Finite Group Databasebeta

← #21423019

Subgroups — #21423019

Order 7552.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#16837552C212yesmaximal, complemented
2#16837311C211yesmaximal, characteristic
3#16837139C212yesmaximal, complemented
4#16837126C211yesmaximal, characteristic, complemented
5C59×#543C211yesmaximal, characteristic, complemented
6C22×D236C2211yescharacteristic
7C22×C22⋊D59—21no—
8C2×#11745671—22no—
9C2×(C2×D59)⋊C4—21no—
10#14273647—22no—
11C22⋊(C59⋊C8)C2212yes—
12C59⋊#95C2211yescharacteristic
13C22⋊Q8×C59—22no—
14C22⋊Q8×C59—21no—
15C22⋊Q8×C59—22no—
16C22⋊Q8×C59C2212yes—
17C2×C22⋊C4×C59—21no—
18C2×C22⋊C4×C59C2211yescharacteristic
19C24×C59D411yescharacteristic
20C23×D59—21no—
21C2×Q8×C59—21no—
22C2×Q8×C59D412yescomplemented
23C22×C4×C59—21no—
24C22×C4×C59—21no—
25C22×C4×C59D411yescharacteristic
26C22×C4×C59C2311yescharacteristic
27C2×C22⋊D59—42no—
28C2×C22⋊D59—42no—
29C2×C22⋊D59—42no—
30C22×Dic59—21no—
31C2×D236—22no—
32C2×D236—42no—
33C4⋊C4×C59—21no—
34C4⋊C4×C59—22no—
35C4⋊C4×C59—22no—
36C4⋊C4×C59D412yescomplemented
37C22⋊C4×C59—22no—
38C22⋊C4×C59—22no—
39C22⋊C4×C59—22no—
40#11745671—44no—
41(C2×D59)⋊C4—42no—
42C2×C59⋊C8—22no—
43C23×C59—21no—
44C23×C59—21no—
45C23×C59—41no—
46C23×C59—41no—
47C23×C59SD1612yescomplemented
48C23×C59C2×D411yescharacteristic
49C22×D59—22no—
50C22×D59—41no—
51C22×D59—41no—
52C22×D59—41no—
53C22×D59—41no—
54C22×D59—41no—
55C22×D59—41no—
56Q8×C59—42no—
57Q8×C59—81no—
58C2×C4×C59—22no—
59C2×C4×C59—21no—
60C2×C4×C59—22no—
61C2×C4×C59—41no—
62C2×C4×C59—41no—
63C2×C4×C59—81no—
64C2×C4×C59—81no—
65C2×C4×C59C2×D412yes—
66C22⋊D59—48no—
67C2×Dic59—22no—
68C2×Dic59—41no—
69C2×Dic59—41no—
70D236—24no—
71D236—42no—
72C59⋊C8—42no—
73C22×C59—21no—
74C22×C59—21no—
75C22×C59—22no—
76C22×C59—42no—
77C22×C59—41no—
78C22×C59—41no—
79C22×C59—81no—
80C22×C59#12011yescharacteristic
81C22×C59C2×SD1611yescharacteristic
82C22×C59#13711yescharacteristic
83C2×D59—42no—
84C2×D59—42no—
85C2×D59—42no—
86C2×D59—81no—
87C2×D59—81no—
88C2×D59—81no—
89C2×D59—81no—
90C4×C59—22no—
91C4×C59—41no—
92C4×C59—42no—
93C4×C59—42no—
94Dic59—42no—
95#1596—591nomaximal, Sylow
96C2×C59—21no—
97C2×C59—21no—
98C2×C59—42no—
99C2×C59#44811yescharacteristic
100C2×C59#45311yescharacteristic
101C2×C59B(4,𝔽2)11yescharacteristic
102D59—82no—
103#543D5911yescharacteristic, complemented
104#450—592no—
105#392—591no—
106#328—592no—
107#323—591no—
108C59#159611yescharacteristic, complemented, Sylow
109C22×D4—591no—
110C22×D4—1181no—
111C2×SD16—1182no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D5912yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—1181no—
118C2×C22⋊C4C2×D5911yescharacteristic
119#102—1182no—
120C22⋊C8—592no—
121#95—591no—
122C24—1181no—
123C24C22⋊D5911yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D5912yescomplemented
126C2×D4—1182no—
127C2×D4—2362no—
128C2×D4—2362no—
129C2×D4—2362no—
130C2×D4—2362no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—1181no—
134C22×C4C22⋊D5911yescharacteristic
135C22×C4C22×D5911yescharacteristic
136SD16—2364no—
137C2×C8—1182no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D5912yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—2362no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—1182no—
151C23—2361no—
152C23—2361no—
153C23—2361no—
154C23—2361no—
155C23—2361no—
156C23—2361no—
157C23#1174567112yescomplemented
158C23C2×C22⋊D5911yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—1184no—
162D4—2362no—
163D4—2368no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—1182no—
172C2×C4—2361no—
173C2×C4—2361no—
174C2×C4C2×C22⋊D5912yes—
175C8—2362no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—4721no—
184C22—4721no—
185C22—4721no—
186C22—4721no—
187C22—2362no—
188C22—2362no—
189C22—2362no—
190C22#1427369611yescharacteristic
191C22C2×#1174567111yescharacteristic
192C22#1427375811yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—2362no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—4722no—
202C2#1683754111yescharacteristic
203C2#1683757011yescharacteristic
204C2#1683758911yescharacteristic
subgroupG/H
centerC22#14273696
commutatorC22×C4×C59C23
FrattiniC22×C4C22×D59
FittingC59×#543C2
socleC22×C59#120
hypercenter#543D59
hypocenterC59#1596
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #16837552C2semidirect
3. #16837139C2semidirect
4. #16837126C2semidirect
5. C59×#543C2semidirect
22. C2×Q8×C59D4semidirect
36. C4⋊C4×C59D4semidirect
47. C23×C59SD16semidirect
103. #543D59semidirect
108. C59#1596semidirect
125. C2×Q8C22⋊D59semidirect
141. C4⋊C4C22⋊D59semidirect
157. C23#11745671semidirect