Finite Group Databasebeta

← #21435174

Subgroups — #21435174

Order 7552.

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

normal subgroups only

This group has 364 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#16837972C211yesmaximal, characteristic
3#16837781C211yesmaximal, characteristic, complemented
4#16837771C211yesmaximal, characteristic, complemented
5#16837756C211yesmaximal, characteristic, complemented
6#16837622C212yesmaximal, complemented
7#16837610C212yesmaximal, complemented
8#16837561C212yesmaximal
9#16837556C212yesmaximal, complemented
10#16837464C211yesmaximal, characteristic, complemented
11C59×#497C211yesmaximal, characteristic, complemented
12C2×#11745687C2211yescharacteristic
13C2×Q8×D59C2211yescharacteristic
14C2×#11745672C2212yes—
15C2×#11745672C2212yes—
16#14273727C2211yescharacteristic
17#14273725C2211yescharacteristic
18#14273722—22no—
19#14273719C2211yescharacteristic
20#14273717—22no—
21#14273714C2211yescharacteristic
22#14273709—22no—
23#14273704—22no—
24#14273703C2212yes—
25Q8×Dic59C2212yes—
26#14273684C2211yescharacteristic
27Dic59⋊C8C2212yes—
28#14273652C2212yes—
29#14273652C2212yes—
30#14273651C2212yes—
31#14273648C2212yes—
32(C2×D59)⋊C8C2212yes—
33C4×C59⋊C8C2211yescharacteristic
34C2×Q16×C59C2212yes—
35C2×SD16×C59C2212yes—
36C4⋊Q8×C59C2211yescharacteristic
37#124×C59C2211yescharacteristic
38#97×C59C2211yescharacteristic
39C2×Q8×C59C2311yescharacteristic
40C2×Q8×C59C2312yes—
41C2×D4×C59C2311yescharacteristic
42C2×C22⋊D59—21no—
43C22×Dic59—21no—
44Q8×D59—42no—
45#11745687—42no—
46C2×C4×D59—21no—
47C2×C4×D59C2311yescharacteristic
48C2×Dic118—21no—
49C2×Dic118C2311yescharacteristic
50C2×Dic118C2311yescharacteristic
51Q16×C59—42no—
52SD16×C59—42no—
53C2×C8×C59C2312yes—
54C4⋊C4×C59—22no—
55C22⋊C4×C59—22no—
56C42×C59C2311yescharacteristic
57C22⋊Dic59—22no—
58#11745672—42no—
59#11745672—42no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4D412yes—
62C4⋊Dic59—22no—
63C4⋊Dic59—22no—
64C4⋊Dic59C2312yes—
65Dic59⋊C4—22no—
66Dic59⋊C4—22no—
67Dic59⋊C4—22no—
68Dic59⋊C4D412yes—
69C4×Dic59—22no—
70C4×Dic59C2311yescharacteristic
71C2×C59⋊C8C2312yes—
72C23×C59—21no—
73C22×D59C2×D411yescharacteristic
74Q8×C59—22no—
75Q8×C59—22no—
76Q8×C59—22no—
77D4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—21no—
80C2×C4×C59C2×D412yes—
81C2×C4×C59C2411yescharacteristic
82C22⋊D59—42no—
83C2×Dic59—22no—
84C2×Dic59—21no—
85C2×Dic59—21no—
86C2×Dic59—42no—
87C2×Dic59C2×D412yes—
88C2×Dic59C2×D411yescharacteristic
89C4×D59—22no—
90C4×D59—42no—
91Dic118—22no—
92Dic118—22no—
93Dic118—42no—
94C8×C59—22no—
95C59⋊C8—22no—
96C22×C59—41no—
97C22×C59—42no—
98C22×C59C22×D411yescharacteristic
99C2×D59—21no—
100C2×D59—22no—
101C4×C59—22no—
102C4×C59—41no—
