Finite Group Databasebeta

← #21430014

Subgroups — #21430014

Order 7552.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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×C4×C22⋊D59C211yesmaximal, characteristic
2#16837333C211yesmaximal, characteristic, complemented
3#16837330C212yesmaximal, complemented
4#16837328C211yesmaximal, characteristic, complemented
5#16837321C211yesmaximal, characteristic, complemented
6#16837317C212yesmaximal
7#16837313C211yesmaximal, characteristic, complemented
8#16837283C212yesmaximal, complemented
9#16837273C212yesmaximal, complemented
10#16837265C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
12C22×C22⋊D59C2211yescharacteristic
13C22×C4×D59C2211yescharacteristic
14C4×C22⋊D59—24no—
15C2×Dic59⋊C4C2212yes—
16C2×Dic59⋊C4C2211yescharacteristic
17C2×Dic59⋊C4C2212yes—
18C2×C22⋊Dic59C2211yescharacteristic
19C2×C22⋊Dic59C2212yes—
20C2×(C2×D59)⋊C4C2212yes—
21C2×(C2×D59)⋊C4C2211yescharacteristic
22C2×(C2×D59)⋊C4C2212yes—
23C2×C4×Dic59C2211yescharacteristic
24C59⋊#95C2212yes—
25C59⋊#95C2211yescharacteristic
26C59⋊#95C2211yescharacteristic
27C59⋊#95C2212yes—
28C59⋊#95C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C22×Q8×C59C2211yescharacteristic
32C23×C4×C59C2211yescharacteristic
33C22⋊Q8×C59—24no—
34C2×C4⋊C4×C59C2212yes—
35C2×C4⋊C4×C59C2211yescharacteristic
36C2×C22⋊C4×C59C2212yes—
37C24×C59C2311yescharacteristic
38C23×D59C2311yescharacteristic
39C2×Q8×C59—22no—
40C2×Q8×C59—42no—
41C22×C4×C59—42no—
42C22×C4×C59D414yes—
43C22×C4×C59C2312yes—
44C22×C4×C59C2312yes—
45C22×C4×C59C2311yescharacteristic
46C22×C4×C59C2311yescharacteristic
47C2×C22⋊D59—22no—
48C2×C22⋊D59—22no—
49C2×C22⋊D59—22no—
50C22×Dic59C2311yescharacteristic
51C22×Dic59C2312yes—
52C22×Dic59C2312yes—
53C22×Dic59C2311yescharacteristic
54C22×Dic59C2311yescharacteristic
55C2×C4×D59—22no—
56C2×C4×D59—42no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—24no—
59C22⋊C4×C59—24no—
60C22⋊Dic59—22no—
61C22⋊Dic59—42no—
62(C2×D59)⋊C4—24no—
63(C2×D59)⋊C4—22no—
64(C2×D59)⋊C4—42no—
65Dic59⋊C4—22no—
66Dic59⋊C4—24no—
67Dic59⋊C4—42no—
68C4×Dic59—22no—
69C23×C59—21no—
70C23×C59—21no—
71C23×C59—22no—
72C23×C59—22no—
73C23×C59C2×D412yes—
74C23×C59C2411yescharacteristic
75C22×D59—21no—
76C22×D59—21no—
77C22×D59—22no—
78C22×D59—22no—
79C22×D59#4112yes—
80Q8×C59—44no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—22no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—44no—
90C2×C4×C59C2×D412yes—
91C2×C4×C59C2×D412yes—
92C22⋊D59—28no—
93C2×Dic59—21no—
94C2×Dic59—21no—
95C2×Dic59—21no—
96C2×Dic59—21no—
97C2×Dic59—22no—
98C2×Dic59—22no—
99C2×Dic59—22no—
100C2×Dic59—22no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59#4112yes—
106C4×D59—44no—
107C22×C59—22no—
108C22×C59—22no—
109C22×C59—22no—
110C22×C59—42no—
111C22×C59—42no—
112C22×C59C22×D411yescharacteristic
113C22×C59C2×#4111yescharacteristic
114C22×C59C2×#4111yescharacteristic
115C22×C59H(5,𝔽2)12yes—
116C22×C592−1+412yes—
117C2×D59—22no—
118C2×D59—24no—
119C2×D59—41no—
120C2×D59—41no—
121C2×D59—42no—
122C4×C59—22no—
123C4×C59—22no—
124C4×C59—42no—
125C4×C59—42no—
126Dic59—22no—
127Dic59—41no—
128Dic59—41no—
