Finite Group Databasebeta

← #21436955

Subgroups — #21436955

Order 7552.

370 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic118C211yesmaximal, characteristic, complemented
2#16838067C211yesmaximal, characteristic
3#16837799C212yesmaximal, complemented
4#16837798C211yesmaximal, characteristic, complemented
5#16837693C211yesmaximal, characteristic
6#16837691C212yesmaximal
7#16837685C212yesmaximal, complemented
8#16837526C211yesmaximal, characteristic, complemented
9C4×#11745671C211yesmaximal, characteristic
10#16837487C212yesmaximal
11C59×#475C211yesmaximal, characteristic, complemented
12C2×Q8×D59—21no—
13C2×#11745671C2211yescharacteristic
14#14273727—21no—
15#14273727C2211yescharacteristic
16#14273726—22no—
17#14273726C2212yes—
18#14273726C2212yes—
19#14273705—22no—
20#14273703—21no—
21Q8×Dic59C2212yes—
22C4×Dic118—21no—
23C4×Dic118C2211yescharacteristic
24C4×D236C2211yescharacteristic
25#14273689C2212yes—
26#14273659C2212yes—
27#14273658C2212yes—
28#14273658C2212yes—
29#14273658C2211yescharacteristic
30Dic59⋊C8C2212yes—
31#14273651C2211yescharacteristic
32#14273649C2212yes—
33#14273647C2211yescharacteristic
34C4×C59⋊C8C2211yescharacteristic
35C8×Dic59C2212yes—
36C4⋊Q8×C59C2211yescharacteristic
37C4×Q8×C59C2211yescharacteristic
38#106×C59C2212yes—
39C4⋊C8×C59C2211yescharacteristic
40#103×C59C2212yes—
41C2×Q8×C59—21no—
42C2×Q8×C59C2311yescharacteristic
43Q8×D59—44no—
44C2×D236C2311yescharacteristic
45C2×C4×D59—21no—
46C2×C4×D59—22no—
47C2×Dic118—22no—
48C2×Dic118—21no—
49C2×Dic118—22no—
50C2×Dic118D411yescharacteristic
51C2×C8×C59C2312yes—
52C4⋊C4×C59—21no—
53C4⋊C4×C59—21no—
54C4⋊C4×C59C2312yes—
55C4⋊C4×C59C2311yescharacteristic
56C42×C59—21no—
57C42×C59C2311yescharacteristic
58#11745671Q814yescomplemented
59(C2×D59)⋊C4—21no—
60(C2×D59)⋊C4—22no—
61C4⋊Dic59—21no—
62C4⋊Dic59—22no—
63C4⋊Dic59D411yescharacteristic
64C4⋊Dic59C2312yes—
65C4⋊Dic59C2311yescharacteristic
66Dic59⋊C4—22no—
67Dic59⋊C4—22no—
68Dic59⋊C4D412yes—
69C4×Dic59—22no—
70C4×Dic59C2312yes—
71C2×C59⋊C8C2311yescharacteristic
72C2×C59⋊C8C2311yescharacteristic
73C22×D59—21no—
74Q8×C59—21no—
75Q8×C59—42no—
76Q8×C59C2×Q812yescomplemented
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—22no—
80C2×C4×C59C2×D411yescharacteristic
81C2×C4×C59C2×D411yescharacteristic
82C2×C4×C59C2411yescharacteristic
83C2×Dic59—22no—
84C2×Dic59—21no—
85C2×Dic59C2×D412yes—
86C2×Dic59C2×D412yes—
87D236—21no—
88D236C2×Q812yes—
89C4×D59—42no—
90C4×D59—44no—
91Dic118—44no—
92Dic118—82no—
93Dic118—81no—
94Dic118SD1614yescomplemented
95C8×C59—22no—
96C59⋊C8—21no—
97C59⋊C8C2×Q812yescomplemented
98C22×C59C22×D411yescharacteristic
99C2×D59—22no—
100C2×D59—41no—
101C2×D59—41no—
102C4×C59—22no—
103C4×C59—21no—
104C4×C59—42no—
105C4×C59—41no—
106C4×C59C2×SD1612yes—
107C4×C59C22×Q811yescharacteristic
108C4×C592−1+411yescharacteristic
109Dic59—22no—
110Dic59—41no—
111Dic59—42no—
