Finite Group Databasebeta

← #21424593

Subgroups — #21424593

Order 7552.

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

normal subgroups only

This group has 310 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×#14273757C211yesmaximal, characteristic
2#16837879C211yesmaximal, characteristic, complemented
3#16837419C211yesmaximal, characteristic
4#16837410C211yesmaximal, characteristic, complemented
5#16837172C211yesmaximal, characteristic, complemented
6#16837158C211yesmaximal, characteristic, complemented
7C59×#410C211yesmaximal, characteristic
8C22×Dic118C2211yescharacteristic
9#14273757—22no—
10#14273757—22no—
11C2×#11745672C412yescomplemented
12C2×#11745670C412yescomplemented
13C2×#11745664C2211yescharacteristic
14C2×#11745664C2211yescharacteristic
15C4×Dic118—22no—
16#14273692—22no—
17#14273686C2211yescharacteristic
18#14273683—22no—
19#14273654—22no—
20#14273650—22no—
21C2×#41×C59C2211yescharacteristic
22#117×C59C2211yescharacteristic
23#117×C59C2211yescharacteristic
24#99×C59—22no—
25#41×C59—21no—
26#41×C59—21no—
27#41×C59—21no—
28#41×C59—21no—
29C2×Q8×C59C2×C411yescharacteristic
30C2×D4×C59—21no—
31C2×D4×C59C2×C411yescharacteristic
32C22×C4×C59—21no—
33C22×C4×C59C2311yescharacteristic
34C22×Dic59—21no—
35C2×Dic118—21no—
36C2×Dic118—21no—
37C2×Dic118—21no—
38C2×Dic118C2×C412yes—
39C2×Dic118D412yes—
40C2×Dic118D412yes—
41C4⋊C4×C59—21no—
42C4⋊C4×C59—21no—
43C22⋊C4×C59—21no—
44C22⋊C4×C59D412yes—
45C42×C59—21no—
46C42×C59—21no—
47C22⋊Dic59—21no—
48#11745672—24no—
49#11745670—24no—
50C4⋊Dic59—21no—
51Dic59⋊C4—24no—
52C4×Dic59—24no—
53C4×Dic59D412yes—
54#11745664—21no—
55#11745664—21no—
56#11745664—21no—
57#11745664—21no—
58C2×C59⋊C8—21no—
59C2×C59⋊C8C2×C412yes—
60C23×C59—21no—
61C23×C59C2×D411yescharacteristic
62Q8×C59—21no—
63Q8×C59—21no—
64D4×C59—21no—
65D4×C59—21no—
66D4×C59—41no—
67D4×C59—41no—
68C2×C4×C59—21no—
69C2×C4×C59—22no—
70C2×C4×C59—41no—
71C2×C4×C59—41no—
72C2×C4×C59—41no—
73C2×C4×C59C22×C411yescharacteristic
74C2×C4×C59C2×D411yescharacteristic
75C2×C4×C59C2×D411yescharacteristic
76C2×C4×C59C2×D411yescharacteristic
77C2×C4×C59#4111yescharacteristic
78C2×C4×C59#4111yescharacteristic
79C2×Dic59—22no—
80C2×Dic59—22no—
81C2×Dic59—22no—
82C2×Dic59—22no—
83Dic118—22no—
84Dic118—22no—
85Dic118—22no—
86Dic118C22⋊C414yes—
87C59⋊C8—22no—
88C59⋊C8—41no—
89C22×C59—41no—
90C22×C59—41no—
91C22×C59—81no—
92C22×C59C4×D411yescharacteristic
93C22×C59#12011yescharacteristic
94C22×C59#12111yescharacteristic
95C4×C59—42no—
96C4×C59—41no—
97C4×C59—81no—
98C4×C59C2×C22⋊C411yescharacteristic
99C4×C59C4×D411yescharacteristic
100C4×C59#12111yescharacteristic
101C4×C59#12311yescharacteristic
102Dic59—24no—
103Dic59—42no—
104#1905—591nomaximal, Sylow
105C2×C59—21no—
106C2×C59—21no—
107C2×C59—21no—
108C2×C59—81no—
109C2×C59#38611yescharacteristic
110C2×#137—591no—
111#519—591no—
112#421—591no—
113#419—591no—
114#410D5911yescharacteristic
115#344—591no—
116H(3,ℤ/4)—591no—
117C59#190511yescharacteristic, complemented, Sylow
118C2×#41C2×D5911yescharacteristic
119C22×Q8—591no—
120#137—1182no—
121#137—1182no—
122C2×Q16—592no—
123C2×SD16—592no—
124C2×M16—591no—
125C2×M16—591no—
126C4×Q8—1182no—
127C4×Q8—1182no—
128#117—591no—
129#117—1182no—
130#117C2×D5911yescharacteristic
131#117C2×D5911yescharacteristic
132#104—1182no—
133#103—1182no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—592no—
140C2×Q8—592no—
141C2×Q8—592no—
142C2×Q8—1181no—
143C2×Q8—1181no—
144C2×Q8—1181no—
145C2×Q8C4×D5911yescharacteristic
146C2×D4—21no—
147C2×D4C4×D5911yescharacteristic
148C22×C4—21no—
149C22×C4—1181no—
150C22×C4C22×D5911yescharacteristic
151Q16—1184no—
152SD16—1184no—
153M16—1181no—
154M16—1181no—
155M16—1181no—
156M16—1181no—
157C2×C8—592no—
158C2×C8—1181no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1184no—
162C4⋊C4—1181no—
163C22⋊C4—21no—
164C22⋊C4—1181no—
165C22⋊C4C22⋊D5912yes—
166C42—21no—
167C42—21no—
168C42—592no—
169C42—1184no—
170C23—21no—
171C23C2×C22⋊D5911yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—594no—
175Q8—1182no—
176Q8—1182no—
177Q8—1182no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—1182no—
188C2×C4—1182no—
189C2×C4—1182no—
190C2×C4—1182no—
191C2×C4C2×C4×D5911yescharacteristic
192C2×C4#1174568511yescharacteristic
193C2×C4D4×D5911yescharacteristic
194C2×C4D4×D5911yescharacteristic
195C2×C4D4×D5911yescharacteristic
196C2×C4#1174568711yescharacteristic
197C8—1182no—
198C8—2361no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D5911yescharacteristic
203C22#1427369511yescharacteristic
204C22#1427369811yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—1184no—
209C4—2362no—
210C4C22⋊C4×D5911yescharacteristic
211C4(C22⋊D59)⋊C411yescharacteristic
212C4#1427370011yescharacteristic
213C4#1427370911yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#1683727811yescharacteristic
subgroupG/H
centerC2#16837278
commutatorC2×C4×C59C22×C4
FrattiniC22×C4C22×D59
FittingC59×#410C2
socleC2×C59#386
hypercenter#410D59
hypocenterC59#1905
LCSG ▷ C2×C4×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
4. #16837410C2semidirect
5. #16837172C2semidirect
6. #16837158C2semidirect
11. C2×#11745672C4semidirect
12. C2×#11745670C4semidirect
117. C59#1905semidirect