129Dic59—42no—
130Dic59—42no—
131#2638—591nomaximal, Sylow
132C2×C59—42no—
133C2×C59#52511yescharacteristic
134C2×C59#53411yescharacteristic
135C2×C59#53611yescharacteristic
136C2×C59#53912yes—
137C2×C59#54212yes—
138D59—42no—
139C2×C22⋊Q8D5911yescharacteristic, complemented
140C2×C4×D4—591no—
141#398—591no—
142#397—591no—
143#397—592no—
144#395—591no—
145#394—592no—
146#393—591no—
147#387—592no—
148#385—592no—
149#381—591no—
150C59#263811yescharacteristic, complemented, Sylow
151C22×Q8C2×D5911yescharacteristic
152C22×D4—591no—
153C23×C4—591no—
154C23×C4C2×D5911yescharacteristic
155C22⋊Q8—24no—
156C4×D4—1184no—
157C2×C4⋊C4—591no—
158C2×C4⋊C4—592no—
159C2×C4⋊C4—592no—
160C2×C4⋊C4C2×D5912yes—
161C2×C4⋊C4C2×D5911yescharacteristic
162C2×C22⋊C4—591no—
163C2×C22⋊C4—591no—
164C2×C22⋊C4—592no—
165C2×C22⋊C4—592no—
166C2×C22⋊C4—592no—
167C2×C22⋊C4C2×D5912yes—
168C2×C42—591no—
169#95—592no—
170#95—591no—
171#95—591no—
172#95—592no—
173#95—592no—
174#95—592no—
175#95—592no—
176C24—591no—
177C24C22×D5911yescharacteristic
178C2×Q8—22no—
179C2×Q8—42no—
180C2×D4—1182no—
181C2×D4—1182no—
182C2×D4—1182no—
183C22×C4—42no—
184C22×C4—591no—
185C22×C4—591no—
186C22×C4—592no—
187C22×C4—592no—
188C22×C4—591no—
189C22×C4—1182no—
190C22×C4—2362no—
191C22×C4C22⋊D5914yes—
192C22×C4C22×D5912yes—
193C22×C4C22×D5912yes—
194C22×C4C22×D5911yescharacteristic
195C22×C4C22×D5911yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—1184no—
199C4⋊C4—1182no—
200C4⋊C4—2362no—
201C22⋊C4—24no—
202C22⋊C4—1184no—
203C22⋊C4—1182no—
204C22⋊C4—1182no—
205C22⋊C4—2362no—
206C22⋊C4—2362no—
207C42—1182no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—22no—
212C23—592no—
213C23—1181no—
214C23—1181no—
215C23—1182no—
216C23—1182no—
217C23C2×C22⋊D5912yes—
218C23C23×D5911yescharacteristic
219Q8—44no—
220D4—1188no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—592no—
231C2×C4—1181no—
232C2×C4—1181no—
233C2×C4—1181no—
234C2×C4—1181no—
235C2×C4—1182no—
236C2×C4—1182no—
237C2×C4—1182no—
238C2×C4—1182no—
239C2×C4—1182no—
240C2×C4—1182no—
241C2×C4—1182no—
242C2×C4—1182no—
243C2×C4—2364no—
244C2×C4C2×C22⋊D5912yes—
245C2×C4C2×C22⋊D5912yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—1182no—
252C22—1184no—
253C22—2361no—
254C22—2361no—
255C22—2362no—
256C22C22×C22⋊D5911yescharacteristic
257C22#41×D5911yescharacteristic
258C22#41×D5911yescharacteristic
259C22#1427377212yes—
260C22#1427377512yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—1182no—
266C4—2361no—
267C4—2361no—
268C4—2362no—
269C4—2362no—
270C2—42no—
271C2—2362no—
272C2#1683790911yescharacteristic
273C2#1683796111yescharacteristic
274C2#1683797611yescharacteristic
275C2#1683799812yes—
276C2#1683802212yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×C22⋊Q8×C59C2
hypercenterC2×C22⋊Q8D59
hypocenterC59#2638
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16837333C2semidirect
3. #16837330C2semidirect
4. #16837328C2semidirect
5. #16837321C2semidirect
7. #16837313C2semidirect
8. #16837283C2semidirect
9. #16837273C2semidirect
10. #16837265C2semidirect
11. C2×C22⋊Q8×C59C2semidirect
139. C2×C22⋊Q8D59semidirect
150. C59#2638semidirect