Finite Group Databasebeta

← #21427386

Subgroups — #21427386

Order 7552.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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×Q8×Dic59C211yesmaximal, characteristic
2C2×#14273692C211yesmaximal, characteristic
3#16837335C212yesmaximal
4#16837334C211yesmaximal, characteristic
5#16837332C211yesmaximal, characteristic
6#16837323C212yesmaximal
7#16837321C211yesmaximal, characteristic
8C4⋊C4×Dic59C212yesmaximal
9#16837253C212yesmaximal
10#16837247C211yesmaximal, characteristic
11C59×#395C211yesmaximal, characteristic
12C22×Dic118C2211yescharacteristic
13Q8×Dic59—24no—
14#14273692—24no—
15C2×C4⋊Dic59C2212yes—
16C2×C4⋊Dic59C2212yes—
17C2×C4⋊Dic59C2211yescharacteristic
18C2×Dic59⋊C4C2212yes—
19C2×Dic59⋊C4C2212yes—
20C2×Dic59⋊C4C2211yescharacteristic
21C2×Dic59⋊C4C2211yescharacteristic
22C2×C4×Dic59C2211yescharacteristic
23C2×C4×Dic59C2212yes—
24C2×C4×Dic59C2211yescharacteristic
25C2×C4×Dic59C2211yescharacteristic
26C2×C4×Dic59C2212yes—
27C59⋊#95C2211yescharacteristic
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C22×Q8×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2212yes—
34C2×C4⋊C4×C59C2211yescharacteristic
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C2×Q8×C59—22no—
38C2×Q8×C59—24no—
39C22×C4×C59C2311yescharacteristic
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2312yes—
44C22×Dic59C2311yescharacteristic
45C22×Dic59C2312yes—
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2312yes—
48C22×Dic59C2311yescharacteristic
49C22×Dic59C2311yescharacteristic
50C2×Dic118—22no—
51C2×Dic118—22no—
52C2×Dic118—22no—
53C4⋊C4×C59—22no—
54C4⋊C4×C59—24no—
55C4⋊Dic59—22no—
56C4⋊Dic59—24no—
57C4⋊Dic59—24no—
58Dic59⋊C4—22no—
59Dic59⋊C4—24no—
60Dic59⋊C4—24no—
61Dic59⋊C4—22no—
62C4×Dic59—22no—
63C4×Dic59—22no—
64C4×Dic59—24no—
65C4×Dic59D414yes—
66C4×Dic59Q818yescomplemented
67C23×C59C2411yescharacteristic
68Q8×C59—28no—
69C2×C4×C59—22no—
70C2×C4×C59—21no—
71C2×C4×C59—22no—
72C2×C4×C59—22no—
73C2×C4×C59—21no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—21no—
77C2×C4×C59—22no—
78C2×C4×C59C2×D412yes—
79C2×C4×C59C2×Q814yes—
80C2×C4×C59#4114yes—
81C2×C4×C59#4112yes—
82C2×Dic59—21no—
83C2×Dic59—22no—
84C2×Dic59—22no—
85C2×Dic59—21no—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—21no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—22no—
93C2×Dic59C2×D412yes—
94C2×Dic59C2×D412yes—
95C2×Dic59C2×Q814yes—
96C2×Dic59C2×Q814yes—
97C2×Dic59#4112yes—
98Dic118—28no—
99C22×C59C22×D411yescharacteristic
100C22×C59C22×Q812yes—
101C22×C59C2×#4111yescharacteristic
102C22×C59C2×#4112yes—
103C22×C59C2×#4111yescharacteristic
104C4×C59—22no—
105C4×C59—22no—
106C4×C59—24no—
107C4×C59—24no—
108C4×C59—41no—
109Dic59—22no—
110Dic59—22no—
111Dic59—24no—
112Dic59—41no—
113Dic59—42no—
114Dic59C4⋊Q814yes—
115#2399—591nomaximal, Sylow
116C2×C59#52511yescharacteristic
117C2×C59#52511yescharacteristic
118C2×C59C2×C4⋊Q811yescharacteristic
119C2×C59#53312yes—
120C2×C59#53312yes—
121C2×C4×Q8—591no—
122C2×C4×Q8—591no—
123#398—591no—
124#398—592no—
125#398—591no—
126#395—591no—
127#395—592no—
128#395D5911yescharacteristic
129C4×C4⋊C4—592no—
130C4×C4⋊C4—592no—
131#376—591no—
132C59#239911yescharacteristic, complemented, Sylow
133C22×Q8—591no—
134C22×Q8C2×D5911yescharacteristic
135C4×Q8—1184no—
136C4×Q8—1184no—
137C2×C4⋊C4—591no—
138C2×C4⋊C4—592no—
139C2×C4⋊C4—591no—
140C2×C4⋊C4—591no—
141C2×C4⋊C4—592no—
142C2×C4⋊C4—592no—
143C2×C4⋊C4—592no—
144C2×C4⋊C4C2×D5911yescharacteristic
145C2×C4⋊C4C2×D5912yes—
146C2×C42—591no—
147C2×C42—592no—
148C2×C42—591no—
149C2×C42—592no—
150C2×C42—591no—
151#95—591no—
152#95—592no—
153#95—592no—
154#95—592no—
155#95—592no—
156#95C2×D5911yescharacteristic
157#95C2×D5912yes—
158C2×Q8—22no—
159C2×Q8—24no—
160C2×Q8—1182no—
161C2×Q8—1182no—
162C2×Q8—1182no—
163C22×C4—591no—
164C22×C4—592no—
165C22×C4—591no—
166C22×C4—591no—
167C22×C4—592no—
168C22×C4—591no—
169C22×C4C22×D5911yescharacteristic
170C22×C4C22×D5911yescharacteristic
171C22×C4C22×D5912yes—
172C22×C4C22×D5912yes—
173C22×C4C22×D5911yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—1182no—
177C4⋊C4—1184no—
178C4⋊C4—1184no—
179C4⋊C4—1182no—
180C4⋊C4—1182no—
181C4⋊C4—1184no—
182C4⋊C4—1184no—
183C42—594no—
184C42—598no—
185C42—1182no—
186C42—1184no—
187C42—1182no—
188C23C23×D5911yescharacteristic
189Q8—28no—
190Q8—1188no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—592no—
201C2×C4—594no—
202C2×C4—592no—
203C2×C4—594no—
204C2×C4—592no—
205C2×C4—1181no—
206C2×C4—1181no—
207C2×C4—1182no—
208C2×C4—1182no—
209C2×C4—1181no—
210C2×C4—1181no—
211C2×C4—1181no—
212C2×C4—1182no—
213C2×C4—1182no—
214C2×C4—1182no—
215C2×C4—1182no—
216C2×C4D4×D5912yes—
217C2×C4#1174568714yes—
218C2×C4Q8×D5914yes—
219C2×C4#1174568912yes—
220C22C2×D4×D5911yescharacteristic
221C22C2×Q8×D5912yes—
222C22#41×D5911yescharacteristic
223C22C2×#1174568712yes—
224C22C2×#1174568911yescharacteristic
225C4—22no—
226C4—24no—
227C4—22no—
228C4—24no—
229C4—41no—
230C4—594no—
231C4—1182no—
232C4—1182no—
233C4—1184no—
234C4—2361no—
235C4—2362no—
236C2#1683790811yescharacteristic
237C2#1683791511yescharacteristic
238C2C4⋊Q8×D5911yescharacteristic
239C2#1683795412yes—
240C2#1683795712yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#395C2
hypercenter#395D59
hypocenterC59#2399
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
66. C4×Dic59Q8semidirect
132. C59#2399semidirect