—
33C2×#41×C53C2×C411yescharacteristic
34C23×C4×C53C2×C411yescharacteristic
35#123×C53—21no—
36C22⋊Q8×C53—21no—
37#121×C53—21no—
38#120×C53—21no—
39C4×D4×C53—22no—
40#117×C53C2×C411yescharacteristic
41C24×C53M1611yescharacteristic
42#41×C53—41no—
43#41×C53—41no—
44C2×Q8×C53M1611yescharacteristic
45C2×D4×C53—21no—
46C2×D4×C53—22no—
47C2×D4×C53M1611yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53M1611yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C4⋊C4×C53—22no—
56C4⋊C4×C53—21no—
57C22⋊C4×C53—22no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—21no—
60C42×C53C22⋊C412yes—
61#11719917—41no—
62#11719917—42no—
63#11719917—42no—
64#11719917—42no—
65#11719917—41no—
66C2×#88322—22no—
67C2×#88322—21no—
68C2×#88322—21no—
69C2×#88322—42no—
70C2×#88322—41no—
71C2×#88322C22⋊C411yescharacteristic
72C2×#88322C22⋊C411yescharacteristic
73C2×#88322C2×D412yes—
74#11719910—44no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—41no—
78C23×C53—81no—
79C23×C53C2×M1611yescharacteristic
80Q8×C53—41no—
81D4×C53—41no—
82D4×C53—42no—
83D4×C53—82no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53C2×M1611yescharacteristic
97#88322—22no—
98#88322—21no—
99#88322—41no—
100#88322—41no—
101#88322#10012yes—
102#11538—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—42no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—41no—
110C22×C53—82no—
111C22×C53#40711yescharacteristic
112C4×C53—21no—
113C4×C53—42no—
114C4×C53—41no—
115C4×C53—41no—
116C4×C53#41011yescharacteristic
117C4×C53C2×#10011yescharacteristic
118#2930—531no—
119#2166—532no—
120#1321—532no—
121#1320—532no—
122C2×C53—21no—
123C2×C53—41no—
124C2×C53—81no—
125C2×C53—81no—
126C2×C53#146611yescharacteristic
127C2×#131—531no—
128C22×M16—531no—
129#525#615711yescharacteristic
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×#41C2×#615711yescharacteristic
141C23×C4C2×#615711yescharacteristic
142#131—2122no—
143C2×M16—531no—
144C2×M16—531no—
145C2×M16—1061no—
146C2×M16—1061no—
147C2×M16—1061no—
148C2×M16—2122no—
149C22×C8—531no—
150C22×C8—531no—
151C22×C8—1061no—
152#123—21no—
153C22⋊Q8—21no—
154#121—21no—
155#120—21no—
156C4×D4—22no—
157#117C2×#615711yescharacteristic
158M32—1064no—
159C4⋊C8—1062no—
160C22⋊C8—1062no—
161C22⋊C8—1062no—
162#97—532no—
163C4×C8—532no—
164C24#1171994011yescharacteristic
165#41—41no—
166#41—41no—
167C2×Q8#1171994011yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4#1171994011yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4#1171994011yescharacteristic
177C22×C4C22×#615711yescharacteristic
178M16—2122no—
179M16—2122no—
180M16—2121no—
181M16—2122no—
182M16—2121no—
183C2×C8—531no—
184C2×C8—532no—
185C2×C8—531no—
186C2×C8—1062no—
187C2×C8—1061no—
188C2×C8—1061no—
189C2×C8—2122no—
190C2×C8—2121no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—21no—
196C42C22⋊#615712yes—
197C16—2124no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23—81no—
202C23C2×#1171994011yescharacteristic
203Q8—41no—
204D4—41no—
205D4—42no—
206D4—82no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4C2×C22⋊#615711yescharacteristic
219C2×C4C2×#1171994011yescharacteristic
220C8—532no—
221C8—1062no—
222C8—1061no—
223C8—2121no—
224C8—2121no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—82no—
233C22#1681159311yescharacteristic
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4#1681160311yescharacteristic
239C4D53⋊#10011yescharacteristic
240C2—21no—
241C2—41no—
242C2—81no—
243C2—81no—
244C2#2028997511yescharacteristic
subgroupG/H
centerC4#16811603
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC53×#525C4
socleC2×C53#1466
cosocleC2×#11719917C23
hypercenter#525#6157
hypocenterC53#11538
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ #117 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20291088C2semidirect
2. #20289649C2semidirect
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
#202910882
#202896492
#202896472
#202834192
#1153853