Finite Group Databasebeta

← #21435123

Subgroups — #21435123

Order 7552.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#16837936C211yesmaximal, characteristic
2#16837794C212yesmaximal, complemented
3#16837770C212yesmaximal, complemented
4#16837609C212yesmaximal, complemented
5#16837604C212yesmaximal, complemented
6#16837558C212yesmaximal, complemented
7#16837541C212yesmaximal
8#16837459C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C59C211yesmaximal, characteristic, complemented
10C2×#11745672C2214yes—
11C2×#11745671C2214yes—
12#14273721—21no—
13#14273719C2211yescharacteristic
14#14273715C2211yescharacteristic
15#14273710—21no—
16#14273706C2211yescharacteristic
17#14273701C2211yescharacteristic
18C4×Dic118C2211yescharacteristic
19C4×D236C2211yescharacteristic
20C4×C22⋊D59—21no—
21#14273684—21no—
22C4⋊(C59⋊C8)C2212yes—
23#14273651C2214yes—
24#14273651C2214yes—
25C22⋊(C59⋊C8)C2212yes—
26#14273638C2212yes—
27C22×Q8×C59C2212yes—
28C4⋊Q8×C59—21no—
29C4⋊Q8×C59—21no—
30C4⋊Q8×C59C2212yes—
31C4⋊Q8×C59C2212yes—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59—21no—
34C2×C42×C59C2211yescharacteristic
35C2×Q8×C59—22no—
36C2×Q8×C59—44no—
37C2×Q8×C59C2314yes—
38C22×C4×C59—22no—
39C22×C4×C59D411yescharacteristic
40C22×C4×C59D411yescharacteristic
41C22×C4×C59C2311yescharacteristic
42C2×C22⋊D59—21no—
43C2×D236C2311yescharacteristic
44C2×C4×D59—21no—
45C2×Dic118C2311yescharacteristic
46C4⋊C4×C59—22no—
47C4⋊C4×C59—22no—
48C4⋊C4×C59—22no—
49C4⋊C4×C59—22no—
50C42×C59D411yescharacteristic
51C42×C59D411yescharacteristic
52C42×C59C2311yescharacteristic
53C42×C59C2311yescharacteristic
54C22⋊Dic59—21no—
55#11745672—44no—
56#11745671—44no—
57(C2×D59)⋊C4—21no—
58(C2×D59)⋊C4—21no—
59(C2×D59)⋊C4—21no—
60(C2×D59)⋊C4—21no—
61C4⋊Dic59C2311yescharacteristic
62C4⋊Dic59C2311yescharacteristic
63Dic59⋊C4—21no—
64Dic59⋊C4—21no—
65Dic59⋊C4—21no—
66Dic59⋊C4—21no—
67C4×Dic59—21no—
68C2×C59⋊C8C2314yes—
69C23×C59C2×D411yescharacteristic
70C22×D59—21no—
71Q8×C59—24no—
72Q8×C59—24no—
73Q8×C59—44no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—24no—
78C2×C4×C59—44no—
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2×D411yescharacteristic
81C2×C4×C59C2×D411yescharacteristic
82C2×C4×C59C2×D411yescharacteristic
83C2×C4×C59C2×D411yescharacteristic
84C2×C4×C59C2411yescharacteristic
85C22⋊D59—42no—
86C2×Dic59—21no—
87C2×Dic59—21no—
88C2×Dic59—21no—
89D236—22no—
90C4×D59—42no—
91Dic118—22no—
92C59⋊C8—24no—
93C22×C59—21no—
94C22×C59—21no—
95C22×C59#13712yes—
96C22×C59C22×D411yescharacteristic
97C2×D59—41no—
98C2×D59—41no—
99C2×D59—41no—
100C4×C59—21no—
101C4×C59—21no—
102C4×C59—21no—
103C4×C59—21no—
104C4×C59—44no—
105C4×C59#13712yes—
106C4×C59H(5,𝔽2)12yes—
107Dic59—41no—
108Dic59—41no—
109Dic59—41no—
110#3218—591nomaximal, Sylow
