Finite Group Databasebeta

← #21432444

Subgroups — #21432444

Order 7552.

644 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 201 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×#14273725C211yesmaximal, characteristic, complemented
2C2×#14273706C211yesmaximal, characteristic
3C22×C4×Dic59C211yesmaximal, characteristic
4#16837309C211yesmaximal, characteristic, complemented
5#16837307C211yesmaximal, characteristic, complemented
6#16837297C214yesmaximal, complemented
7#16837271C211yesmaximal, characteristic, complemented
8#16837260C214yesmaximal
9C2×C22⋊Q8×C59C211yesmaximal, characteristic
10C22×Dic118C2211yescharacteristic
11C23×Dic59C2212yes—
12#14273725—24no—
13#14273706—24no—
14C2×C4⋊Dic59C2211yescharacteristic
15C2×Dic59⋊C4C2212yes—
16C2×Dic59⋊C4C2214yes—
17C2×Dic59⋊C4C2214yes—
18C2×C22⋊Dic59C2212yes—
19C2×C4×Dic59—24no—
20C2×C4×Dic59C2212yes—
21C2×C4×Dic59C2211yescharacteristic
22C2×C4×Dic59C2211yescharacteristic
23C59⋊#95C2214yes—
24C59⋊#95C2214yes—
25C22×Q8×C59C2211yescharacteristic
26C23×C4×C59C2211yescharacteristic
27C22⋊Q8×C59—24no—
28C2×C4⋊C4×C59C2211yescharacteristic
29C2×C4⋊C4×C59C2212yes—
30C2×C22⋊C4×C59C2212yes—
31C24×C59C2311yescharacteristic
32C2×Q8×C59—22no—
33C2×Q8×C59—22no—
34C2×Q8×C59—22no—
35C22×C4×C59—22no—
36C22×C4×C59—22no—
37C22×C4×C59D414yes—
38C22×C4×C59C2312yes—
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C22×Dic59—24no—
43C22×Dic59—24no—
44C22×Dic59Q818yescomplemented
45C22×Dic59C2312yes—
46C22×Dic59C2312yes—
47C22×Dic59C2312yes—
48C22×Dic59C2312yes—
49C2×Dic118—22no—
50C2×Dic118—22no—
51C2×Dic118—22no—
52C4⋊C4×C59—22no—
53C4⋊C4×C59—24no—
54C22⋊C4×C59—24no—
55C22⋊Dic59—24no—
56C4⋊Dic59—22no—
57Dic59⋊C4—24no—
58Dic59⋊C4—28no—
59Dic59⋊C4—28no—
60C4×Dic59—22no—
61C4×Dic59—24no—
62C4×Dic59D414yes—
63C23×C59—21no—
64C23×C59—22no—
65C23×C59—21no—
66C23×C59C2×D412yes—
67C23×C59C2×Q814yes—
68C23×C59C2411yescharacteristic
69Q8×C59—44no—
70C2×C4×C59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—28no—
73C2×C4×C59—22no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—24no—
77C2×C4×C59—22no—
78C2×C4×C59C2×D412yes—
79C2×C4×C59C2×D412yes—
80C2×C4×C59C2×D412yes—
81C2×C4×C59#4112yes—
82C2×Dic59—22no—
83C2×Dic59—22no—
84C2×Dic59—216no—
85C2×Dic59—22no—
86C2×Dic59—22no—
87C2×Dic59—22no—
88C2×Dic59—24no—
89C2×Dic59—22no—
90C2×Dic59C2×D414yes—
91C2×Dic59C2×Q814yes—
92C2×Dic59C2×Q814yes—
93C2×Dic59#4114yes—
94Dic118—44no—
95C22×C59—22no—
96C22×C59—24no—
97C22×C59—22no—
98C22×C59—24no—
99C22×C59C4⋊Q814yescomplemented
100C22×C59C22×D411yescharacteristic
101C22×C59C22×D411yescharacteristic
102C22×C59C22×Q812yes—
103C22×C59C2×#4112yes—
104C22×C59C2×#4111yescharacteristic
105C4×C59—22no—
106C4×C59—42no—
107C4×C59—42no—
108C4×C59#12114yes—
109Dic59—24no—
110Dic59—42no—
111Dic59—42no—
112Dic59C22⋊Q818yescomplemented
113#2857—591nomaximal, Sylow
