Finite Group Databasebeta

← #21435161

Subgroups — #21435161

Order 7552.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#16838013C211yesmaximal, characteristic, complemented
2#16837965C211yesmaximal, characteristic
3#16837794C211yesmaximal, characteristic, complemented
4#16837766C211yesmaximal, characteristic, complemented
5#16837757C211yesmaximal, characteristic, complemented
6#16837622C212yesmaximal, complemented
7#16837600C212yesmaximal, complemented
8#16837561C212yesmaximal, complemented
9#16837551C212yesmaximal, complemented
10#16837461C211yesmaximal, characteristic, complemented
11C59×#492C211yesmaximal, characteristic, complemented
12C2×#11745687C2211yescharacteristic
13C2×D4×D59C2211yescharacteristic
14C2×#11745672C2212yescomplemented
15C2×#11745671C2212yes—
16#14273723C2211yescharacteristic
17#14273722—22no—
18#14273719C2211yescharacteristic, complemented
19#14273717—22no—
20#14273715C2211yescharacteristic
21#14273714C2211yescharacteristic
22#14273703C2212yes—
23#14273699—22no—
24#14273695—22no—
25Q8×Dic59C2212yescomplemented
26#14273684C2211yescharacteristic
27Dic59⋊C8C2212yescomplemented
28#14273652C2212yescomplemented
29#14273649C2212yes—
30#14273648C2212yes—
31#14273648C2212yescomplemented
32(C2×D59)⋊C8C2212yes—
33#14273638C2211yescharacteristic, complemented
34C2×SD16×C59C2212yes—
35C2×SD16×C59C2212yescomplemented
36C4⋊Q8×C59C2211yescharacteristic, complemented
37#127×C59C2211yescharacteristic
38C4×C8×C59C2211yescharacteristic, complemented
39C23×D59—21no—
40C2×Q8×C59C2312yes—
41C2×D4×C59—22no—
42C2×D4×C59C2311yescharacteristic
43C2×D4×C59C2311yescharacteristic
44C2×C22⋊D59—21no—
45C2×C22⋊D59—21no—
46C2×C22⋊D59—22no—
47C22×Dic59—21no—
48#11745687—42no—
49D4×D59—42no—
50C2×D236C2311yescharacteristic
51C2×C4×D59C2311yescharacteristic
52C2×Dic118C2311yescharacteristic
53SD16×C59—42no—
54SD16×C59—42no—
55C2×C8×C59C2312yes—
56C4⋊C4×C59—22no—
57C42×C59C2311yescharacteristic
58C22⋊Dic59—22no—
59#11745672—42no—
60#11745671—42no—
61(C2×D59)⋊C4—22no—
62(C2×D59)⋊C4—22no—
63(C2×D59)⋊C4D412yes—
64C4⋊Dic59—22no—
65C4⋊Dic59C2312yes—
66Dic59⋊C4—22no—
67Dic59⋊C4—22no—
68Dic59⋊C4D412yes—
69C4×Dic59—22no—
70C4×Dic59C2311yescharacteristic
71C2×C59⋊C8C2312yes—
72C23×C59—21no—
73C23×C59—21no—
74C22×D59—21no—
75C22×D59—42no—
76C22×D59—82no—
77C22×D59C2×D411yescharacteristic
78Q8×C59—22no—
79Q8×C59—22no—
80D4×C59—22no—
81D4×C59—22no—
82D4×C59—82no—
83C2×C4×C59—22no—
84C2×C4×C59C2×D412yes—
85C2×C4×C59C2411yescharacteristic
86C22⋊D59—42no—
87C22⋊D59—42no—
88C22⋊D59—82no—
89C2×Dic59—22no—
90C2×Dic59—21no—
91C2×Dic59—42no—
92C2×Dic59C2×D412yes—
93C2×Dic59C2×D411yescharacteristic
94D236—22no—
95C4×D59—22no—
96Dic118—22no—
97C8×C59—22no—
98C59⋊C8—22no—
99C22×C59—41no—
100C22×C59—41no—
101C22×C59—42no—
102C22×C59—42no—
103C22×C59C22×D411yescharacteristic
104C2×D59—21no—
105C2×D59—22no—
106C2×D59—41no—
107C2×D59—42no—
108C2×D59—84no—
109C4×C59—22no—
110C4×C59—42no—
111C4×C59H(5,𝔽2)12yes—
112Dic59—21no—
113Dic59—22no—
114Dic59—41no—
115Dic59—42no—
116#3222—591nomaximal, Sylow
117C2×C59—81no—
118C2×C59—81no—