112Dic59C2×SD1614yes—
113Q8×SD16—591nomaximal, Sylow
114C2×C59D4×Q811yescharacteristic
115C2×C59C22×SD1611yescharacteristic
116C2×C59#57711yescharacteristic
117D59—42no—
118Q82—591no—
119D4×Q8—591no—
120#498—591no—
121#498—592no—
122#476—592no—
123#476—591no—
124#475—592no—
125#475D5911yescharacteristic, complemented
126C8×Q8—591no—
127C4×SD16—591no—
128C4×SD16—592no—
129C59Q8×SD1611yescharacteristic, complemented, Sylow
130C22×Q8—1181no—
131C2×SD16—591no—
132C4⋊Q8—592no—
133C4⋊Q8—592no—
134C4⋊Q8—591no—
135C4⋊Q8—1181no—
136C4⋊Q8—1182no—
137C4⋊Q8C2×D5911yescharacteristic
138C22⋊Q8—1182no—
139C22⋊Q8—1181no—
140C4×Q8—591no—
141C4×Q8—592no—
142C4×Q8—1181no—
143C4×Q8C2×D5911yescharacteristic
144C4×D4—592no—
145C4×D4—591no—
146#106—592no—
147#106—592no—
148#106—591no—
149#106—592no—
150#106C2×D5912yes—
151C4⋊C8—592no—
152C4⋊C8C2×D5911yescharacteristic
153#103—591no—
154#103C2×D5912yes—
155#102—592no—
156#102—591no—
157C4×C8—592no—
158C4×C8—591no—
159C2×Q8—21no—
160C2×Q8—591no—
161C2×Q8—1182no—
162C2×Q8—1182no—
163C2×Q8—1181no—
164C2×Q8—2364no—
165C2×Q8C22×D5911yescharacteristic
166C2×D4—591no—
167C22×C4—1182no—
168C22×C4—1181no—
169SD16—594no—
170C2×C8—591no—
171C2×C8—591no—
172C2×C8C22×D5912yes—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—592no—
176C4⋊C4—591no—
177C4⋊C4—592no—
178C4⋊C4—591no—
179C4⋊C4—1182no—
180C4⋊C4—1182no—
181C4⋊C4—1182no—
182C4⋊C4—1181no—
183C4⋊C4C22×D5911yescharacteristic
184C4⋊C4C22×D5912yes—
185C22⋊C4—1182no—
186C22⋊C4—1181no—
187C42—21no—
188C42—592no—
189C42—1182no—
190C42C22×D5911yescharacteristic
191C23—1181no—
192Q8—21no—
193Q8—42no—
194Q8—594no—
195Q8—4721no—
196Q8—4722no—
197Q8—2364no—
198Q8Q8×D5912yescomplemented
199D4—592no—
200D4—1181no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—592no—
205C2×C4—592no—
206C2×C4—1182no—
207C2×C4—1181no—
208C2×C4—2364no—
209C2×C4—2362no—
210C2×C4D4×D5911yescharacteristic
211C2×C4D4×D5911yescharacteristic
212C2×C4C23×D5911yescharacteristic
213C8—22no—
214C8—592no—
215C8—1181no—
216C22—1182no—
217C22—2361no—
218C22—2361no—
219C22C2×D4×D5911yescharacteristic
220C4—21no—
221C4—22no—
222C4—42no—
223C4—41no—
224C4—594no—
225C4—1182no—
226C4—2361no—
227C4—2362no—
228C4SD16×D5912yes—
229C4C2×Q8×D5911yescharacteristic
230C4#1427377411yescharacteristic
231C2—2362no—
232C2#1683806511yescharacteristic
233C2C2×SD16×D5911yescharacteristic
234C2#1683821111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#475C2
socleC22×C59C22×D4
hypercenter#475D59
hypocenterC59Q8×SD16
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. Q8×Dic118C2semidirect
3. #16837799C2semidirect
4. #16837798C2semidirect
7. #16837685C2semidirect
8. #16837526C2semidirect
11. C59×#475C2semidirect
58. #11745671Q8semidirect
76. Q8×C59C2×Q8semidirect
94. Dic118SD16semidirect
97. C59⋊C8C2×Q8semidirect
125. #475D59semidirect
129. C59Q8×SD16semidirect
198. Q8Q8×D59semidirect