103C4×C59—42no—
104C4×C59H(5,𝔽2)12yes—
105Dic59—21no—
106Dic59—22no—
107Dic59—41no—
108Dic59—41no—
109Dic59—42no—
110#3223—591nomaximal, Sylow
111C2×C59—81no—
112C2×C59#53511yescharacteristic
113C2×C59#57812yes—
114D59—41no—
115#541—591no—
116#536—591no—
117#497D5911yescharacteristic, complemented
118#494—591no—
119#491—591no—
120#488—591no—
121#464—592no—
122#462—592no—
123#452—592no—
124#451—592no—
125#433—591no—
126C59#322311yescharacteristic, complemented, Sylow
127C2×#41—591no—
128C22×Q8—591no—
129C2×Q16—592no—
130C2×Q16—592no—
131C2×Q16C2×D5912yes—
132C2×SD16C2×D5912yes—
133C4⋊Q8—591no—
134C4⋊Q8—591no—
135C4⋊Q8C2×D5911yescharacteristic
136#126—1182no—
137#125—591no—
138#125—1182no—
139#124—591no—
140#124C2×D5911yescharacteristic
141#123—1182no—
142C22⋊Q8—592no—
143C22⋊Q8—1182no—
144C4×Q8—592no—
145#117—591no—
146C4⋊C8—592no—
147#103—592no—
148#103—592no—
149#103—592no—
150#102—592no—
151C22⋊C8—592no—
152#97C2×D5911yescharacteristic
153C4×C8—591no—
154#41—2362no—
155C2×Q8—591no—
156C2×Q8—591no—
157C2×Q8—1181no—
158C2×Q8—2362no—
159C2×Q8C22×D5911yescharacteristic
160C2×Q8C22×D5912yes—
161C2×D4—1181no—
162C2×D4C22×D5911yescharacteristic
163C22×C4—591no—
164C22×C4—1181no—
165C22×C4—1181no—
166Q16—42no—
167Q16—2362no—
168Q16—2362no—
169SD16—42no—
170C2×C8—592no—
171C2×C8C22×D5912yes—
172C4⋊C4—22no—
173C4⋊C4—592no—
174C4⋊C4—592no—
175C4⋊C4—1182no—
176C4⋊C4—1182no—
177C4⋊C4—1182no—
178C4⋊C4—1182no—
179C4⋊C4—1182no—
180C22⋊C4—22no—
181C22⋊C4—592no—
182C22⋊C4—1182no—
183C22⋊C4—1182no—
184C42—591no—
185C42—1182no—
186C42C22×D5911yescharacteristic
187C23—21no—
188C23—591no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—1182no—
193Q8—1182no—
194Q8—2362no—
195D4—22no—
196D4—2362no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—592no—
200C2×C4—591no—
201C2×C4—1182no—
202C2×C4—1182no—
203C2×C4—1181no—
204C2×C4—1181no—
205C2×C4—2362no—
206C2×C4—2362no—
207C2×C4D4×D5912yes—
208C2×C4C23×D5911yescharacteristic
209C8—22no—
210C8—1182no—
211C22—42no—
212C22—41no—
213C22—1182no—
214C22—1181no—
215C22C2×D4×D5911yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—1182no—
220C4—1181no—
221C4—2362no—
222C4—2361no—
223C4—2361no—
224C4#1427377212yes—
225C2—81no—
226C2—2361no—
227C2#1683796511yescharacteristic
228C2#1683821812yes—
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#497C2
socleC22×C59C22×D4
hypercenter#497D59
hypocenterC59#3223
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16838013C2semidirect
3. #16837781C2semidirect
4. #16837771C2semidirect
5. #16837756C2semidirect
6. #16837622C2semidirect
7. #16837610C2semidirect
9. #16837556C2semidirect
10. #16837464C2semidirect
11. C59×#497C2semidirect
117. #497D59semidirect
126. C59#3223semidirect