Finite Group Databasebeta

← #21436677

Subgroups — #21436677

Order 7552.

374 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 115 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#16838126C211yesmaximal, characteristic, complemented
2#16838057C211yesmaximal, characteristic
3#16837805C211yesmaximal, characteristic, complemented
4#16837739C212yesmaximal, complemented
5#16837725C212yesmaximal, complemented
6#16837688C211yesmaximal, characteristic, complemented
7#16837647C212yesmaximal, complemented
8C4×#11745670C211yesmaximal, characteristic, complemented
9#16837479C211yesmaximal, characteristic, complemented
10#16837443C212yesmaximal, complemented
11C59×#474C211yesmaximal, characteristic, complemented
12C2×#11745685—21no—
13C2×#11745670C2211yescharacteristic
14C2×#11745659C2212yes—
15#14273727C2211yescharacteristic
16#14273722—22no—
17#14273716—21no—
18#14273709—22no—
19#14273707C2212yes—
20#14273706—21no—
21#14273698C2212yes—
22Q8×Dic59—21no—
23C4×Dic118C2211yescharacteristic
24C4×D236C2211yescharacteristic
25#14273685C2212yes—
26C4⋊C4×D59C2212yes—
27#14273660C2212yes—
28#14273660C2212yes—
29#14273659C2212yes—
30#14273658C2211yescharacteristic
31#14273652C2212yes—
32#14273650C2211yescharacteristic
33#14273648C2211yescharacteristic
34(C2×D59)⋊C8C2212yes—
35C4×C59⋊C8C2211yescharacteristic
36C4⋊Q8×C59C2211yescharacteristic
37C4×D4×C59C2211yescharacteristic
38#107×C59C2212yes—
39C4⋊C8×C59C2211yescharacteristic
40#102×C59C2212yes—
41C2×Q8×C59—21no—
42C2×D4×C59C2311yescharacteristic
43C22×C4×C59—21no—
44C2×C22⋊D59—22no—
45#11745685—44no—
46C2×D236D411yescharacteristic
47C2×C4×D59—22no—
48C2×C4×D59C2312yes—
49C2×Dic118—21no—
50C2×Dic118C2311yescharacteristic
51C2×C8×C59C2312yes—
52C4⋊C4×C59—21no—
53C4⋊C4×C59C2312yes—
54C4⋊C4×C59C2311yescharacteristic
55C22⋊C4×C59—21no—
56C42×C59C2311yescharacteristic
57C22⋊Dic59—22no—
58#11745670—22no—
59(C2×D59)⋊C4—22no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4D412yes—
62C4⋊Dic59—21no—
63C4⋊Dic59—22no—
64C4⋊Dic59D411yescharacteristic
65C4⋊Dic59C2311yescharacteristic
66C4⋊Dic59C2312yes—
67Dic59⋊C4—21no—
68Dic59⋊C4—22no—
69C4×Dic59—21no—
70C4×Dic59—22no—
71C2×C59⋊C8C2311yescharacteristic
72C2×C59⋊C8C2311yescharacteristic
73#11745659—24no—
74C23×C59—21no—
75C22×D59C2×D412yes—
76Q8×C59—42no—
77D4×C59—21no—
78D4×C59—21no—
79C2×C4×C59—22no—
80C2×C4×C59—21no—
81C2×C4×C59—42no—
82C2×C4×C59C2×D411yescharacteristic
83C2×C4×C59C2×D411yescharacteristic
84C2×C4×C59C2411yescharacteristic
85C22⋊D59—44no—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—22no—
89C2×Dic59C2×D412yes—
90D236—22no—
91C4×D59—22no—
92C4×D59—22no—
93C4×D59—44no—
94Dic118—21no—
95Dic118—21no—
96Dic118—42no—
97C8×C59—22no—
98C59⋊C8—21no—
99C59⋊C8#4112yescomplemented
100C22×C59—41no—
101C22×C59—41no—
102C22×C59—41no—
103C22×C59C22×D411yescharacteristic
104C2×D59—22no—
105C2×D59—22no—
106C2×D59—22no—
107C4×C59—21no—
108C4×C59—41no—
109C4×C59—42no—
110C4×C59AGL(1,ℤ/8)12yes—
111C4×C59C2×#4111yescharacteristic
