Finite Group Databasebeta

← #21434916

Subgroups — #21434916

Order 7552.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22⋊Q8×D59C212yesmaximal
2#16837734C212yesmaximal, complemented
3#16837731C212yesmaximal, complemented
4#16837660C211yesmaximal, characteristic, complemented
5#16837658C212yesmaximal, complemented
6#16837559C212yesmaximal, complemented
7#16837556C212yesmaximal, complemented
8#16837365C211yesmaximal, characteristic
9C59×#470C211yesmaximal, characteristic, complemented
10C2×Q8×D59C2212yes—
11C22×C4×D59C2211yescharacteristic
12C2×Dic236C2211yescharacteristic, complemented
13C2×#11745672C2212yescomplemented
14C2×#11745664C2211yescharacteristic
15#14273707C2212yes—
16#14273706C2212yes—
17#14273705—22no—
18#14273704—22no—
19#14273703C2212yes—
20C4⋊C4×D59—22no—
21C4⋊C4×D59C2212yes—
22C22⋊C4×D59—22no—
23#14273659C2211yescharacteristic, complemented
24#14273658C2212yescomplemented
25#14273652C2212yescomplemented
26#14273651C2212yescomplemented
27#14273650C2212yescomplemented
28(C2×D59)⋊C8C2212yes—
29(C2×D59)⋊C8C2212yes—
30C2×Q16×C59C2211yescharacteristic, complemented
31C2×M16×C59C2211yescharacteristic
32C22⋊Q8×C59C2212yes—
33#106×C59C2211yescharacteristic, complemented
34#103×C59C2212yescomplemented
35C23×D59D411yescharacteristic
36C2×Q8×C59C2312yes—
37C22×C4×C59C2311yescharacteristic
38C22×Dic59D411yescharacteristic
39Q8×D59—42no—
40Q8×D59—42no—
41C2×C4×D59—22no—
42C2×C4×D59—22no—
43C2×C4×D59—41no—
44C2×C4×D59—41no—
45C2×C4×D59D412yes—
46C2×C4×D59C2312yes—
47C2×Dic118—22no—
48C2×Dic118C2312yes—
49Q16×C59—41no—
50M16×C59—41no—
51C2×C8×C59C2311yescharacteristic
52C2×C8×C59C2311yescharacteristic
53C4⋊C4×C59—22no—
54C4⋊C4×C59C2312yes—
55C22⋊C4×C59—22no—
56C22⋊Dic59—22no—
57#11745672—42no—
58(C2×D59)⋊C4—22no—
59(C2×D59)⋊C4—22no—
60C4⋊Dic59—22no—
61C4⋊Dic59C2312yes—
62Dic59⋊C4—22no—
63Dic59⋊C4—22no—
64Dic59⋊C4—22no—
65#11745664—41no—
66C2×C59⋊C8C2312yes—
67Dic236—41no—
68C23×C59C2×D411yescharacteristic
69C22×D59—24no—
70C22×D59—41no—
71C22×D59C2×D412yes—
72Q8×C59—22no—
73Q8×C59—22no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2411yescharacteristic
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—41no—
83C2×Dic59C2×D412yes—
84C4×D59—22no—
85C4×D59—22no—
86C4×D59—42no—
87C4×D59—44no—
88C4×D59—44no—
89Dic118—22no—
90Dic118—22no—
91Dic118—44no—
92C8×C59—21no—
93C8×C59—21no—
94C59⋊C8—22no—
95C22×C59—21no—
96C22×C59—22no—
97C22×C59C22×D411yescharacteristic
98C2×D59—22no—
99C2×D59—22no—
100C2×D59—44no—
101C2×D59#13714yes—
102C4×C59—21no—
103C4×C59—42no—
104C4×C59—42no—
105C4×C59H(5,𝔽2)11yescharacteristic
106C4×C59H(5,𝔽2)11yescharacteristic
107Dic59—22no—
108Dic59—42no—
109Dic59—42no—
110#3195—591nomaximal, Sylow
111C2×C59—41no—
112C2×C59#53411yescharacteristic
113C2×C59C2×#13712yes—
114D59—24no—
115C2×C22⋊Q8—592no—
116#483—592no—
117#483—592no—
118#470—591no—