119C2×C59#53511yescharacteristic
120C2×C59#57712yes—
121D59—41no—
122D59—81no—
123#541—591no—
124#535—591no—
125#497—591no—
126#492D5911yescharacteristic, complemented
127#490—591no—
128#488—591no—
129#464—592no—
130#460—592no—
131#452—592no—
132#450—592no—
133#433—591no—
134C59#322211yescharacteristic, complemented, Sylow
135C2×#41—591no—
136C22×D4—591no—
137C2×Q16—592no—
138C2×SD16—592no—
139C2×SD16C2×D5912yes—
140C2×SD16C2×D5912yescomplemented
141C4⋊Q8C2×D5911yescharacteristic, complemented
142#127—591no—
143#127C2×D5911yescharacteristic
144#126—1182no—
145#125—591no—
146#125—1182no—
147#124—591no—
148#124—591no—
149C22⋊Q8—592no—
150#121—1182no—
151#120—1182no—
152C4×Q8—592no—
153#117—591no—
154C4⋊C8—592no—
155#103—592no—
156#102—592no—
157#102—592no—
158#102—592no—
159C22⋊C8—592no—
160#97—591no—
161C4×C8C2×D5911yescharacteristic, complemented
162C24—1181no—
163#41—2362no—
164C2×Q8—591no—
165C2×Q8C22×D5912yes—
166C2×D4—22no—
167C2×D4—591no—
168C2×D4—1182no—
169C2×D4—1181no—
170C2×D4—1181no—
171C2×D4—2362no—
172C2×D4C22×D5911yescharacteristic
173C2×D4C22×D5911yescharacteristic
174C22×C4—591no—
175C22×C4—1181no—
176Q16—2362no—
177SD16—42no—
178SD16—42no—
179SD16—2362no—
180C2×C8—592no—
181C2×C8C22×D5912yes—
182C4⋊C4—22no—
183C4⋊C4—592no—
184C4⋊C4—592no—
185C4⋊C4—1182no—
186C4⋊C4—1182no—
187C4⋊C4—1182no—
188C22⋊C4—592no—
189C22⋊C4—1182no—
190C22⋊C4—1182no—
191C22⋊C4—1182no—
192C42—591no—
193C42—1182no—
194C42C22×D5911yescharacteristic
195C23—21no—
196C23—21no—
197C23—591no—
198C23—1181no—
199C23—4722no—
200C23—2362no—
201Q8—22no—
202Q8—22no—
203Q8—1182no—
204D4—22no—
205D4—22no—
206D4—82no—
207D4—1182no—
208D4—4722no—
209D4—2362no—
210D4—2362no—
211C2×C4—22no—
212C2×C4—592no—
213C2×C4—591no—
214C2×C4—1182no—
215C2×C4—1182no—
216C2×C4—1181no—
217C2×C4—2362no—
218C2×C4D4×D5912yes—
219C2×C4C23×D5911yescharacteristic
220C8—22no—
221C8—1182no—
222C22—42no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—1182no—
227C22—1181no—
228C22—4724no—
229C22—2362no—
230C22—2361no—
231C22C2×D4×D5911yescharacteristic
232C4—22no—
233C4—42no—
234C4—1182no—
235C4—1181no—
236C4—2362no—
237C4—2361no—
238C4#1427377212yes—
239C2—81no—
240C2—81no—
241C2—4721no—
242C2—2361no—
243C2#1683796511yescharacteristic
244C2#1683820812yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#492C2
socleC22×C59C22×D4
hypercenter#492D59
hypocenterC59#3222
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #16838013C2semidirect
3. #16837794C2semidirect
4. #16837766C2semidirect
5. #16837757C2semidirect
6. #16837622C2semidirect
7. #16837600C2semidirect
8. #16837561C2semidirect
9. #16837551C2semidirect
10. #16837461C2semidirect
11. C59×#492C2semidirect
14. C2×#11745672C22semidirect
18. #14273719C22semidirect
25. Q8×Dic59C22semidirect
27. Dic59⋊C8C22semidirect
28. #14273652C22semidirect
31. #14273648C22semidirect
33. #14273638C22semidirect
35. C2×SD16×C59C22semidirect
36. C4⋊Q8×C59C22semidirect
38. C4×C8×C59C22semidirect
126. #492D59semidirect
134. C59#3222semidirect
140. C2×SD16C2×D59semidirect
141. C4⋊Q8C2×D59semidirect
161. C4×C8C2×D59semidirect