Finite Group Databasebeta

← #21432558

Subgroups — #21432558

Order 7552.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#14273774C211yesmaximal, characteristic
2#16837879C212yesmaximal, complemented
3#16837376C211yesmaximal, characteristic, complemented
4#16837375C212yesmaximal, complemented
5#16837374C218yesmaximal, complemented
6C59×#410C211yesmaximal, characteristic, complemented
7#14273774—24no—
8#14273774—24no—
9C2×#11745685C412yescomplemented
10C2×#11745685C2211yescharacteristic
11C2×#11745687C412yescomplemented
12C2×#11745687C2214yes—
13C22×Dic118C412yescomplemented
14C22×Dic118C2211yescharacteristic
15C2×Q8×D59C412yescomplemented
16C4×Dic118—24no—
17#14273692—24no—
18#14273686C2212yes—
19#14273683C2218yes—
20#14273642C2214yes—
21#14273641C2218yes—
22C2×#41×C59C2211yescharacteristic
23#117×C59C2212yes—
24#99×C59C2214yes—
25#41×C59—22no—
26#41×C59—22no—
27C2×Q8×C59C2×C411yescharacteristic
28C2×D4×C59C2×C411yescharacteristic
29C2×D4×C59C2312yes—
30C22×C4×C59C2×C412yes—
31C22×C4×C59C2311yescharacteristic
32C2×C22⋊D59C2×C414yes—
33C2×C22⋊D59C2312yes—
34C22×Dic59C2×C414yes—
35C22×Dic59C2312yes—
36Q8×D59—28no—
37#11745687—216no—
38#11745687—28no—
39#11745685—22no—
40#11745685—22no—
41#11745685—24no—
42#11745685—24no—
43C2×D236C2×C411yescharacteristic
44C2×C4×D59C2×C412yes—
45C2×C4×D59C2×C414yes—
46C2×Dic118—24no—
47C2×Dic118—24no—
48C2×Dic118C2×C414yes—
49C2×Dic118C2×C411yescharacteristic
50C2×Dic118C2×C412yes—
51C2×Dic118C2×C412yes—
52C2×Dic118D414yes—
53C2×Dic118D414yes—
54C4⋊C4×C59—22no—
55C22⋊C4×C59C2314yes—
56C42×C59—22no—
57C22⋊Dic59C2314yes—
58C4⋊Dic59—22no—
59Dic59⋊C4—28no—
60C4×Dic59—22no—
61C4×Dic59—28no—
62C23×C59C22×C412yes—
63C23×C59C2411yescharacteristic
64C22×D59C22×C412yes—
65Q8×C59—22no—
66D4×C59—24no—
67D4×C59—22no—
68C2×C4×C59—22no—
69C2×C4×C59—22no—
70C2×C4×C59—24no—
71C2×C4×C59C22×C412yes—
72C2×C4×C59C22×C411yescharacteristic
73C2×C4×C59C22×C411yescharacteristic
74C2×C4×C59C2×D412yes—
75C2×C4×C59C2×D412yes—
76C22⋊D59—24no—
77C22⋊D59—28no—
78C2×Dic59—28no—
79C2×Dic59—24no—
80C2×Dic59C22×C412yes—
81C2×Dic59C22×C414yes—
82C2×Dic59C2×D418yes—
83D236—21no—
84D236—21no—
85C4×D59—24no—
86C4×D59—28no—
87Dic118—21no—
88Dic118—21no—
89Dic118—28no—
90Dic118C22⋊C414yes—
91Dic118C22⋊C414yes—
92Dic118C22⋊C418yes—
93C22×C59—22no—
94C22×C59—41no—
95C22×C59—42no—
96C22×C59C23×C411yescharacteristic
97C22×C59C22×D412yes—
98C2×D59—22no—
99C2×D59—42no—
100C4×C59—22no—
101C4×C59—44no—
102C4×C59C2×C22⋊C411yescharacteristic
103C4×C59C2×C22⋊C411yescharacteristic
104C4×C59C2×C22⋊C412yes—
105Dic59—22no—
106Dic59—44no—
107Dic59C2×C22⋊C418yes—
108#2884—591nomaximal, Sylow
109C2×C59—22no—
110C2×C59—21no—