114C2×C59—24no—
115C2×C59C2×#12111yescharacteristic
116C2×C59C2×C22⋊Q812yes—
117C2×C59C2×C4⋊Q811yescharacteristic
118C2×C59#53211yescharacteristic
119C2×C59#53312yes—
120C2×C4⋊Q8—591no—
121C2×C22⋊Q8—591no—
122C2×C22⋊Q8D5911yescharacteristic
123C22×C42—591no—
124#391—591no—
125#391—591no—
126#389—594no—
127#384—591no—
128#380—594no—
129C59#285711yescharacteristic, complemented, Sylow
130C22×Q8—591no—
131C22×Q8C2×D5911yescharacteristic
132C23×C4—592no—
133C23×C4C2×D5911yescharacteristic
134C4⋊Q8—1184no—
135C22⋊Q8—24no—
136C22⋊Q8—1184no—
137C2×C4⋊C4—591no—
138C2×C4⋊C4—592no—
139C2×C4⋊C4—594no—
140C2×C4⋊C4—594no—
141C2×C4⋊C4C2×D5911yescharacteristic
142C2×C4⋊C4C2×D5912yes—
143C2×C22⋊C4—592no—
144C2×C22⋊C4C2×D5912yes—
145C2×C42—591no—
146C2×C42—592no—
147C2×C42—591no—
148C2×C42—1184no—
149#95—594no—
150#95—594no—
151C24C22×D5911yescharacteristic
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—1182no—
156C2×Q8—1182no—
157C2×Q8—1182no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—592no—
161C22×C4—598no—
162C22×C4—592no—
163C22×C4—592no—
164C22×C4—592no—
165C22×C4—1184no—
166C22×C4—1184no—
167C22×C4C22⋊D5914yes—
168C22×C4C22×D5912yes—
169C22×C4C22×D5912yes—
170C22×C4C22×D5911yescharacteristic
171C22×C4C22×D5911yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—1188no—
175C4⋊C4—1188no—
176C4⋊C4—1184no—
177C4⋊C4—1182no—
178C22⋊C4—24no—
179C22⋊C4—1184no—
180C42—594no—
181C42—1184no—
182C42—1182no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23Q8×D5914yes—
187C23C2×C22⋊D5912yes—
188C23C23×D5911yescharacteristic
189Q8—44no—
190Q8—2364no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—594no—
200C2×C4—594no—
201C2×C4—594no—
202C2×C4—594no—
203C2×C4—11816no—
204C2×C4—1182no—
205C2×C4—1182no—
206C2×C4—1182no—
207C2×C4—1182no—
208C2×C4—1182no—
209C2×C4—1184no—
210C2×C4—1182no—
211C2×C4D4×D5912yes—
212C2×C4#1174568712yes—
213C2×C4C2×C22⋊D5912yes—
214C2×C4C2×C22⋊D5912yes—
215C22—22no—
216C22—24no—
217C22—22no—
218C22—24no—
219C22#1427372514yescomplemented
220C22C2×D4×D5911yescharacteristic
221C22C22×C22⋊D5911yescharacteristic
222C22C2×Q8×D5912yes—
223C22#41×D5912yes—
224C22C2×#1174568711yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—598no—
229C4—1184no—
230C4—2362no—
231C4—2362no—
232C4#1427369914yes—
233C2—24no—
234C2C2×#1427369911yescharacteristic
235C2C22⋊Q8×D5912yes—
236C2C2×#1427372511yescharacteristic
237C2#1683794811yescharacteristic
238C2#1683795412yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×C22⋊Q8×C59C2
hypercenterC2×C22⋊Q8D59
hypocenterC59#2857
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14273725C2semidirect
4. #16837309C2semidirect
5. #16837307C2semidirect
6. #16837297C2semidirect
7. #16837271C2semidirect
44. C22×Dic59Q8semidirect
99. C22×C59C4⋊Q8semidirect
112. Dic59C22⋊Q8semidirect
129. C59#2857semidirect
219. C22#14273725semidirect