Finite Group Databasebeta

← #21431104

Subgroups — #21431104

Order 7552.

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

normal subgroups only

This group has 492 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×#14273725C211yesmaximal, characteristic, complemented
2#16837325C212yesmaximal, complemented
3#16837316C212yesmaximal
4#16837314C212yesmaximal, complemented
5#16837309C211yesmaximal, characteristic, complemented
6#16837296C212yesmaximal, complemented
7#16837290C212yesmaximal, complemented
8#16837286C212yesmaximal, complemented
9C2×#124×C59C211yesmaximal, characteristic
10C22×Dic118C2211yescharacteristic
11#14273725—24no—
12C2×C4⋊Dic59C2211yescharacteristic
13C2×Dic59⋊C4C2212yes—
14C2×Dic59⋊C4C2212yes—
15C2×Dic59⋊C4C2212yes—
16C2×C22⋊Dic59C2212yes—
17C2×C22⋊Dic59C2212yes—
18C2×C22⋊Dic59C2212yes—
19C2×C22⋊Dic59C2212yes—
20C2×C4×Dic59C2211yescharacteristic
21C2×C4×Dic59C2211yescharacteristic
22C59⋊#95C2212yes—
23C59⋊#95C2212yes—
24C59⋊#95C2212yes—
25C59⋊#95C2212yes—
26C59⋊#95C2212yes—
27C22×Q8×C59C2211yescharacteristic
28C22×D4×C59C2211yescharacteristic
29#124×C59—24no—
30C2×C22⋊C4×C59C2212yes—
31C2×C22⋊C4×C59C2212yes—
32C2×C42×C59C2211yescharacteristic
33C24×C59C2312yes—
34C2×Q8×C59—41no—
35C2×Q8×C59—41no—
36C2×Q8×C59D414yes—
37C2×D4×C59—22no—
38C2×D4×C59—41no—
39C2×D4×C59—41no—
40C22×C4×C59C2312yes—
41C22×C4×C59C2311yescharacteristic
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2312yes—
46C22×Dic59C2312yes—
47C22×Dic59C2312yes—
48C2×Dic118—22no—
49C2×Dic118—41no—
50C2×Dic118—41no—
51C22⋊C4×C59—24no—
52C22⋊C4×C59—24no—
53C42×C59—22no—
54C22⋊Dic59—24no—
55C22⋊Dic59—24no—
56C22⋊Dic59—42no—
57C22⋊Dic59—42no—
58C4⋊Dic59—41no—
59Dic59⋊C4—24no—
60Dic59⋊C4—24no—
61Dic59⋊C4—24no—
62C4×Dic59—22no—
63C4×Dic59—41no—
64C23×C59—22no—
65C23×C59—22no—
66C23×C59—22no—
67C23×C59—22no—
68C23×C59—22no—
69C23×C59—22no—
70C23×C59—22no—
71C23×C59C2411yescharacteristic
72Q8×C59—44no—
73D4×C59—44no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59C2×D412yes—
84C2×C4×C59C2×D412yes—
85C2×C4×C59C2×D412yes—
86C2×Dic59—22no—
87C2×Dic59—22no—
88C2×Dic59—22no—
89C2×Dic59—22no—
90C2×Dic59—22no—
91C2×Dic59—22no—
92C2×Dic59—22no—
93C2×Dic59—22no—
94C2×Dic59—22no—
95C2×Dic59—22no—
96C2×Dic59—22no—
97C2×Dic59#4114yes—
98Dic118—44no—
99C22×C59—42no—
100C22×C59—42no—
101C22×C59—42no—
102C22×C59—42no—
103C22×C59—42no—
104C22×C59—42no—
105C22×C59—42no—
106C22×C59C22×D411yescharacteristic
107C22×C59C2×#4112yes—
108C22×C59H(5,𝔽2)11yescharacteristic
109C22×C59H(5,𝔽2)11yescharacteristic
110C22×C592−1+412yes—
111C4×C59—22no—
112C4×C59—22no—
113C4×C59—22no—
114C4×C59—42no—
115Dic59—24no—
116Dic59—42no—
117Dic59—42no—
118Dic59—42no—
