Finite Group Databasebeta

← #21424580

Subgroups — #21424580

Order 7552.

346 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#14273752C211yesmaximal, characteristic
2#16837876C211yesmaximal, characteristic, complemented
3#16837408C211yesmaximal, characteristic, complemented
4#16837376C211yesmaximal, characteristic, complemented
5#16837174C211yesmaximal, characteristic, complemented
6#16837158C211yesmaximal, characteristic, complemented
7C59×#421C211yesmaximal, characteristic, complemented
8C22×D236C2211yescharacteristic
9#14273752—22no—
10#14273752—22no—
11C2×#11745671C412yescomplemented
12C2×#11745669C412yescomplemented
13C2×#11745664C2211yescharacteristic
14C4×D236—22no—
15#14273689—22no—
16#14273686C2211yescharacteristic
17#14273686C2211yescharacteristic
18C22⋊C4×D59—22no—
19#14273649—22no—
20#14273642—22no—
21C2×#41×C59C2211yescharacteristic
22C2×M16×C59C2211yescharacteristic
23#117×C59C2211yescharacteristic
24#104×C59—22no—
25C23×D59—21no—
26#41×C59—21no—
27#41×C59—21no—
28#41×C59—21no—
29#41×C59—21no—
30C2×Q8×C59C2×C411yescharacteristic, complemented
31C2×D4×C59—21no—
32C2×D4×C59C2×C411yescharacteristic, complemented
33C22×C4×C59—21no—
34C22×C4×C59C2311yescharacteristic
35C2×D236—21no—
36C2×D236—21no—
37C2×D236—21no—
38C2×D236C2×C412yes—
39C2×D236D412yes—
40C2×D236D412yes—
41C2×C4×D59—24no—
42M16×C59—21no—
43M16×C59—21no—
44C2×C8×C59—21no—
45C4⋊C4×C59—21no—
46C22⋊C4×C59—21no—
47C42×C59D412yes—
48C22⋊Dic59—21no—
49C22⋊Dic59D412yes—
50#11745671—24no—
51#11745669—24no—
52(C2×D59)⋊C4—24no—
53C4⋊Dic59—21no—
54C4⋊Dic59—21no—
55C4×Dic59—21no—
56C4×Dic59—21no—
57#11745664—21no—
58#11745664—21no—
59C2×C59⋊C8C2×C412yescomplemented
60C23×C59—21no—
61C23×C59C2×D411yescharacteristic
62C22×D59—22no—
63C22×D59—22no—
64C22×D59—22no—
65C22×D59—82no—
66Q8×C59—21no—
67Q8×C59—21no—
68D4×C59—21no—
69D4×C59—21no—
70D4×C59—41no—
71D4×C59—41no—
72C2×C4×C59—21no—
73C2×C4×C59—22no—
74C2×C4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59C22×C411yescharacteristic
77C2×C4×C59C2×D411yescharacteristic
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59#4111yescharacteristic
81C2×C4×C59#4111yescharacteristic
82C2×Dic59—22no—
83C2×Dic59—41no—
84D236—22no—
85D236—22no—
86D236—22no—
87D236C22⋊C414yes—
88C4×D59—84no—
89C8×C59—41no—
90C59⋊C8—22no—
91C22×C59—41no—
92C22×C59—41no—
93C22×C59—81no—
94C22×C59C4×D411yescharacteristic
95C22×C59#12011yescharacteristic
96C22×C59#12111yescharacteristic
97C2×D59—24no—
98C2×D59—44no—
99C2×D59—82no—
100C2×D59—82no—
101C4×C59—42no—
102C4×C59—41no—
103C4×C59C2×C22⋊C411yescharacteristic
104C4×C59C4×D411yescharacteristic
105C4×C59#12111yescharacteristic
106C4×C59#12311yescharacteristic
107Dic59—42no—
108Dic59—81no—
109#1904—591nomaximal, Sylow
110C2×C59—21no—
111C2×C59—21no—
112C2×C59—21no—
113C2×C59—81no—
114C2×C59#38611yescharacteristic
115D59—44no—
