Finite Group Databasebeta

← #21430412

Subgroups — #21430412

Order 7552.

544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14273726C212yesmaximal, complemented
2C2×#14273704C212yesmaximal
3#16837333C212yesmaximal, complemented
4#16837322C212yesmaximal, complemented
5#16837314C211yesmaximal, characteristic
6#16837290C212yesmaximal
7#16837287C212yesmaximal, complemented
8#16837274C211yesmaximal, characteristic
9C59×#399C211yesmaximal, characteristic
10C22×Dic118C2212yes—
11C22×Dic118C2211yescharacteristic
12C23×Dic59C2211yescharacteristic
13#14273726—28no—
14#14273704—28no—
15C2×C4⋊Dic59C2212yes—
16C2×Dic59⋊C4C2212yes—
17C2×Dic59⋊C4C2212yes—
18C2×Dic59⋊C4C2212yes—
19C2×Dic59⋊C4C2212yes—
20C2×C22⋊Dic59C2212yes—
21C2×C22⋊Dic59C2211yescharacteristic
22C2×C4×Dic59C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2212yes—
26C59⋊#95C2212yes—
27C59⋊#95C2211yescharacteristic
28C2×C4⋊C4×C59C2212yes—
29C2×C4⋊C4×C59C2211yescharacteristic
30C2×C22⋊C4×C59C2212yes—
31C2×C22⋊C4×C59C2211yescharacteristic
32#95×C59C2211yescharacteristic
33C24×C59C2311yescharacteristic
34C22×C4×C59C2311yescharacteristic
35C22×C4×C59C2312yes—
36C22×C4×C59C2311yescharacteristic
37C22×C4×C59C2312yes—
38C22×Dic59—24no—
39C22×Dic59—41no—
40C22×Dic59C2312yes—
41C22×Dic59C2312yes—
42C22×Dic59C2312yes—
43C22×Dic59C2312yes—
44C2×Dic118—24no—
45C2×Dic118—24no—
46C2×Dic118—24no—
47C2×Dic118—41no—
48C2×Dic118—42no—
49C4⋊C4×C59—24no—
50C4⋊C4×C59—41no—
51C22⋊C4×C59—24no—
52C22⋊C4×C59—41no—
53C22⋊Dic59—24no—
54C22⋊Dic59Q814yescomplemented
55C4⋊Dic59D418yes—
56Dic59⋊C4—24no—
57Dic59⋊C4—24no—
58Dic59⋊C4—24no—
59Dic59⋊C4—24no—
60C4×Dic59—24no—
61C23×C59—22no—
62C23×C59—21no—
63C23×C59—21no—
64C23×C59—22no—
65C23×C59C2×Q812yes—
66C23×C59C2411yescharacteristic
67C2×C4×C59—22no—
68C2×C4×C59—22no—
69C2×C4×C59—22no—
70C2×C4×C59—22no—
71C2×C4×C59—22no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59C2×D414yes—
77C2×Dic59—22no—
78C2×Dic59—22no—
79C2×Dic59—22no—
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—22no—
83C2×Dic59—22no—
84C2×Dic59—22no—
85C2×Dic59—22no—
86C2×Dic59—44no—
87C2×Dic59C2×D414yes—
88C2×Dic59C2×D414yes—
89C2×Dic59C2×Q814yes—
90Dic118—44no—
91Dic118—48no—
92C22×C59—22no—
93C22×C59—24no—
94C22×C59—42no—
95C22×C59—41no—
96C22×C59—41no—
97C22×C59C22×D412yes—
98C22×C59C22×Q811yescharacteristic
99C22×C59H(5,𝔽2)11yescharacteristic
100C22×C592−1+411yescharacteristic
101C22×C592−1+412yes—
102C4×C59—24no—
103C4×C59—42no—
104C4×C59—41no—
105C4×C59—41no—
106Dic59—24no—
107Dic59—24no—
108Dic59—24no—
109Dic59—42no—
110#2677—591nomaximal, Sylow
111C2×C59—42no—
112C2×C59#53612yes—
113C2×C59#54411yescharacteristic
114C2×C59D4211yescharacteristic
115C2×C59D4×Q812yes—
116C2×C59#56311yescharacteristic
117C2×C4⋊Q8—592no—
118C2×C22⋊Q8—592no—
119#399D5911yescharacteristic
120#398—592no—
121#395—592no—
122#393—591no—
123#388—592no—
124#387—592no—
125#385—591no—
126C59#267711yescharacteristic, complemented, Sylow
127C22×Q8—592no—
128C22×Q8—591no—
129C23×C4—591no—
130C4⋊Q8—1188no—
131C22⋊Q8—1188no—
132C2×C4⋊C4—592no—
133C2×C4⋊C4—592no—
134C2×C4⋊C4—592no—
135C2×C4⋊C4—592no—
136C2×C4⋊C4—592no—
137C2×C4⋊C4C2×D5912yes—
138C2×C4⋊C4C2×D5911yescharacteristic
139C2×C22⋊C4—591no—
140C2×C22⋊C4—592no—
141C2×C22⋊C4C2×D5912yes—
142C2×C22⋊C4C2×D5911yescharacteristic
143C2×C42—592no—
144#95—592no—
145#95—592no—
146#95—592no—
147#95—592no—
148#95—591no—
149#95C2×D5911yescharacteristic
150C24C22×D5911yescharacteristic
151C2×Q8—1184no—
152C2×Q8—1184no—
153C2×Q8—1184no—
154C2×Q8—2361no—
155C2×Q8—2362no—
156C22×C4—592no—
157C22×C4—592no—
158C22×C4—592no—
159C22×C4—592no—
160C22×C4—1184no—
161C22×C4—2361no—
162C22×C4C22×D5911yescharacteristic
163C22×C4C22×D5912yes—
164C22×C4C22×D5911yescharacteristic
165C22×C4C22×D5912yes—
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—598no—
169C4⋊C4—1184no—
170C4⋊C4—1184no—
171C4⋊C4—1184no—
172C4⋊C4—1184no—
173C22⋊C4—24no—
174C22⋊C4—41no—
175C22⋊C4—594no—
176C22⋊C4—1184no—
177C42—1184no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23Q8×D5912yes—
183C23C23×D5911yescharacteristic
184Q8—2368no—
185Q8—2364no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—594no—
196C2×C4—594no—
197C2×C4—594no—
198C2×C4—1182no—
199C2×C4—1182no—
200C2×C4—1182no—
201C2×C4—1182no—
202C2×C4—1182no—
203C2×C4—1182no—
204C2×C4—1182no—
205C2×C4—1182no—
206C2×C4—1182no—
207C2×C4—2364no—
208C2×C4D4×D5914yes—
209C22—22no—
210C22—24no—
211C22—41no—
212C22—42no—
213C22—41no—
214C22C2×D4×D5912yes—
215C22C2×Q8×D5911yescharacteristic
216C22#1427377211yescharacteristic
217C22#1427377412yes—
218C22#1427377411yescharacteristic
219C4—24no—
220C4—41no—
221C4—42no—
222C4—41no—
223C4—1184no—
224C4—1184no—
225C4—1184no—
226C4—2362no—
227C2—42no—
228C2#1683797412yes—
229C2#1683803111yescharacteristic
230C2#1683803711yescharacteristic
231C2#1683806512yes—
232C2#1683815511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#399C2
hypercenter#399D59
hypocenterC59#2677
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14273726C2semidirect
3. #16837333C2semidirect
4. #16837322C2semidirect
7. #16837287C2semidirect
54. C22⋊Dic59Q8semidirect
126. C59#2677semidirect