Finite Group Databasebeta

← #21427499

Subgroups — #21427499

Order 7552.

610 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14273708C211yesmaximal, characteristic
2C2×#14273683C211yesmaximal, characteristic, complemented
3C22×Dic59⋊C4C211yesmaximal, characteristic, complemented
4#16837333C212yesmaximal, complemented
5#16837325C212yesmaximal, complemented
6#16837298C212yesmaximal, complemented
7#16837290C212yesmaximal, complemented
8#16837282C211yesmaximal, characteristic
9#16837277C211yesmaximal, characteristic, complemented
10#16837265C211yesmaximal, characteristic, complemented
11C2×#123×C59C211yesmaximal, characteristic
12C23×Dic59C2211yescharacteristic
13C23×Dic59C2211yescharacteristic
14#14273708—24no—
15#14273683—24no—
16C2×C4⋊Dic59C2212yes—
17C2×Dic59⋊C4—24no—
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2211yescharacteristic
20C2×Dic59⋊C4C2212yes—
21C2×Dic59⋊C4C2212yes—
22C2×Dic59⋊C4C2212yes—
23C2×C22⋊Dic59C2211yescharacteristic
24C2×C22⋊Dic59C2211yescharacteristic
25C2×C22⋊Dic59C2211yescharacteristic
26C2×C22⋊Dic59C2211yescharacteristic
27C2×C22⋊Dic59C2212yes—
28C2×C4×Dic59C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2212yes—
31C59⋊#95C2212yes—
32C59⋊#95C2212yes—
33C22×D4×C59C2211yescharacteristic
34C23×C4×C59C2211yescharacteristic
35#123×C59—24no—
36C2×C4⋊C4×C59C2212yes—
37C2×C22⋊C4×C59C2212yes—
38C2×C22⋊C4×C59C2211yescharacteristic
39C24×C59C2311yescharacteristic
40C24×C59C2311yescharacteristic
41C2×D4×C59—22no—
42C2×D4×C59—22no—
43C2×D4×C59—22no—
44C22×C4×C59—22no—
45C22×C4×C59—22no—
46C22×C4×C59—22no—
47C22×C4×C59C2312yes—
48C22×C4×C59C2311yescharacteristic
49C22×C4×C59C2312yes—
50C22×Dic59—22no—
51C22×Dic59—22no—
52C22×Dic59—22no—
53C22×Dic59—24no—
54C22×Dic59D414yes—
55C22×Dic59C2311yescharacteristic
56C22×Dic59C2312yes—
57C22×Dic59C2311yescharacteristic
58C22×Dic59C2311yescharacteristic
59C22×Dic59C2311yescharacteristic
60C22×Dic59C2312yes—
61C4⋊C4×C59—24no—
62C22⋊C4×C59—24no—
63C22⋊C4×C59D414yes—
64C22⋊Dic59—22no—
65C22⋊Dic59—22no—
66C22⋊Dic59—22no—
67C22⋊Dic59—22no—
68C22⋊Dic59—24no—
69C4⋊Dic59—24no—
70Dic59⋊C4—24no—
71Dic59⋊C4—24no—
72Dic59⋊C4—24no—
73Dic59⋊C4—22no—
74Dic59⋊C4—22no—
75C4×Dic59—24no—
76C23×C59—22no—
77C23×C59—22no—
78C23×C59—21no—
79C23×C59—21no—
80C23×C59—22no—
81C23×C59—21no—
82C23×C59—22no—
83C23×C59C2×D412yes—
84C23×C59C2×D412yes—
85C23×C59#4112yes—
86C23×C59C2411yescharacteristic
87D4×C59—44no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—44no—
97C2×C4×C59C2×D414yes—
98C2×Dic59—28no—
99C2×Dic59—28no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59—22no—
104C2×Dic59—21no—
105C2×Dic59—21no—
106C2×Dic59—22no—
107C2×Dic59—21no—
108C2×Dic59—21no—
109C2×Dic59—22no—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—22no—
113C2×Dic59C2×D412yes—
114C2×Dic59C2×D412yes—
115C2×Dic59#4114yes—
116C22×C59—22no—
117C22×C59—22no—
118C22×C59—24no—
119C22×C59—24no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59—41no—
123C22×C59—42no—
124C22×C59—42no—
125C22×C59#12114yes—
126C22×C59C22×D411yescharacteristic
127C22×C59C22×D411yescharacteristic
128C22×C59C2×#4112yes—
129C22×C59C2×#4111yescharacteristic
130C22×C592−1+412yes—
131C4×C59—24no—
132C4×C59—42no—
133C4×C59—41no—