119#470—592no—
120#470D5911yescharacteristic, complemented
121#451—592no—
122#451—592no—
123#407—591no—
124C59#319511yescharacteristic, complemented, Sylow
125C22×Q8—592no—
126C23×C4—591no—
127C2×Q16—592no—
128C2×Q16—591no—
129C2×Q16C2×D5911yescharacteristic, complemented
130C2×M16—591no—
131C2×M16C2×D5911yescharacteristic
132C22⋊Q8—592no—
133C22⋊Q8—592no—
134C22⋊Q8—592no—
135C22⋊Q8—1182no—
136C22⋊Q8—1182no—
137C22⋊Q8C2×D5912yes—
138C2×C4⋊C4—592no—
139C2×C4⋊C4—1182no—
140C2×C22⋊C4—1182no—
141#106—591no—
142#106—592no—
143#106C2×D5911yescharacteristic, complemented
144#103—592no—
145#103—592no—
146#103—592no—
147#103C2×D5912yescomplemented
148C22⋊C8—592no—
149C22⋊C8—592no—
150C24—591no—
151C2×Q8—592no—
152C2×Q8—1182no—
153C2×Q8—2362no—
154C2×Q8—2362no—
155C2×Q8C22×D5912yes—
156C22×C4—591no—
157C22×C4—592no—
158C22×C4—592no—
159C22×C4—1182no—
160C22×C4—1182no—
161C22×C4—2361no—
162C22×C4—2361no—
163C22×C4C22×D5911yescharacteristic
164Q16—41no—
165Q16—2361no—
166Q16—2362no—
167M16—41no—
168M16—2361no—
169C2×C8—592no—
170C2×C8C22×D5911yescharacteristic
171C2×C8C22×D5911yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—592no—
174C4⋊C4—1182no—
175C4⋊C4—1182no—
176C4⋊C4—1182no—
177C4⋊C4—1182no—
178C4⋊C4C22×D5912yes—
179C22⋊C4—22no—
180C22⋊C4—1182no—
181C22⋊C4—1182no—
182C22⋊C4—1182no—
183C23—592no—
184C23—1184no—
185C23—2361no—
186C23D4×D5911yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—1182no—
190Q8—1182no—
191Q8—2364no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—592no—
197C2×C4—1182no—
198C2×C4—1182no—
199C2×C4—1182no—
200C2×C4—1182no—
201C2×C4—2362no—
202C2×C4—2364no—
203C2×C4—2364no—
204C2×C4—2361no—
205C2×C4D4×D5911yescharacteristic
206C2×C4C23×D5911yescharacteristic
207C8—21no—
208C8—21no—
209C8—1182no—
210C22—22no—
211C22—21no—
212C22—594no—
213C22—1182no—
214C22—1182no—
215C22—2364no—
216C22C2×D4×D5911yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—1182no—
221C4—2362no—
222C4—2362no—
223C4#1427377211yescharacteristic
224C4#1427377311yescharacteristic
225C2—41no—
226C2—1184no—
227C2#1683796011yescharacteristic
228C2#137×D5912yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#470C2
socleC22×C59C22×D4
hypercenter#470D59
hypocenterC59#3195
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #16837734C2semidirect
3. #16837731C2semidirect
4. #16837660C2semidirect
5. #16837658C2semidirect
6. #16837559C2semidirect
7. #16837556C2semidirect
9. C59×#470C2semidirect
12. C2×Dic236C22semidirect
13. C2×#11745672C22semidirect
23. #14273659C22semidirect
24. #14273658C22semidirect
25. #14273652C22semidirect
26. #14273651C22semidirect
27. #14273650C22semidirect
30. C2×Q16×C59C22semidirect
33. #106×C59C22semidirect
34. #103×C59C22semidirect
120. #470D59semidirect
124. C59#3195semidirect
129. C2×Q16C2×D59semidirect
143. #106C2×D59semidirect
147. #103C2×D59semidirect