Finite Group Databasebeta

← #21436023

Subgroups — #21436023

Order 7552.

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

normal subgroups only

This group has 398 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#16838118C211yesmaximal, characteristic, complemented
2#16838055C211yesmaximal, characteristic, complemented
3#16837742C212yesmaximal, complemented
4#16837689C211yesmaximal, characteristic, complemented
5#16837634C212yesmaximal, complemented
6#16837604C211yesmaximal, characteristic, complemented
7#16837550C212yesmaximal, complemented
8C4×D472C211yesmaximal, characteristic, complemented
9C8×D236C211yesmaximal, characteristic, complemented
10#16837397C212yesmaximal, complemented
11C59×#487C211yesmaximal, characteristic, complemented
12C2×#11745689C2212yes—
13C2×#11745671C2212yes—
14C2×D472C2211yescharacteristic
15C2×C8×D59C2212yes—
16#14273716—22no—
17#14273712—22no—
18#14273703C2212yes—
19#14273702—21no—
20#14273702—21no—
21#14273702C2212yes—
22C4×D236C2211yescharacteristic
23C4×D236C2211yescharacteristic
24C4×D236C2211yescharacteristic
25#14273685—21no—
26#14273685C2212yes—
27C4⋊C4×D59—21no—
28#14273661C2211yescharacteristic
29#14273658C2212yes—
30C4⋊(C59⋊C8)C2211yescharacteristic
31#14273651C2212yes—
32#14273649C2211yescharacteristic
33#14273649C2211yescharacteristic
34#14273647C2212yes—
35(C2×D59)⋊C8C2212yes—
36C4⋊Q8×C59C2211yescharacteristic
37C4⋊Q8×C59C2211yescharacteristic
38#103×C59C2212yes—
39#103×C59C2212yes—
40C4×C8×C59C2211yescharacteristic
41C2×Q8×C59—21no—
42C2×Q8×C59C2312yes—
43#11745689—24no—
44#11745689—42no—
45C2×D236—22no—
46C2×D236—22no—
47C2×D236D411yescharacteristic
48C2×D236C2311yescharacteristic
49C2×D236C2311yescharacteristic
50C2×C4×D59—21no—
51C2×C4×D59—21no—
52C2×C4×D59—22no—
53C2×C4×D59C2312yes—
54C2×C8×C59C2311yescharacteristic
55C2×C8×C59C2311yescharacteristic
56C4⋊C4×C59—21no—
57C4⋊C4×C59—21no—
58C4⋊C4×C59—21no—
59C4⋊C4×C59C2312yes—
60C42×C59C2311yescharacteristic
61#11745671—42no—
62(C2×D59)⋊C4—22no—
63(C2×D59)⋊C4—22no—
64(C2×D59)⋊C4—21no—
65(C2×D59)⋊C4—21no—
66(C2×D59)⋊C4—22no—
67(C2×D59)⋊C4D412yes—
68C4⋊Dic59D411yescharacteristic
69C4⋊Dic59C2311yescharacteristic
70C4⋊Dic59C2311yescharacteristic
71Dic59⋊C4—22no—
72C4×Dic59—22no—
73C2×C59⋊C8C2312yes—
74D472—22no—
75C8×D59—42no—
76C22×D59—21no—
77C22×D59—21no—
78C22×D59C2×D412yes—
79Q8×C59—22no—
80Q8×C59—22no—
81Q8×C59—81no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59C2×D411yescharacteristic
85C2×C4×C59C2×D411yescharacteristic
86C2×C4×C59C2411yescharacteristic
87C2×Dic59—21no—
88C2×Dic59—21no—
89C2×Dic59C2×D412yes—
90D236—22no—
91D236—21no—
92D236—21no—
93D236—21no—
94D236—44no—
95D236—82no—
96D236#4112yes—
97C4×D59—22no—
98C4×D59—22no—
99C4×D59—44no—
100C4×D59—42no—
101C4×D59—42no—
102C8×C59—21no—
103C8×C59—21no—
104C59⋊C8—22no—
105C22×C59C22×D411yescharacteristic
106C2×D59—22no—
107C2×D59—22no—
108C2×D59—22no—
109C2×D59—22no—
110C2×D59—41no—
111C2×D59—41no—
112C2×D59—41no—
113C2×D59—41no—
114C2×D59—41no—
115C4×C59—21no—
116C4×C59—42no—
117C4×C59—42no—
118C4×C59#13512yes—
119C4×C59C2×#4111yescharacteristic
