Finite Group Databasebeta

← #21430268

Subgroups — #21430268

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#16837325C212yesmaximal
3#16837323C212yesmaximal, complemented
4#16837314C212yesmaximal
5#16837306C212yesmaximal, complemented
6#16837285C212yesmaximal, complemented
7#16837262C211yesmaximal, characteristic
8C4×C22⋊Dic59C211yesmaximal, characteristic
9C59×#381C211yesmaximal, characteristic
10C22×Dic118C2212yes—
11C2×C4⋊Dic59C2212yes—
12C2×C4⋊Dic59C2212yes—
13C2×Dic59⋊C4C2212yes—
14C2×Dic59⋊C4C2212yes—
15C2×C22⋊Dic59C2212yes—
16C2×C22⋊Dic59C2211yescharacteristic
17C2×C22⋊Dic59C2211yescharacteristic
18C2×C4×Dic59C2212yes—
19C59⋊#95C2211yescharacteristic
20C59⋊#95C2211yescharacteristic
21C59⋊#95C2212yes—
22C59⋊#95C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2212yes—
26C23×C4×C59C2211yescharacteristic
27C2×C22⋊C4×C59C2212yes—
28#95×C59C2211yescharacteristic
29#95×C59C2211yescharacteristic
30#95×C59C2212yes—
31C24×C59C2311yescharacteristic
32C22×C4×C59—42no—
33C22×C4×C59Q814yescomplemented
34C22×C4×C59C2312yes—
35C22×C4×C59C2312yes—
36C22×C4×C59C2311yescharacteristic
37C22×C4×C59C2311yescharacteristic
38C22×Dic59C2312yes—
39C22×Dic59C2312yes—
40C22×Dic59C2312yes—
41C22×Dic59C2312yes—
42C2×Dic118—24no—
43C2×Dic118—42no—
44C2×Dic118—42no—
45C22⋊C4×C59—42no—
46C22⋊Dic59—24no—
47C22⋊Dic59—41no—
48C22⋊Dic59—41no—
49C4⋊Dic59—24no—
50C4⋊Dic59—42no—
51Dic59⋊C4—42no—
52Dic59⋊C4—42no—
53C4×Dic59—42no—
54C23×C59—22no—
55C23×C59—21no—
56C23×C59—22no—
57C23×C59—21no—
58C23×C59C2×Q812yes—
59C23×C59C2411yescharacteristic
60C2×C4×C59—22no—
61C2×C4×C59—22no—
62C2×C4×C59—22no—
63C2×C4×C59—22no—
64C2×C4×C59—22no—
65C2×C4×C59—22no—
66C2×C4×C59—22no—
67C2×C4×C59—22no—
68C2×C4×C59—44no—
69C2×C4×C59C2×Q812yes—
70C2×C4×C59C2×Q812yes—
71C2×Dic59—22no—
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#4114yes—
83Dic118—48no—
84C22×C59—22no—
85C22×C59—22no—
86C22×C59—22no—
87C22×C59—42no—
88C22×C59—42no—
89C22×C59C22×Q811yescharacteristic
90C22×C59C2×#4112yes—
91C22×C59H(5,𝔽2)12yes—
92C22×C59H(5,𝔽2)11yescharacteristic
93C22×C592−1+411yescharacteristic
94C4×C59—22no—
95C4×C59—22no—
96C4×C59—42no—
97C4×C59—42no—
98Dic59—24no—
99Dic59—42no—
100Dic59—42no—
101Dic59—42no—
102#2665—591nomaximal, Sylow
103C2×C59—42no—
104C2×C59#53311yescharacteristic
105C2×C59#53812yes—
106C2×C59#54212yes—
107C2×C59#54311yescharacteristic
108C2×C59#54411yescharacteristic
109#400—592no—
110#396—592no—
111#395—592no—
112#393—592no—
113#391—592no—
114#387—592no—
115#381D5911yescharacteristic
116#380—591no—
117C4×C22⋊C4—591no—
118C59#266511yescharacteristic, complemented, Sylow
119C22×Q8—592no—
120C23×C4C2×D5911yescharacteristic
121C2×C4⋊C4—592no—
122C2×C4⋊C4—592no—
123C2×C4⋊C4—592no—
124C2×C4⋊C4—592no—
125C2×C22⋊C4—592no—
126C2×C22⋊C4—591no—
127C2×C22⋊C4—591no—
128C2×C22⋊C4C2×D5912yes—
129C2×C42—592no—
130#95—591no—
131#95—591no—
132#95—592no—
133#95—592no—
134#95—592no—
135#95—592no—
136#95—592no—
137#95C2×D5911yescharacteristic
138#95C2×D5911yescharacteristic
139#95C2×D5912yes—
140C24C22×D5911yescharacteristic
141C2×Q8—1184no—
142C2×Q8—2362no—
143C2×Q8—2362no—
144C22×C4—42no—
145C22×C4—592no—
146C22×C4—592no—
147C22×C4—592no—
148C22×C4—592no—
149C22×C4Dic11814yescomplemented
150C22×C4C22×D5912yes—
151C22×C4C22×D5912yes—
152C22×C4C22×D5911yescharacteristic
153C22×C4C22×D5911yescharacteristic
154C4⋊C4—1184no—
155C4⋊C4—2362no—
156C4⋊C4—2362no—
157C4⋊C4—2362no—
158C22⋊C4—42no—
159C22⋊C4—1184no—
160C22⋊C4—2361no—
161C22⋊C4—2361no—
162C42—2362no—
163C23—22no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23C2×Dic11812yes—
168C23C23×D5911yescharacteristic
169Q8—2368no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—594no—
180C2×C4—1182no—
181C2×C4—1182no—
182C2×C4—1182no—
183C2×C4—1182no—
184C2×C4—1182no—
185C2×C4—1182no—
186C2×C4—1182no—
187C2×C4—1182no—
188C2×C4—1182no—
189C2×C4—1182no—
190C2×C4—1182no—
191C2×C4C2×Dic11812yes—
192C2×C4C2×Dic11812yes—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C22×Dic11811yescharacteristic
199C22#41×D5912yes—
200C22#1427377212yes—
201C22#1427377311yescharacteristic
202C22#1427377411yescharacteristic
203C4—22no—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—1184no—
208C4—2362no—
209C4—2362no—
210C4—2362no—
211C2—42no—
212C2#1683795511yescharacteristic
213C2#1683798812yes—
214C2#1683802012yes—
215C2#1683802811yescharacteristic
216C2#1683803011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#381C2
hypercenter#381D59
hypocenterC59#2665
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #16837344C2semidirect
3. #16837323C2semidirect
5. #16837306C2semidirect
6. #16837285C2semidirect
33. C22×C4×C59Q8semidirect
118. C59#2665semidirect
149. C22×C4Dic118semidirect