111C2×C59—22no—
112C2×C59#53511yescharacteristic
113C2×C59C2×#13711yescharacteristic
114C2×C59C2×#13711yescharacteristic
115D59—81no—
116C2×C4⋊Q8D5911yescharacteristic, complemented
117#529—591no—
118#497—592no—
119#491—592no—
120#462—592no—
121#461—592no—
122#451—592no—
123#448—592no—
124#432—591no—
125C59#321811yescharacteristic, complemented, Sylow
126C22×Q8C2×D5912yes—
127C2×Q16—594no—
128C2×SD16—594no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8C2×D5912yes—
132C4⋊Q8C2×D5912yes—
133#126—1181no—
134#125—591no—
135#124—591no—
136#123—1181no—
137C22⋊Q8—591no—
138#121—591no—
139C4×Q8—591no—
140C4×D4—591no—
141C4×D4—1181no—
142#117—1181no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C42C2×D5911yescharacteristic
146C4⋊C8—592no—
147#103—594no—
148#103—594no—
149C22⋊C8—592no—
150#97—592no—
151C2×Q8—22no—
152C2×Q8—44no—
153C2×Q8—591no—
154C2×Q8C22×D5914yes—
155C2×D4—591no—
156C2×D4—1181no—
157C22×C4—22no—
158C22×C4—1181no—
159C22×C4C22⋊D5911yescharacteristic
160C22×C4C22⋊D5911yescharacteristic
161C22×C4C22×D5911yescharacteristic
162Q16—2364no—
163SD16—2364no—
164C2×C8—594no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—591no—
170C4⋊C4—591no—
171C4⋊C4—1181no—
172C4⋊C4—1181no—
173C4⋊C4—1181no—
174C4⋊C4—1181no—
175C22⋊C4—1181no—
176C22⋊C4—1181no—
177C22⋊C4—1181no—
178C22⋊C4—1181no—
179C22⋊C4—1181no—
180C42—1181no—
181C42C22⋊D5911yescharacteristic
182C42C22⋊D5911yescharacteristic
183C42C22×D5911yescharacteristic
184C42C22×D5911yescharacteristic
185C23—1181no—
186C23C2×C22⋊D5911yescharacteristic
187Q8—24no—
188Q8—24no—
189Q8—44no—
190Q8—1182no—
191D4—1182no—
192D4—2362no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—1181no—
199C2×C4—1181no—
200C2×C4—1181no—
201C2×C4—2362no—
202C2×C4C2×C22⋊D5911yescharacteristic
203C2×C4C2×C22⋊D5911yescharacteristic
204C2×C4C2×C22⋊D5911yescharacteristic
205C2×C4C2×C22⋊D5911yescharacteristic
206C2×C4C2×C22⋊D5911yescharacteristic
207C2×C4C23×D5911yescharacteristic
208C8—1184no—
209C22—21no—
210C22—21no—
211C22—2361no—
212C22—2361no—
213C22—2361no—
214C22#1427375812yes—
215C22C22×C22⋊D5911yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—44no—
221C4—2361no—
222C4—2361no—
223C4—2361no—
224C4#1427375812yes—
225C4#1427377212yes—
226C2—22no—
227C2—4721no—
228C2#1683796911yescharacteristic
229C2C2×#1427375811yescharacteristic
230C2C2×#1427375811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×C4⋊Q8×C59C2
socleC22×C59C22×D4
hypercenterC2×C4⋊Q8D59
hypocenterC59#3218
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16837794C2semidirect
3. #16837770C2semidirect
4. #16837609C2semidirect
5. #16837604C2semidirect
6. #16837558C2semidirect
8. #16837459C2semidirect
9. C2×C4⋊Q8×C59C2semidirect
116. C2×C4⋊Q8D59semidirect
125. C59#3218semidirect