Finite Group Databasebeta

← #21434917

Subgroups — #21434917

Order 7552.

410 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14273706C211yesmaximal, characteristic
2#16837732C212yesmaximal, complemented
3#16837730C212yesmaximal, complemented
4#16837658C214yesmaximal, complemented
5#16837558C212yesmaximal, complemented
6#16837557C212yesmaximal, complemented
7#16837366C211yesmaximal, characteristic
8C2×C22⋊Q8×C59C211yesmaximal, characteristic
9C22×Dic118C2211yescharacteristic
10C2×#11745672C2214yescomplemented
11C2×#11745664C2212yes—
12#14273706—21no—
13#14273706—21no—
14#14273706C2212yes—
15#14273706C2212yes—
16C2×C4⋊Dic59C2211yescharacteristic
17C2×Dic59⋊C4—21no—
18C2×C22⋊Dic59—21no—
19#14273658C2214yescomplemented
20#14273651C2214yescomplemented
21#14273650C2214yescomplemented
22C22⋊(C59⋊C8)C2212yes—
23C22⋊(C59⋊C8)C2212yes—
24C22×Q8×C59C2211yescharacteristic
25C23×C4×C59C2211yescharacteristic
26C22⋊Q8×C59—21no—
27C22⋊Q8×C59—21no—
28C22⋊Q8×C59C2212yes—
29C22⋊Q8×C59C2212yes—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59C2211yescharacteristic
32C2×C22⋊C4×C59—21no—
33C24×C59D411yescharacteristic
34C2×Q8×C59—21no—
35C2×Q8×C59—42no—
36C2×Q8×C59C2312yes—
37C22×C4×C59—21no—
38C22×C4×C59—21no—
39C22×C4×C59—42no—
40C22×C4×C59D411yescharacteristic
41C22×C4×C59D411yescharacteristic
42C22×C4×C59D411yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×C4×C59C2311yescharacteristic
45C22×C4×C59C2311yescharacteristic
46C22×Dic59—21no—
47C22×Dic59—21no—
48C2×Dic118—21no—
49C2×Dic118—42no—
50C2×Dic118C2312yes—
51C4⋊C4×C59—22no—
52C4⋊C4×C59—22no—
53C4⋊C4×C59—21no—
54C4⋊C4×C59C2312yes—
55C22⋊C4×C59—22no—
56C22⋊C4×C59—22no—
57C22⋊Dic59—22no—
58C22⋊Dic59—22no—
59#11745672—44no—
60C4⋊Dic59—21no—
61C4⋊Dic59C2312yes—
62Dic59⋊C4—22no—
63Dic59⋊C4—22no—
64#11745664—42no—
65C2×C59⋊C8C2314yes—
66C23×C59—22no—
67C23×C59—22no—
68C23×C59—41no—
69C23×C59C2×D411yescharacteristic
70C23×C59C2×D411yescharacteristic
71C23×C59C2×D411yescharacteristic
72Q8×C59—24no—
73Q8×C59—42no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59—41no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59C2×D411yescharacteristic
85C2×C4×C59C2×D411yescharacteristic
86C2×C4×C59C2×D411yescharacteristic
87C2×C4×C59C2411yescharacteristic
88C2×Dic59—22no—
89C2×Dic59—22no—
90C2×Dic59—42no—
91C2×Dic59—42no—
92Dic118—24no—
93Dic118—42no—
94C59⋊C8—24no—
95C22×C59—21no—
96C22×C59—21no—
97C22×C59—21no—
98C22×C59—21no—
99C22×C59—21no—
100C22×C59—21no—
101C22×C59—21no—
102C22×C59—44no—
103C22×C59#13712yes—
104C22×C59#13712yes—
105C22×C59C22×D411yescharacteristic
106C4×C59—21no—
107C4×C59—21no—
108C4×C59—21no—
109C4×C59—42no—
110C4×C59—42no—
111C4×C59H(5,𝔽2)12yes—
112Dic59—42no—
113Dic59—42no—
114#3195—591nomaximal, Sylow
115C2×C59—22no—
116C2×C59—22no—
117C2×C59—41no—
118C2×C59#53411yescharacteristic
119C2×C59C2×#13711yescharacteristic
120C2×C59C2×#13711yescharacteristic
121C2×C22⋊Q8—591no—
122C2×C22⋊Q8D5911yescharacteristic
123#483—592no—
124#483—592no—
125#470—594no—
126#451—592no—
127#451—592no—
128#407—591no—
129C59#319511yescharacteristic, complemented, Sylow
130C22×Q8—591no—
131C22×Q8C2×D5911yescharacteristic
132C23×C4C2×D5911yescharacteristic
133C2×Q16—594no—
134C2×M16—592no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—592no—
138C22⋊Q8—592no—
139C22⋊Q8—1181no—
140C22⋊Q8—1181no—
141C22⋊Q8C2×D5912yes—
142C22⋊Q8C2×D5912yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—591no—
145C2×C4⋊C4—1181no—
146C2×C4⋊C4C2×D5911yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—1181no—
149#106—594no—
150#103—594no—
151#103—594no—
152C22⋊C8—592no—
153C22⋊C8—592no—
154C24C22⋊D5911yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—592no—
158C2×Q8—1181no—
159C2×Q8—2362no—
160C2×Q8C22×D5912yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—1181no—
165C22×C4—1181no—
166C22×C4C22⋊D5911yescharacteristic
167C22×C4C22⋊D5911yescharacteristic
168C22×C4C22⋊D5911yescharacteristic
169C22×C4C22×D5911yescharacteristic
170C22×C4C22×D5911yescharacteristic
171C22×C4C22×D5911yescharacteristic
172Q16—2364no—
173M16—2362no—
174C2×C8—594no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—592no—
179C4⋊C4—1181no—
180C4⋊C4—1182no—
181C4⋊C4—1182no—
182C4⋊C4C22×D5912yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—1182no—
186C22⋊C4—1182no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23C2×C22⋊D5911yescharacteristic
191C23C2×C22⋊D5911yescharacteristic
192C23C2×C22⋊D5911yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—1184no—
196Q8—2362no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—1182no—
208C2×C4—1182no—
209C2×C4—2362no—
210C2×C4—2362no—
211C2×C4C2×C22⋊D5911yescharacteristic
212C2×C4C2×C22⋊D5911yescharacteristic
213C2×C4C2×C22⋊D5911yescharacteristic
214C2×C4C23×D5911yescharacteristic
215C8—1184no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—44no—
224C22#1427375712yes—
225C22#1427375812yes—
226C22C22×C22⋊D5911yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—2362no—
233C4—2362no—
234C4#1427377212yes—
235C2—22no—
236C2—22no—
237C2—41no—
238C2#1683796111yescharacteristic
239C2C2×#1427375711yescharacteristic
240C2C2×#1427375811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×C22⋊Q8×C59C2
socleC22×C59C22×D4
hypercenterC2×C22⋊Q8D59
hypocenterC59#3195
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16837732C2semidirect
3. #16837730C2semidirect
4. #16837658C2semidirect
5. #16837558C2semidirect
6. #16837557C2semidirect
10. C2×#11745672C22semidirect
19. #14273658C22semidirect
20. #14273651C22semidirect
21. #14273650C22semidirect
129. C59#3195semidirect