Finite Group Databasebeta

← #21431510

Subgroups — #21431510

Order 7552.

424 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 137 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#16837344C212yesmaximal, complemented
2#16837322C212yesmaximal, complemented
3#16837314C211yesmaximal, characteristic
4#16837314C211yesmaximal, characteristic
5#16837306C212yesmaximal, complemented
6#16837295C212yesmaximal
7#16837275C212yesmaximal
8#16837267C212yesmaximal, complemented
9C59×#385C211yesmaximal, characteristic
10C22×Dic118C2211yescharacteristic
11C22×Dic118C2211yescharacteristic
12C2×C4⋊Dic59C2212yes—
13C2×Dic59⋊C4C2212yes—
14C2×Dic59⋊C4C2212yes—
15C2×C22⋊Dic59C2212yes—
16C2×C22⋊Dic59C2212yes—
17C2×C4×Dic59C2212yes—
18C59⋊#95C2211yescharacteristic
19C59⋊#95C2212yes—
20C59⋊#95C2212yes—
21C59⋊#95C2212yes—
22C59⋊#95C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2211yescharacteristic
26C23×C4×C59C2211yescharacteristic
27C2×C4⋊C4×C59C2212yes—
28C2×C22⋊C4×C59C2211yescharacteristic
29C2×C22⋊C4×C59C2211yescharacteristic
30#95×C59C2212yes—
31C24×C59C2311yescharacteristic
32C22×C4×C59—41no—
33C22×C4×C59—42no—
34C22×C4×C59C2311yescharacteristic
35C22×C4×C59C2312yes—
36C22×C4×C59C2311yescharacteristic
37C22×C4×C59C2312yes—
38C22×Dic59C2312yes—
39C22×Dic59C2312yes—
40C22×Dic59C2312yes—
41C22×Dic59C2312yes—
42C2×Dic118—24no—
43C2×Dic118—41no—
44C2×Dic118—41no—
45C2×Dic118—42no—
46C4⋊C4×C59—42no—
47C22⋊C4×C59—41no—
48C22⋊C4×C59Q814yescomplemented
49C22⋊Dic59—24no—
50C22⋊Dic59—42no—
51C4⋊Dic59—42no—
52Dic59⋊C4—42no—
53Dic59⋊C4—42no—
54C4×Dic59—24no—
55C23×C59—22no—
56C23×C59—22no—
57C23×C59—21no—
58C23×C59—21no—
59C23×C59C2×Q812yes—
60C23×C59C2411yescharacteristic
61C2×C4×C59—22no—
62C2×C4×C59—22no—
63C2×C4×C59—22no—
64C2×C4×C59—22no—
65C2×C4×C59—22no—
66C2×C4×C59—21no—
67C2×C4×C59—21no—
68C2×C4×C59—22no—
69C2×C4×C59—22no—
70C2×C4×C59—44no—
71C2×C4×C59C2×Q814yes—
72C2×Dic59—22no—
73C2×Dic59—22no—
74C2×Dic59—22no—
75C2×Dic59—22no—
76C2×Dic59—22no—
77C2×Dic59—22no—
78C2×Dic59—22no—
79C2×Dic59—22no—
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—22no—
83C2×Dic59#4114yes—
84Dic118—44no—
85Dic118—44no—
86C22×C59—22no—
87C22×C59—24no—
88C22×C59—42no—
89C22×C59—41no—
90C22×C59—41no—
91C22×C59C22×Q811yescharacteristic
92C22×C59C2×#4112yes—
93C22×C59H(5,𝔽2)12yes—
94C22×C59H(5,𝔽2)11yescharacteristic
95C22×C592−1+411yescharacteristic
96C4×C59—24no—
97C4×C59—41no—
98C4×C59—41no—
99C4×C59—42no—
100Dic59—24no—
101Dic59—42no—
102Dic59—42no—
103Dic59—42no—
104#2763—591nomaximal, Sylow
105C2×C59—42no—
106C2×C59#54212yes—
107C2×C59#54411yescharacteristic
108C2×C59#55111yescharacteristic
109C2×C59#55412yes—
110C2×C59H(3,𝔽4)11yescharacteristic
111#400—592no—
112#395—592no—
113#393—591no—
114#393—591no—
115#391—592no—
116#388—592no—
117#385—592no—
118#385D5911yescharacteristic
119#382—592no—
120C59#276311yescharacteristic, complemented, Sylow
121C22×Q8—591no—
122C22×Q8—591no—
123C23×C4C2×D5911yescharacteristic
124C2×C4⋊C4—592no—
125C2×C4⋊C4—592no—
126C2×C4⋊C4—592no—
127C2×C4⋊C4C2×D5912yes—
128C2×C22⋊C4—592no—
129C2×C22⋊C4—592no—
130C2×C22⋊C4C2×D5911yescharacteristic
131C2×C22⋊C4C2×D5911yescharacteristic
132C2×C42—592no—
133#95—591no—
134#95—592no—
135#95—592no—
136#95—592no—
137#95—592no—
138#95—592no—
139#95—592no—
140#95—591no—
141#95C2×D5912yes—
142C24C22×D5911yescharacteristic
143C2×Q8—1184no—
144C2×Q8—2361no—
145C2×Q8—2361no—
146C2×Q8—2362no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—592no—
150C22×C4—592no—
151C22×C4—592no—
152C22×C4—592no—
153C22×C4C22×D5911yescharacteristic
154C22×C4C22×D5911yescharacteristic
155C22×C4C22×D5912yes—
156C22×C4C22×D5912yes—
157C4⋊C4—42no—
158C4⋊C4—2362no—
159C4⋊C4—2362no—
160C4⋊C4—2362no—
161C22⋊C4—41no—
162C22⋊C4—1184no—
163C22⋊C4—2362no—
164C22⋊C4Dic11814yescomplemented
165C42—1184no—
166C23—22no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23C2×Dic11812yes—
171C23C23×D5911yescharacteristic
172Q8—2364no—
173Q8—2364no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—594no—
185C2×C4—1182no—
186C2×C4—1182no—
187C2×C4—1182no—
188C2×C4—1182no—
189C2×C4—1182no—
190C2×C4—1182no—
191C2×C4—1182no—
192C2×C4—1182no—
193C2×C4—1182no—
194C2×C4—1182no—
195C2×C4—1182no—
196C2×C4C2×Dic11814yes—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22×Dic11811yescharacteristic
203C22#41×D5912yes—
204C22#1427377212yes—
205C22#1427377211yescharacteristic
206C22#1427377511yescharacteristic
207C4—24no—
208C4—42no—
209C4—41no—
210C4—41no—
211C4—1184no—
212C4—2362no—
213C4—2362no—
214C4—2362no—
215C2—42no—
216C2#1683802212yes—
217C2#1683803311yescharacteristic
218C2#1683808011yescharacteristic
219C2#1683811112yes—
220C2#1683814511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#385C2
hypercenter#385D59
hypocenterC59#2763
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #16837344C2semidirect
2. #16837322C2semidirect
5. #16837306C2semidirect
8. #16837267C2semidirect
48. C22⋊C4×C59Q8semidirect
120. C59#2763semidirect
164. C22⋊C4Dic118semidirect