Finite Group Databasebeta

← #21433910

Subgroups — #21433910

Order 7552.

416 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C4⋊Q8×D59C211yesmaximal, characteristic, complemented
2C4×#11745689C211yesmaximal, characteristic
3#16837697C211yesmaximal, characteristic, complemented
4#16837693C211yesmaximal, characteristic, complemented
5#16837689C211yesmaximal, characteristic, complemented
6#16837685C212yesmaximal, complemented
7#16837681C212yesmaximal, complemented
8#16837443C212yesmaximal, complemented
9#16837424C211yesmaximal, characteristic, complemented
10#16837402C212yesmaximal
11C59×#477C211yesmaximal, characteristic, complemented
12C2×#11745689C2211yescharacteristic
13C2×Q8×D59—21no—
14C2×#11745659C2212yes—
15#14273727C2211yescharacteristic
16#14273726—21no—
17#14273726C2212yes—
18#14273725—21no—
19Q8×Dic59C2211yescharacteristic
20C4×D236—21no—
21C4×D236C2211yescharacteristic
22#14273689—21no—
23#14273689C2211yescharacteristic
24#14273685—21no—
25#14273685C2211yescharacteristic
26C4⋊C4×D59—21no—
27C4⋊C4×D59C2212yes—
28C42×D59—21no—
29C42×D59C2211yescharacteristic
30#14273660C2212yes—
31#14273659C2212yes—
32#14273658C2212yes—
33C4⋊(C59⋊C8)C2211yescharacteristic
34Dic59⋊C8C2212yes—
35#14273651C2212yes—
36#14273649C2212yes—
37#14273647C2212yes—
38C4⋊Q8×C59C2211yescharacteristic
39C4×Q8×C59C2211yescharacteristic
40#107×C59C2212yes—
41C4⋊C8×C59C2211yescharacteristic
42#103×C59C2212yes—
43C2×Q8×C59—21no—
44C2×Q8×C59C2311yescharacteristic
45#11745689—22no—
46#11745689Q814yescomplemented
47Q8×D59—44no—
48C2×D236—21no—
49C2×D236C2311yescharacteristic
50C2×C4×D59—21no—
51C2×C4×D59—22no—
52C2×C4×D59—21no—
53C2×C4×D59—21no—
54C2×C4×D59D411yescharacteristic
55C2×C4×D59D411yescharacteristic
56C2×C4×D59C2311yescharacteristic
57C2×Dic118—21no—
58C2×Dic118—22no—
59C2×C8×C59C2312yes—
60C4⋊C4×C59—21no—
61C4⋊C4×C59—21no—
62C4⋊C4×C59C2312yes—
63C4⋊C4×C59C2311yescharacteristic
64C42×C59—21no—
65C42×C59C2311yescharacteristic
66(C2×D59)⋊C4—21no—
67(C2×D59)⋊C4—21no—
68(C2×D59)⋊C4—21no—
69C4⋊Dic59—21no—
70C4⋊Dic59—21no—
71C4⋊Dic59C2312yes—
72C4⋊Dic59C2311yescharacteristic
73Dic59⋊C4—22no—
74Dic59⋊C4—22no—
75C4×Dic59—21no—
76C4×Dic59—21no—
77C4×Dic59D411yescharacteristic
78C4×Dic59D411yescharacteristic
79C4×Dic59C2311yescharacteristic
80C2×C59⋊C8C2312yes—
81#11745659—24no—
82C22×D59—21no—
83C22×D59C2×D411yescharacteristic
84Q8×C59—21no—
85Q8×C59—42no—
86Q8×C59C2×Q812yescomplemented
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59C2×D411yescharacteristic
91C2×C4×C59C2×D411yescharacteristic
92C2×C4×C59C2411yescharacteristic
93C2×Dic59—21no—
94C2×Dic59—22no—
95C2×Dic59C2×D411yescharacteristic
96C2×Dic59C2×D411yescharacteristic
97C2×Dic59C2×D411yescharacteristic
98D236—24no—
99D236—21no—
100D236C2×Q812yescomplemented
101C4×D59—22no—
102C4×D59—24no—
103C4×D59—22no—
104C4×D59—44no—
105C4×D59—42no—
106C4×D59—42no—
107C4×D59C2×Q812yes—
108C4×D59#4112yes—
109Dic118—42no—
110Dic118—44no—
111C8×C59—22no—
112C59⋊C8—22no—
113C22×C59C22×D411yescharacteristic
114C2×D59—21no—
115C2×D59—21no—
116C2×D59—22no—
117C2×D59—41no—
118C2×D59—41no—
119C2×D59C22⋊Q812yes—
120C4×C59—22no—
121C4×C59—21no—
122C4×C59—42no—
123C4×C59—41no—
124C4×C59#13712yes—
125C4×C59C22×Q811yescharacteristic
126C4×C59C2×#4111yescharacteristic
127Dic59—21no—
128Dic59—42no—
129Dic59—41no—
