Finite Group Databasebeta

← #21435317

Subgroups — #21435317

Order 7552.

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

normal subgroups only

This group has 362 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#16837808C212yesmaximal, complemented
3#16837764C212yesmaximal, complemented
4#16837732C212yesmaximal, complemented
5#16837715C212yesmaximal
6#16837694C212yesmaximal, complemented
7#16837689C212yesmaximal, complemented
8#16837459C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C59C211yesmaximal, characteristic, complemented
10#14273721—21no—
11#14273719C2211yescharacteristic
12#14273715C2211yescharacteristic
13#14273710—21no—
14#14273706C2211yescharacteristic
15#14273701C2211yescharacteristic
16C4×Dic118C2211yescharacteristic
17C4×D236C2211yescharacteristic
18C4×C22⋊D59—21no—
19#14273684—21no—
20#14273660C2214yes—
21#14273658C2214yes—
22C4⋊(C59⋊C8)C2212yes—
23#14273650C2214yes—
24#14273647C2214yes—
25C22⋊(C59⋊C8)C2212yes—
26#14273638C2212yes—
27C22×Q8×C59—21no—
28C4⋊Q8×C59—21no—
29C4⋊Q8×C59—21no—
30C4⋊Q8×C59C2212yes—
31C4⋊Q8×C59C2212yes—
32C2×C4⋊C4×C59—21no—
33C2×C4⋊C4×C59C2212yes—
34C2×C42×C59C2211yescharacteristic
35C2×Q8×C59—22no—
36C2×Q8×C59—22no—
37C2×Q8×C59—44no—
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×C59C2314yes—
50C42×C59D411yescharacteristic
51C42×C59D411yescharacteristic
52C42×C59C2311yescharacteristic
53C42×C59C2311yescharacteristic
54C22⋊Dic59—21no—
55(C2×D59)⋊C4—21no—
56(C2×D59)⋊C4—21no—
57(C2×D59)⋊C4—21no—
58(C2×D59)⋊C4—21no—
59C4⋊Dic59C2311yescharacteristic
60C4⋊Dic59C2311yescharacteristic
61Dic59⋊C4—21no—
62Dic59⋊C4—21no—
63Dic59⋊C4—21no—
64Dic59⋊C4—21no—
65C4×Dic59—21no—
66C2×C59⋊C8C2314yes—
67C23×C59C2×D411yescharacteristic
68C22×D59—21no—
69Q8×C59—44no—
70Q8×C59—44no—
71C2×C4×C59—22no—
72C2×C4×C59—22no—
73C2×C4×C59—22no—
74C2×C4×C59—24no—
75C2×C4×C59—44no—
76C2×C4×C59C2×D411yescharacteristic
77C2×C4×C59C2×D411yescharacteristic
78C2×C4×C59C2×D411yescharacteristic
79C2×C4×C59C2×D411yescharacteristic
80C2×C4×C59C2×D411yescharacteristic
81C2×C4×C59C2411yescharacteristic
82C22⋊D59—42no—
83C2×Dic59—21no—
84C2×Dic59—21no—
85C2×Dic59—21no—
86D236—22no—
87C4×D59—42no—
88Dic118—22no—
89C59⋊C8—24no—
90C22×C59—21no—
91C22×C59—21no—
92C22×C59AGL(1,ℤ/8)12yes—
93C22×C59C22×D411yescharacteristic
94C2×D59—41no—
95C2×D59—41no—
96C2×D59—41no—
97C4×C59—21no—
98C4×C59—21no—
99C4×C59—21no—
100C4×C59—21no—
101C4×C59—44no—
102C4×C59#13712yes—
103C4×C592−1+412yes—
104Dic59—41no—
105Dic59—41no—
106Dic59—41no—
107#3238—591nomaximal, Sylow
108C2×C59—22no—
109C2×C59#53611yescharacteristic
110C2×C59C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C59C2×#13711yescharacteristic
112D59—81no—
113C2×C4⋊Q8D5911yescharacteristic, complemented
114#529—591no—
115#501—592no—
116#489—592no—
117#483—592no—
118#480—592no—
119#477—592no—
120#476—592no—
121#432—591no—
122C59#323811yescharacteristic, complemented, Sylow
123C22×Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C2×D5912yes—
127C4⋊Q8C2×D5912yes—
128#126—1181no—
129#125—591no—
130#124—591no—
131#123—1181no—
132C22⋊Q8—591no—
133#121—591no—
134C4×Q8—591no—
135C4×D4—591no—
136C4×D4—1181no—
137#117—1181no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C2×D5912yes—
140C2×C42C2×D5911yescharacteristic
141#107—594no—
142#106—594no—
143C4⋊C8—592no—
144#103—594no—
145#102—594no—
146C22⋊C8—592no—
147#97—592no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—44no—
151C2×Q8—591no—
152C2×D4—591no—
153C2×D4—1181no—
154C22×C4—22no—
155C22×C4—1181no—
156C22×C4C22⋊D5911yescharacteristic
157C22×C4C22⋊D5911yescharacteristic
158C22×C4C22×D5911yescharacteristic
159C2×C8—594no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—591no—
164C4⋊C4—591no—
165C4⋊C4—1181no—
166C4⋊C4—1181no—
167C4⋊C4—1181no—
168C4⋊C4—1181no—
169C4⋊C4C22×D5914yes—
170C22⋊C4—1181no—
171C22⋊C4—1181no—
172C22⋊C4—1181no—
173C22⋊C4—1181no—
174C22⋊C4—1181no—
175C42—1181no—
176C42C22⋊D5911yescharacteristic
177C42C22⋊D5911yescharacteristic
178C42C22×D5911yescharacteristic
179C42C22×D5911yescharacteristic
180C23—1181no—
181C23C2×C22⋊D5911yescharacteristic
182Q8—44no—
183Q8—44no—
184Q8—1182no—
185D4—1182no—
186D4—2362no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—44no—
192C2×C4—1181no—
193C2×C4—1181no—
194C2×C4—1181no—
195C2×C4—2362no—
196C2×C4C2×C22⋊D5911yescharacteristic
197C2×C4C2×C22⋊D5911yescharacteristic
198C2×C4C2×C22⋊D5911yescharacteristic
199C2×C4C2×C22⋊D5911yescharacteristic
200C2×C4C2×C22⋊D5911yescharacteristic
201C2×C4C23×D5911yescharacteristic
202C8—1184no—
203C22—21no—
204C22—21no—
205C22—2361no—
206C22—2361no—
207C22—2361no—
208C22#1427375312yes—
209C22C22×C22⋊D5911yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—44no—
215C4—2361no—
216C4—2361no—
217C4—2361no—
218C4#1427375812yes—
219C4#1427377512yes—
220C2—22no—
221C2—4721no—
222C2#1683797611yescharacteristic
223C2C2×#1427375311yescharacteristic
224C2C2×#1427375811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC2×C4⋊Q8×C59C2
socleC22×C59C22×D4
hypercenterC2×C4⋊Q8D59
hypocenterC59#3238
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. #16837808C2semidirect
3. #16837764C2semidirect
4. #16837732C2semidirect
6. #16837694C2semidirect
7. #16837689C2semidirect
8. #16837459C2semidirect
9. C2×C4⋊Q8×C59C2semidirect
113. C2×C4⋊Q8D59semidirect
122. C59#3238semidirect