116C2×AGL(1,ℤ/8)—591no—
117#518—591no—
118#421D5911yescharacteristic, complemented
119#418—591no—
120#410—591no—
121#344—591no—
122H(3,ℤ/4)—591no—
123C59#190411yescharacteristic, complemented, Sylow
124C2×#41C2×D5911yescharacteristic
125C22×D4—591no—
126AGL(1,ℤ/8)—1182no—
127AGL(1,ℤ/8)—1182no—
128C2×SD16—592no—
129C2×D8—592no—
130C2×M16—591no—
131C2×M16C2×D5911yescharacteristic
132C4×D4—1182no—
133C4×D4—1182no—
134#117—591no—
135#117—591no—
136#117C2×D5911yescharacteristic
137C2×C22⋊C4—1182no—
138#104—22no—
139#102—1182no—
140#99—1182no—
141C24—1181no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146C2×Q8C4×D5911yescharacteristic, complemented
147C2×D4—21no—
148C2×D4—592no—
149C2×D4—592no—
150C2×D4—592no—
151C2×D4—1181no—
152C2×D4—1181no—
153C2×D4—1181no—
154C2×D4C4×D5911yescharacteristic, complemented
155C22×C4—21no—
156C22×C4—1184no—
157C22×C4C22×D5911yescharacteristic
158SD16—1184no—
159D8—1184no—
160M16—21no—
161M16—21no—
162M16—1181no—
163M16—1181no—
164C2×C8—21no—
165C2×C8—592no—
166C4⋊C4—21no—
167C4⋊C4—1181no—
168C4⋊C4—1181no—
169C22⋊C4—21no—
170C22⋊C4—592no—
171C22⋊C4—1184no—
172C22⋊C4—1181no—
173C42—1181no—
174C42—1181no—
175C42C22⋊D5912yes—
176C23—21no—
177C23—1182no—
178C23—1182no—
179C23—1182no—
180C23—4722no—
181C23D4×D5911yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—21no—
186D4—41no—
187D4—41no—
188D4—594no—
189D4—1182no—
190D4—1182no—
191D4—1182no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1182no—
197C2×C4—4724no—
198C2×C4—2361no—
199C2×C4C2×C4×D5911yescharacteristic
200C2×C4#1174568511yescharacteristic
201C2×C4D4×D5911yescharacteristic
202C2×C4D4×D5911yescharacteristic
203C2×C4#1174568711yescharacteristic
204C2×C4C2×C22⋊D5911yescharacteristic
205C8—41no—
206C8—1182no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22—1184no—
211C22—4722no—
212C22—4722no—
213C22—2364no—
214C22(C22⋊D59)⋊C411yescharacteristic
215C22#1427369511yescharacteristic
216C22#1427370011yescharacteristic
217C4—42no—
218C4—41no—
219C4—4721no—
220C4—2362no—
221C4C22⋊C4×D5911yescharacteristic
222C4C4×C22⋊D5911yescharacteristic
223C4#1427369811yescharacteristic
224C4#1427370911yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—81no—
229C2—2364no—
230C2#1683727811yescharacteristic
subgroupG/H
centerC2#16837278
commutatorC2×C4×C59C22×C4
FrattiniC22×C4C22×D59
FittingC59×#421C2
socleC2×C59#386
hypercenter#421D59
hypocenterC59#1904
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #16837876C2semidirect
3. #16837408C2semidirect
4. #16837376C2semidirect
5. #16837174C2semidirect
6. #16837158C2semidirect
7. C59×#421C2semidirect
11. C2×#11745671C4semidirect
12. C2×#11745669C4semidirect
30. C2×Q8×C59C2×C4semidirect
32. C2×D4×C59C2×C4semidirect
59. C2×C59⋊C8C2×C4semidirect
118. #421D59semidirect
123. C59#1904semidirect
146. C2×Q8C4×D59semidirect
154. C2×D4C4×D59semidirect