130Dic59C22⋊Q812yes—
131Dic59AGL(1,ℤ/8)12yes—
132#3079—591nomaximal, Sylow
133C2×C59C2×C22⋊Q811yescharacteristic
134C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C59C2×#13711yescharacteristic
136D59—22no—
137D59—42no—
138C2×C4⋊Q8—591no—
139C4×#41—591no—
140#477—591no—
141#477D5911yescharacteristic, complemented
142#476—591no—
143#476—591no—
144#475—592no—
145#474—592no—
146#428—592no—
147#423—591no—
148#417—592no—
149C59#307911yescharacteristic, complemented, Sylow
150C2×#41—591no—
151C22×Q8—1181no—
152C2×M16—592no—
153C4⋊Q8—592no—
154C4⋊Q8—591no—
155C4⋊Q8—1181no—
156C4⋊Q8—1181no—
157C4⋊Q8C2×D5911yescharacteristic
158C4×Q8—591no—
159C4×Q8C2×D5911yescharacteristic
160C4×D4—591no—
161C4×D4—591no—
162C4×D4—1181no—
163C4×D4—1181no—
164#117—591no—
165#117—1181no—
166C2×C4⋊C4—592no—
167C2×C4⋊C4—1181no—
168C2×C42—591no—
169C2×C42—1181no—
170#107—592no—
171#107C2×D5912yes—
172#106—592no—
173#106—592no—
174C4⋊C8—592no—
175C4⋊C8—591no—
176C4⋊C8C2×D5911yescharacteristic
177#103—592no—
178#103C2×D5912yes—
179#102—592no—
180#102—592no—
181#41—594no—
182#41—1182no—
183C2×Q8—21no—
184C2×Q8—1182no—
185C2×Q8—1181no—
186C2×Q8—2364no—
187C2×Q8C22×D5911yescharacteristic
188C2×D4—591no—
189C2×D4—1181no—
190C22×C4—591no—
191C22×C4—591no—
192C22×C4—591no—
193C22×C4—1181no—
194C22×C4—1181no—
195C22×C4—1182no—
196C22×C4—1181no—
197M16—1184no—
198C2×C8—592no—
199C2×C8C22×D5912yes—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—592no—
203C4⋊C4—591no—
204C4⋊C4—1182no—
205C4⋊C4—1182no—
206C4⋊C4—1181no—
207C4⋊C4—1181no—
208C4⋊C4C22×D5912yes—
209C4⋊C4C22×D5911yescharacteristic
210C22⋊C4—1181no—
211C22⋊C4—1181no—
212C22⋊C4—1181no—
213C42—21no—
214C42—591no—
215C42—591no—
216C42—591no—
217C42—1181no—
218C42—1181no—
219C42C22×D5911yescharacteristic
220C23—591no—
221C23—1181no—
222Q8—21no—
223Q8—42no—
224Q8—2362no—
225Q8—2364no—
226Q8Q8×D5912yescomplemented
227D4—592no—
228D4—1184no—
229D4—1181no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—591no—
234C2×C4—591no—
235C2×C4—592no—
236C2×C4—592no—
237C2×C4—591no—
238C2×C4—1182no—
239C2×C4—1184no—
240C2×C4—1182no—
241C2×C4—1181no—
242C2×C4—1182no—
243C2×C4—2362no—
244C2×C4—2364no—
245C2×C4—2362no—
246C2×C4D4×D5911yescharacteristic
247C2×C4D4×D5911yescharacteristic
248C2×C4C23×D5911yescharacteristic
249C8—22no—
250C8—1182no—
251C22—592no—
252C22—1181no—
253C22—1181no—
254C22—1182no—
255C22—2361no—
256C22—2361no—
257C22C2×D4×D5911yescharacteristic
258C4—22no—
259C4—21no—
260C4—42no—
261C4—41no—
262C4—592no—
263C4—592no—
264C4—1181no—
265C4—2362no—
266C4—2361no—
267C4#1427375912yes—
268C4C2×Q8×D5911yescharacteristic
269C4#41×D5911yescharacteristic
270C2—1182no—
271C2—2362no—
272C2C22⋊Q8×D5911yescharacteristic
273C2AGL(1,ℤ/8)×D5911yescharacteristic
274C2C2×#1427375911yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#477C2
socleC22×C59C22×D4
hypercenter#477D59
hypocenterC59#3079
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D59C2semidirect
3. #16837697C2semidirect
4. #16837693C2semidirect
5. #16837689C2semidirect
6. #16837685C2semidirect
7. #16837681C2semidirect
8. #16837443C2semidirect
9. #16837424C2semidirect
11. C59×#477C2semidirect
46. #11745689Q8semidirect
86. Q8×C59C2×Q8semidirect
100. D236C2×Q8semidirect
141. #477D59semidirect
149. C59#3079semidirect
226. Q8Q8×D59semidirect