119#2731—591nomaximal, Sylow
120C2×C59—82no—
121C2×C59#53411yescharacteristic
122C2×C59#54212yes—
123C2×C59#54812yes—
124C2×C59#55511yescharacteristic
125C2×C59#56311yescharacteristic
126C2×C4⋊Q8—591no—
127C2×#124D5911yescharacteristic
128#396—592no—
129#394—592no—
130#393—592no—
131#391—591no—
132#388—592no—
133#388—592no—
134#387—592no—
135C59#273111yescharacteristic, complemented, Sylow
136C22×Q8—591no—
137C22×Q8C2×D5911yescharacteristic
138C22×D4C2×D5911yescharacteristic
139C4⋊Q8—1184no—
140#124—24no—
141C2×C4⋊C4—592no—
142C2×C4⋊C4—592no—
143C2×C4⋊C4—592no—
144C2×C4⋊C4—591no—
145C2×C22⋊C4—592no—
146C2×C22⋊C4—592no—
147C2×C22⋊C4—592no—
148C2×C22⋊C4—592no—
149C2×C22⋊C4C2×D5912yes—
150C2×C22⋊C4C2×D5912yes—
151C2×C42—591no—
152C2×C42—591no—
153C2×C42C2×D5911yescharacteristic
154#95—592no—
155#95—592no—
156#95—592no—
157#95—592no—
158#95—592no—
159C24C22×D5912yes—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—1182no—
163C2×Q8—2361no—
164C2×Q8—2361no—
165C2×Q8C22⋊D5914yes—
166C2×D4—22no—
167C2×D4—41no—
168C2×D4—41no—
169C22×C4—592no—
170C22×C4—592no—
171C22×C4—592no—
172C22×C4—592no—
173C22×C4C22×D5912yes—
174C22×C4C22×D5911yescharacteristic
175C22×C4C22×D5911yescharacteristic
176C22×C4C22×D5911yescharacteristic
177C4⋊C4—1184no—
178C4⋊C4—1184no—
179C4⋊C4—1184no—
180C4⋊C4—2361no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—1184no—
184C22⋊C4—1184no—
185C22⋊C4—2362no—
186C22⋊C4—2362no—
187C42—22no—
188C42—1182no—
189C42—2361no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23C23×D5911yescharacteristic
198Q8—44no—
199Q8—2364no—
200D4—44no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—594no—
211C2×C4—1182no—
212C2×C4—1182no—
213C2×C4—1182no—
214C2×C4—1182no—
215C2×C4—1182no—
216C2×C4—1182no—
217C2×C4—1182no—
218C2×C4—1182no—
219C2×C4—1182no—
220C2×C4—1182no—
221C2×C4—1182no—
222C2×C4C2×C22⋊D5912yes—
223C2×C4C2×C22⋊D5912yes—
224C2×C4C2×C22⋊D5912yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C22×C22⋊D5911yescharacteristic
233C22#41×D5912yes—
234C22#1427377211yescharacteristic
235C22#1427377211yescharacteristic
236C22#1427377412yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—1184no—
242C4—2362no—
243C4—2362no—
244C4—2362no—
245C2—82no—
246C2#1683796111yescharacteristic
247C2#1683801612yes—
248C2#1683805912yes—
249C2#1683811411yescharacteristic
250C2#1683815211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#124×C59C2
hypercenterC2×#124D59
hypocenterC59#2731
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#14273725C2semidirect
2. #16837325C2semidirect
4. #16837314C2semidirect
5. #16837309C2semidirect
6. #16837296C2semidirect
7. #16837290C2semidirect
8. #16837286C2semidirect
135. C59#2731semidirect