120C4×C59H(5,𝔽2)11yescharacteristic
121Dic59—22no—
122Dic59—41no—
123Dic59—41no—
124#3303—591nomaximal, Sylow
125C2×C59#54611yescharacteristic
126C2×C59C2×#13511yescharacteristic
127C2×C59#57711yescharacteristic
128D59—42no—
129D59—42no—
130D59—81no—
131#555—591no—
132#547—591no—
133#487D5911yescharacteristic, complemented
134#485—592no—
135#476—591no—
136#465—592no—
137#461—591no—
138#449—592no—
139C4×D8—591no—
140C8×D4—591no—
141#416—592no—
142C59#330311yescharacteristic, complemented, Sylow
143C2×#41—592no—
144C2×SD16—592no—
145C2×D8—591no—
146C22×C8—592no—
147C4⋊Q8C2×D5911yescharacteristic
148C4⋊Q8C2×D5911yescharacteristic
149#124—1182no—
150#123—1182no—
151C22⋊Q8—592no—
152#121—592no—
153#121—1181no—
154#121—1181no—
155C4×D4—591no—
156C4×D4—591no—
157C4×D4—591no—
158#117—592no—
159#117—1181no—
160C2×C4⋊C4—1181no—
161#107—591no—
162#106—592no—
163C4⋊C8—591no—
164#103—592no—
165#103C2×D5912yes—
166#103C2×D5912yes—
167#102—591no—
168#102—591no—
169#102—592no—
170C22⋊C8—592no—
171C4×C8C2×D5911yescharacteristic
172#41—1184no—
173#41—2362no—
174C2×Q8—21no—
175C2×Q8C22×D5912yes—
176C2×D4—591no—
177C2×D4—591no—
178C2×D4—591no—
179C2×D4—1182no—
180C2×D4—1182no—
181C22×C4—592no—
182C22×C4—1182no—
183C22×C4—1181no—
184C22×C4—1181no—
185SD16—2362no—
186D8—1182no—
187C2×C8—592no—
188C2×C8—2362no—
189C2×C8C22×D5911yescharacteristic
190C2×C8C22×D5911yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—591no—
195C4⋊C4—591no—
196C4⋊C4—591no—
197C4⋊C4—1182no—
198C4⋊C4C22×D5912yes—
199C22⋊C4—592no—
200C22⋊C4—1182no—
201C22⋊C4—1182no—
202C22⋊C4—1182no—
203C22⋊C4—1181no—
204C22⋊C4—1181no—
205C42—1182no—
206C42C22×D5911yescharacteristic
207C23—592no—
208C23—1181no—
209C23—1181no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—592no—
214D4—1182no—
215D4—1181no—
216D4—1181no—
217D4—1181no—
218D4—4722no—
219D4—2364no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—592no—
223C2×C4—1182no—
224C2×C4—1182no—
225C2×C4—1181no—
226C2×C4—1181no—
227C2×C4—2364no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4D4×D5911yescharacteristic
231C2×C4D4×D5911yescharacteristic
232C2×C4C23×D5911yescharacteristic
233C8—21no—
234C8—21no—
235C8—1182no—
236C22—1182no—
237C22—1182no—
238C22—1182no—
239C22—1182no—
240C22—2361no—
241C22—2361no—
242C22—2361no—
243C22—2361no—
244C22—2361no—
245C22C2×D4×D5911yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—1182no—
250C4—2361no—
251C4—2361no—
252C4#1427375012yes—
253C4#41×D5911yescharacteristic
254C4#1427377311yescharacteristic
255C2—4721no—
256C2—2362no—
257C2—2362no—
258C2#1683804511yescharacteristic
259C2C2×#1427375011yescharacteristic
260C2#1683820811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#487C2
socleC22×C59C22×D4
hypercenter#487D59
hypocenterC59#3303
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #16838118C2semidirect
2. #16838055C2semidirect
3. #16837742C2semidirect
4. #16837689C2semidirect
5. #16837634C2semidirect
6. #16837604C2semidirect
7. #16837550C2semidirect
8. C4×D472C2semidirect
9. C8×D236C2semidirect
10. #16837397C2semidirect
11. C59×#487C2semidirect
133. #487D59semidirect
142. C59#3303semidirect