134Dic59—22no—
135Dic59—22no—
136Dic59—24no—
137Dic59—41no—
138Dic59—41no—
139Dic59—42no—
140#2413—591nomaximal, Sylow
141C2×C59—24no—
142C2×C59—81no—
143C2×C59C2×#12111yescharacteristic
144C2×C59#52511yescharacteristic
145C2×C59#53611yescharacteristic
146C2×C59#54712yes—
147C2×C59#55212yes—
148C2×#123—591no—
149C2×#123D5911yescharacteristic
150C2×#117—591no—
151C22×C4⋊C4—591no—
152#398—592no—
153#396—592no—
154#389—592no—
155#388—592no—
156#386—591no—
157#385—591no—
158#381—591no—
159C59#241311yescharacteristic, complemented, Sylow
160C22×D4C2×D5911yescharacteristic
161C23×C4—591no—
162C23×C4—591no—
163C23×C4C2×D5911yescharacteristic
164#123—24no—
165#123—1184no—
166#117—1184no—
167C2×C4⋊C4—592no—
168C2×C4⋊C4—592no—
169C2×C4⋊C4—592no—
170C2×C4⋊C4—591no—
171C2×C4⋊C4—591no—
172C2×C4⋊C4—592no—
173C2×C4⋊C4—1184no—
174C2×C4⋊C4C2×D5912yes—
175C2×C22⋊C4—591no—
176C2×C22⋊C4—591no—
177C2×C22⋊C4—591no—
178C2×C22⋊C4—591no—
179C2×C22⋊C4—592no—
180C2×C22⋊C4C2×D5912yes—
181C2×C22⋊C4C2×D5911yescharacteristic
182C2×C42—592no—
183#95—592no—
184#95—592no—
185#95—592no—
186#95—592no—
187C24C22×D5911yescharacteristic
188C24C22×D5911yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—594no—
196C22×C4—592no—
197C22×C4—591no—
198C22×C4—591no—
199C22×C4—591no—
200C22×C4—592no—
201C22×C4—591no—
202C22×C4—1184no—
203C22×C4—1182no—
204C22×C4—1182no—
205C22×C4—1182no—
206C22×C4C22×D5912yes—
207C22×C4C22×D5911yescharacteristic
208C22×C4C22×D5912yes—
209C4⋊C4—24no—
210C4⋊C4—1184no—
211C4⋊C4—1184no—
212C4⋊C4—1182no—
213C4⋊C4—1182no—
214C4⋊C4—1184no—
215C4⋊C4—1184no—
216C22⋊C4—24no—
217C22⋊C4—1182no—
218C22⋊C4—1182no—
219C22⋊C4—1184no—
220C22⋊C4—1182no—
221C22⋊C4—1182no—
222C22⋊C4C22⋊D5914yes—
223C42—1184no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—22no—
231C23D4×D5912yes—
232C23#1174568712yes—
233C23C2×C22⋊D5912yes—
234C23C23×D5911yescharacteristic
235D4—44no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—592no—
246C2×C4—592no—
247C2×C4—594no—
248C2×C4—1188no—
249C2×C4—1188no—
250C2×C4—1182no—
251C2×C4—1181no—
252C2×C4—1181no—
253C2×C4—1182no—
254C2×C4—1181no—
255C2×C4—1181no—
256C2×C4—1181no—
257C2×C4—1181no—
258C2×C4—1182no—
259C2×C4—1182no—
260C2×C4—1182no—
261C2×C4—1182no—
262C2×C4—1182no—
263C2×C4C2×C22⋊D5914yes—
264C22—22no—
265C22—22no—
266C22—24no—
267C22—24no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22#1427369814yes—
274C22C2×D4×D5911yescharacteristic
275C22C22×C22⋊D5911yescharacteristic
276C22#41×D5912yes—
277C22C2×#1174568711yescharacteristic
278C22#1427377412yes—
279C4—24no—
280C4—42no—
281C4—41no—
282C4—1182no—
283C4—1182no—
284C4—1184no—
285C4—2361no—
286C4—2361no—
287C4—2362no—
288C2—24no—
289C2—81no—
290C2C2×#1427369811yescharacteristic
291C2#1683791011yescharacteristic
292C2#1683797411yescharacteristic
293C2#1683805112yes—
294C2#1683808912yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#123×C59C2
hypercenterC2×#123D59
hypocenterC59#2413
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#14273683C2semidirect
3. C22×Dic59⋊C4C2semidirect
4. #16837333C2semidirect
5. #16837325C2semidirect
6. #16837298C2semidirect
7. #16837290C2semidirect
9. #16837277C2semidirect
10. #16837265C2semidirect
159. C59#2413semidirect