112C4×C592−1+411yescharacteristic
113Dic59—22no—
114Dic59—41no—
115Dic59—42no—
116Dic59—41no—
117#3353—591nomaximal, Sylow
118C2×C59—81no—
119C2×C59#54711yescharacteristic
120C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C59#57811yescharacteristic
122D59—42no—
123#556—591no—
124#547—591no—
125#500—591no—
126#484—592no—
127#482—592no—
128#475—591no—
129#474D5911yescharacteristic, complemented
130#468—592no—
131C4×SD16—591no—
132#436—591no—
133#428—592no—
134C59#335311yescharacteristic, complemented, Sylow
135C2×#41—1181no—
136C2×SD16—591no—
137C2×M16—592no—
138C4⋊Q8—591no—
139C4⋊Q8C2×D5911yescharacteristic
140#126—1182no—
141#124—1181no—
142#123—1182no—
143C22⋊Q8—592no—
144C22⋊Q8—1181no—
145#121—592no—
146C4×Q8—591no—
147C4×Q8—1181no—
148C4×D4—591no—
149C4×D4C2×D5911yescharacteristic
150#117—592no—
151C2×C4⋊C4—592no—
152#107—592no—
153#107—592no—
154#107C2×D5912yes—
155#106—592no—
156#106—591no—
157C4⋊C8C2×D5911yescharacteristic
158#103—592no—
159#103—591no—
160#102—591no—
161#102C2×D5912yes—
162C22⋊C8—592no—
163C4×C8—591no—
164#41—2364no—
165C2×Q8—21no—
166C2×Q8—591no—
167C2×Q8—1181no—
168C2×D4—591no—
169C2×D4—1182no—
170C2×D4C22×D5911yescharacteristic
171C22×C4—21no—
172C22×C4—592no—
173C22×C4—1182no—
174SD16—1182no—
175M16—1184no—
176C2×C8—591no—
177C2×C8—591no—
178C2×C8C22×D5912yes—
179C4⋊C4—21no—
180C4⋊C4—591no—
181C4⋊C4—591no—
182C4⋊C4—592no—
183C4⋊C4—1182no—
184C4⋊C4—1182no—
185C4⋊C4—1181no—
186C4⋊C4—1181no—
187C4⋊C4C22×D5912yes—
188C4⋊C4C22×D5911yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—592no—
191C22⋊C4—1182no—
192C22⋊C4—1182no—
193C22⋊C4—1182no—
194C42—1182no—
195C42—1181no—
196C42C22×D5911yescharacteristic
197C23—21no—
198C23—592no—
199Q8—42no—
200Q8—1181no—
201Q8—1181no—
202Q8—2362no—
203D4—21no—
204D4—21no—
205D4—1182no—
206D4—2364no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—592no—
211C2×C4—1182no—
212C2×C4—1182no—
213C2×C4—1181no—
214C2×C4—1181no—
215C2×C4—1182no—
216C2×C4—2364no—
217C2×C4D4×D5911yescharacteristic
218C2×C4D4×D5911yescharacteristic
219C2×C4C23×D5911yescharacteristic
220C8—22no—
221C8—592no—
222C8—1181no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—1182no—
227C22—1182no—
228C22—1182no—
229C22C2×D4×D5911yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—1182no—
234C4—2362no—
235C4—2361no—
236C4—2361no—
237C4#1427375112yes—
238C4#41×D5911yescharacteristic
239C4#1427377511yescharacteristic
240C2—81no—
241C2—2362no—
242C2#1683805411yescharacteristic
243C2C2×#1427375111yescharacteristic
244C2#1683821811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#474C2
socleC22×C59C22×D4
hypercenter#474D59
hypocenterC59#3353
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16838126C2semidirect
3. #16837805C2semidirect
4. #16837739C2semidirect
5. #16837725C2semidirect
6. #16837688C2semidirect
7. #16837647C2semidirect
8. C4×#11745670C2semidirect
9. #16837479C2semidirect
10. #16837443C2semidirect
11. C59×#474C2semidirect
99. C59⋊C8#41semidirect
129. #474D59semidirect
134. C59#3353semidirect