111C2×C59—42no—
112C2×C59C22×C22⋊C411yescharacteristic
113D59—42no—
114C2×2−1+4—591no—
115#519—592no—
116#410—591no—
117#410—592no—
118#410—598no—
119#410D5911yescharacteristic, complemented
120C59#288411yescharacteristic, complemented, Sylow
1212−1+4—1184no—
1222−1+4—1184no—
123C2×#41—592no—
124C2×#41—594no—
125C2×#41—592no—
126C2×#41—591no—
127C2×#41C2×D5911yescharacteristic
128C22×Q8—592no—
129C22×Q8—592no—
130C22×Q8—591no—
131C4×Q8—1184no—
132C4×Q8—1184no—
133#117—592no—
134#117—598no—
135#117C2×D5912yes—
136#99—594no—
137#99—598no—
138#99C2×D5914yes—
139#41—22no—
140#41—22no—
141#41—1184no—
142#41—1184no—
143#41—11816no—
144#41—1188no—
145#41—1182no—
146#41—1182no—
147C2×Q8—592no—
148C2×Q8—592no—
149C2×Q8—594no—
150C2×Q8—591no—
151C2×Q8—594no—
152C2×Q8—594no—
153C2×Q8—1184no—
154C2×Q8—1184no—
155C2×Q8—1188no—
156C2×Q8C4×D5911yescharacteristic
157C2×D4—594no—
158C2×D4—592no—
159C2×D4—591no—
160C2×D4C4×D5911yescharacteristic
161C2×D4C22×D5912yes—
162C22×C4—592no—
163C22×C4—594no—
164C22×C4—594no—
165C22×C4—592no—
166C22×C4C4×D5912yes—
167C22×C4C22×D5911yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—1188no—
170C4⋊C4—1182no—
171C22⋊C4—594no—
172C22⋊C4C22×D5914yes—
173C42—22no—
174C42—1188no—
175C42—1182no—
176C23—592no—
177C23C2×C4×D5912yes—
178C23C23×D5911yescharacteristic
179Q8—22no—
180Q8—594no—
181Q8—594no—
182Q8—598no—
183Q8—1188no—
184Q8—1181no—
185Q8—1181no—
186D4—24no—
187D4—22no—
188D4—1188no—
189D4—1184no—
190D4—1181no—
191D4—1181no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—594no—
196C2×C4—598no—
197C2×C4—592no—
198C2×C4—1188no—
199C2×C4—1188no—
200C2×C4—1184no—
201C2×C4—1184no—
202C2×C4C2×C4×D5912yes—
203C2×C4C2×C4×D5911yescharacteristic
204C2×C4C2×C4×D5911yescharacteristic
205C2×C4D4×D5912yes—
206C2×C4D4×D5912yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—1182no—
211C22—2362no—
212C22C22×C4×D5911yescharacteristic
213C22C2×D4×D5912yes—
214C4—22no—
215C4—44no—
216C4—598no—
217C4—1182no—
218C4—2364no—
219C4C22⋊C4×D5911yescharacteristic
220C4C22⋊C4×D5911yescharacteristic
221C4C22⋊C4×D5912yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—2362no—
226C2C2×C22⋊C4×D5911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D59
commutatorC22×C59C23×C4
FrattiniC23C23×D59
FittingC59×#410C2
socleC2×C59C22×C22⋊C4
hypercenter#410D59
hypocenterC59#2884
LCSG ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #16837879C2semidirect
3. #16837376C2semidirect
4. #16837375C2semidirect
5. #16837374C2semidirect
6. C59×#410C2semidirect
9. C2×#11745685C4semidirect
11. C2×#11745687C4semidirect
13. C22×Dic118C4semidirect
15. C2×Q8×D59C4semidirect
119. #410D59semidirect
120. C59#2884semidirect