Finite Group Databasebeta

← #20279919

Subgroups — #20279919

Order 6784.

500 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#13133215C211yesmaximal, characteristic
2C2×#13133182C211yesmaximal, characteristic
3#16810544C211yesmaximal, characteristic
4#16810543C211yesmaximal, characteristic
5#16810542C211yesmaximal, characteristic
6#16810527C212yesmaximal
7#16810520C212yesmaximal
8#16810518C212yesmaximal
9#16810492C212yesmaximal
10#16810474C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14#13133215—24no—
15#13133182—24no—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C2×Q8×C53—22no—
38C2×Q8×C53—42no—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2312yes—
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2312yes—
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×Dic106—22no—
51C2×Dic106—22no—
52C2×Dic106—22no—
53C2×Dic106—42no—
54C2×Dic106Q814yescomplemented
55C4⋊C4×C53—22no—
56C4⋊C4×C53—42no—
57C4⋊Dic53—42no—
58#11719919—22no—
59#11719919—22no—
60#11719919—24no—
61#11719919—24no—
62#11719919—42no—
63#11719919—42no—
64#11719919Q818yescomplemented
65C4×Dic53—22no—
66C4×Dic53—22no—
67C4×Dic53—24no—
68C4×Dic53D414yes—
69C23×C53C2411yescharacteristic
70Q8×C53—44no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53C2×D412yes—
82C2×C4×C53C2×Q812yes—
83C2×C4×C53C2×Q814yes—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53C2×D412yes—
94C2×Dic53C2×D412yes—
95C2×Dic53C2×Q812yes—
96C2×Dic53C2×Q814yes—
97C2×Dic53C2×Q812yes—
98C2×Dic53C2×Q814yes—
99C2×Dic53#4112yes—
100Dic106—28no—
101Dic106—44no—
102C22×C53C22×D411yescharacteristic
103C22×C53C22×Q811yescharacteristic
104C22×C53C22×Q812yes—
105C22×C53C2×#4111yescharacteristic
106C22×C53H(5,𝔽2)12yes—
107C4×C53—22no—
108C4×C53—22no—
109C4×C53—24no—
110C4×C53—41no—
111C4×C53—42no—
112Dic53—22no—
113Dic53—24no—
114Dic53—42no—
115Dic53—41no—
116Dic53C22⋊Q814yes—
117Dic53C4⋊Q814yes—
118#2567—531nomaximal, Sylow
119C2×C53C2×C22⋊Q811yescharacteristic
120C2×C53C2×C4⋊Q811yescharacteristic
121C2×C53#53511yescharacteristic
122C2×C53#55412yes—
123C2×C53Q8212yes—
124C2×C4⋊Q8—531no—
125C2×C4×Q8—531no—
126#395—531no—
127#395—531no—
128#395—531no—
129#395D5311yescharacteristic
130#391—532no—
131#389—532no—
132#389—532no—
133#384—532no—
134C4×C4⋊C4—531no—
135C53#256711yescharacteristic, complemented, Sylow
136C22×Q8—531no—
137C22×Q8—531no—
138C22×Q8C2×D5311yescharacteristic
139C4⋊Q8—1064no—
140C4×Q8—1064no—
141C2×C4⋊C4—532no—
142C2×C4⋊C4—531no—
143C2×C4⋊C4—531no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—532no—
146C2×C4⋊C4—532no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—532no—
149C2×C4⋊C4C2×D5311yescharacteristic
150C2×C4⋊C4C2×D5312yes—
151C2×C42—531no—
152C2×C42—531no—
153C2×C42—531no—
154C2×C42—532no—
155#95—531no—
156#95—532no—
157#95—532no—
158#95—532no—
159#95C2×D5311yescharacteristic
160#95C2×D5312yes—
161C2×Q8—22no—
162C2×Q8—42no—
163C2×Q8—534no—
164C2×Q8—1062no—
165C2×Q8—1062no—
166C2×Q8—1062no—
167C2×Q8—2122no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—532no—
172C22×C4—532no—
173C22×C4—531no—
174C22×C4C22×D5312yes—
175C22×C4C22×D5311yescharacteristic
176C22×C4C22×D5311yescharacteristic
177C22×C4C22×D5311yescharacteristic
178C22×C4C22×D5312yes—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—538no—
182C4⋊C4—1062no—
183C4⋊C4—1064no—
184C4⋊C4—1064no—
185C4⋊C4—1062no—
186C4⋊C4—2122no—
187C4⋊C4—2122no—
188C4⋊C4—2122no—
189C42—534no—
190C42—1062no—
191C42—1062no—
192C42—1064no—
193C23C23×D5311yescharacteristic
194Q8—44no—
195Q8—1068no—
196Q8—2124no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—532no—
208C2×C4—532no—
209C2×C4—534no—
210C2×C4—532no—
211C2×C4—532no—
212C2×C4—534no—
213C2×C4—532no—
214C2×C4—1061no—
215C2×C4—1062no—
216C2×C4—1061no—
217C2×C4—1062no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4D4×D5312yes—
224C2×C4Q8×D5314yes—
225C2×C4Q8×D5312yes—
226C22C2×D4×D5311yescharacteristic
227C22C2×Q8×D5312yes—
228C22C2×Q8×D5311yescharacteristic
229C22#41×D5311yescharacteristic
230C22#1313326212yes—
231C4—24no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—42no—
236C4—534no—
237C4—534no—
238C4—1064no—
239C4—1062no—
240C4—2121no—
241C4—2122no—
242C2C22⋊Q8×D5311yescharacteristic
243C2C4⋊Q8×D5311yescharacteristic
244C2#1681118811yescharacteristic
245C2#1681132712yes—
246C2#1681135312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2567
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
54. C2×Dic106Q8semidirect
64. #11719919Q8semidirect
135. C53#2567semidirect