Finite Group Databasebeta

← #21424049

Subgroups — #21424049

Order 7552.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14273772C211yesmaximal, characteristic
2C2×#14273655C212yesmaximal, complemented
3#16837401C212yesmaximal, complemented
4C2×#14273644C211yesmaximal, characteristic, complemented
5C59×#390C211yesmaximal, characteristic, complemented
6#14273772—22no—
7#14273772—24no—
8#14273772C414yescomplemented
9C2×#11745685C2212yes—
10C2×#11745687—22no—
11C22×C22⋊D59—22no—
12C2×D4×D59—22no—
13C2×#11745664C2212yescomplemented
14#14273655—24no—
15#14273650—22no—
16#14273647—22no—
17#14273644—22no—
18C22×D4×C59C2211yescharacteristic
19C2×C4⋊C4×C59C2211yescharacteristic, complemented
20C2×C22⋊C4×C59—22no—
21C2×C42×C59C2211yescharacteristic, complemented
22C24×C59D412yes—
23C23×D59—22no—
24C2×D4×C59—22no—
25C2×D4×C59—22no—
26C2×D4×C59C2×C412yes—
27C2×D4×C59D412yes—
28C22×C4×C59—21no—
29C22×C4×C59—21no—
30C22×C4×C59C2311yescharacteristic
31C2×C22⋊D59—24no—
32C2×C22⋊D59—28no—
33C2×C22⋊D59—28no—
34C2×C22⋊D59—22no—
35C22×Dic59—22no—
36#11745687—28no—
37#11745687—28no—
38D4×D59—28no—
39D4×D59—28no—
40#11745685—22no—
41#11745685—22no—
42#11745685C2×C414yes—
43#11745685D414yes—
44C2×D236D412yes—
45C2×C4×D59—22no—
46C2×Dic118D412yes—
47C4⋊C4×C59—22no—
48C22⋊C4×C59—44no—
49C42×C59—22no—
50#11745664—22no—
51#11745664—22no—
52C2×C59⋊C8—22no—
53C23×C59—22no—
54C23×C59—22no—
55C23×C59—42no—
56C23×C59—42no—
57C23×C59C22⋊C414yes—
58C23×C59C2×D411yescharacteristic
59C22×D59—22no—
60C22×D59—28no—
61C22×D59—82no—
62C22×D59—82no—
63D4×C59—24no—
64D4×C59—24no—
65C2×C4×C59—22no—
66C2×C4×C59—41no—
67C2×C4×C59—41no—
68C2×C4×C59—42no—
69C2×C4×C59—41no—
70C2×C4×C59C22×C411yescharacteristic
71C2×C4×C59C2×D411yescharacteristic
72C2×C4×C59C2×D412yes—
73C2×C4×C59C2×D412yes—
74C22⋊D59—216no—
75C22⋊D59—24no—
76C22⋊D59—24no—
77C2×Dic59—28no—
78C2×Dic59—22no—
79D236—22no—
80D236C22⋊C414yescomplemented
81C4×D59—24no—
82C4×D59—24no—
83Dic118—22no—
84Dic118C22⋊C414yescomplemented
85C59⋊C8—42no—
86C22×C59—21no—
87C22×C59—21no—
88C22×C59—24no—
89C22×C59—44no—
90C22×C59—42no—
91C22×C59—42no—
92C22×C59—82no—
93C22×C59C2×C22⋊C411yescharacteristic
94C22×C59#12011yescharacteristic
95C22×C59#12011yescharacteristic
96C2×D59—24no—
97C2×D59—42no—
98C2×D59—42no—
99C2×D59—88no—
100C4×C59—42no—
101C4×C59—81no—
102C4×C59C2×C22⋊C412yes—
103C4×C59#12012yes—
104Dic59—24no—
105#1792—591nomaximal, Sylow
106C2×C59—21no—
107C2×C59—21no—
108C2×C59—44no—
109C2×C59#37911yescharacteristic
110C2×C59#45311yescharacteristic
111C2×C59#45511yescharacteristic
112D59—44no—
113C2×H(5,𝔽2)—591no—
114C2×#104—592no—
115#417—592no—
116C2×#100—591no—
117#390D5911yescharacteristic, complemented
118C59#179211yescharacteristic, complemented, Sylow
119H(5,𝔽2)—594no—
120H(5,𝔽2)—1184no—
121H(5,𝔽2)—1182no—
122C2×#41—592no—
123C2×#41—1182no—
124C22×D4—1182no—
125C22×D4—1182no—
126C22×D4C2×D5911yescharacteristic
127C2×M16—592no—
128C2×C4⋊C4C2×D5911yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D5911yescharacteristic, complemented
131#104—1184no—
132#103—1182no—
133#102—1182no—
134#100—1182no—
135C24—1182no—
136C24C22⋊D5912yes—
137#41—594no—
138#41—594no—
139#41—1188no—
140#41—1188no—
141#41—1182no—
142#41—1182no—
143C2×Q8—592no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—592no—
147C2×D4—1188no—
148C2×D4—1188no—
149C2×D4—1184no—
150C2×D4—1188no—
151C2×D4—1188no—
152C2×D4—1182no—
153C2×D4C4×D5912yes—
154C2×D4D23612yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—1182no—
158C22×C4—1182no—
159C22×C4C22×D5911yescharacteristic
160M16—1182no—
161M16—1182no—
162C2×C8—1182no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—1188no—
171C23—1182no—
172C23—4722no—
173C23—4722no—
174C23(C2×D59)⋊C414yes—
175C23C2×C22⋊D5911yescharacteristic
176Q8—594no—
177Q8—1182no—
178D4—24no—
179D4—24no—
180D4—594no—
181D4—11816no—
182D4—1184no—
183D4—1184no—
184D4—1182no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—1188no—
191C2×C4—1182no—
192C2×C4—1184no—
193C2×C4—1184no—
194C2×C4C2×C4×D5911yescharacteristic
195C2×C4C2×D23611yescharacteristic
196C2×C4D4×D5912yes—
197C2×C4D4×D5912yes—
198C8—2362no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—1184no—
207C22—4728no—
208C22—2362no—
209C22—2362no—
210C22C2×(C2×D59)⋊C411yescharacteristic
211C22#1427369511yescharacteristic
212C22#1427369511yescharacteristic
213C4—42no—
214C4—81no—
215C4—1184no—
216C4C22⋊C4×D5912yes—
217C4#1427369712yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—2364no—
222C2#1683725711yescharacteristic
223C2#1683756911yescharacteristic
224C2#1683757911yescharacteristic
subgroupG/H
centerC22#14273695
commutatorC2×C4×C59C22×C4
FrattiniC22×C4C22×D59
FittingC59×#390C2
socleC22×C59#120
hypercenter#390D59
hypocenterC59#1792
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#14273655C2semidirect
3. #16837401C2semidirect
4. C2×#14273644C2semidirect
5. C59×#390C2semidirect
8. #14273772C4semidirect
13. C2×#11745664C22semidirect
19. C2×C4⋊C4×C59C22semidirect
21. C2×C42×C59C22semidirect
80. D236C22⋊C4semidirect
84. Dic118C22⋊C4semidirect
117. #390D59semidirect
118. C59#1792semidirect
128. C2×C4⋊C4C2×D59semidirect
130. C2×C42C2×D59semidirect