Finite Group Databasebeta

← #21423313

Subgroups — #21423313

Order 7552.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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#16838109C211yesmaximal, characteristic
2#16837753C211yesmaximal, characteristic, complemented
3#16837682C211yesmaximal, characteristic
4#16837613C211yesmaximal, characteristic, complemented
5#16837150C211yesmaximal, characteristic, complemented
6#16837134C211yesmaximal, characteristic, complemented
7C59×D4⋊C8C211yesmaximal, characteristic
8C2×Dic236—21no—
9#14273727C2211yescharacteristic
10#14273727C2211yescharacteristic
11#14273718—21no—
12#14273706—21no—
13#14273704—22no—
14C4×Dic118C2211yescharacteristic
15D4×Dic59—21no—
16C2×C4⋊Dic59—21no—
17#14273660—21no—
18C4⋊(C59⋊C8)C2211yescharacteristic
19#14273652—22no—
20#14273652—21no—
21#14273648—21no—
22C4×D4×C59C2211yescharacteristic
23C4⋊C8×C59C2211yescharacteristic
24C4×C8×C59C2211yescharacteristic
25C2×D4×C59D411yescharacteristic
26C22×C4×C59—21no—
27C22×Dic59—22no—
28C2×Dic118—21no—
29C2×Dic118D411yescharacteristic
30C2×Dic118D412yes—
31C2×C8×C59—21no—
32C2×C8×C59—21no—
33C4⋊C4×C59D411yescharacteristic
34C22⋊C4×C59—21no—
35C42×C59C2311yescharacteristic
36C22⋊Dic59—22no—
37C4⋊Dic59—22no—
38C4⋊Dic59—21no—
39C4⋊Dic59—21no—
40C4⋊Dic59—21no—
41C4⋊Dic59—21no—
42C4⋊Dic59D411yescharacteristic
43Dic59⋊C4—22no—
44Dic59⋊C4—21no—
45C4×Dic59—21no—
46C2×C59⋊C8—21no—
47Dic236—42no—
48C23×C59—21no—
49D4×C59—21no—
50D4×C59Q1612yescomplemented
51C2×C4×C59—21no—
52C2×C4×C59—41no—
53C2×C4×C59—41no—
54C2×C4×C59C2×D411yescharacteristic
55C2×C4×C59C2×D411yescharacteristic
56C2×C4×C59C2×D411yescharacteristic
57C2×Dic59—21no—
58C2×Dic59—21no—
59C2×Dic59—22no—
60C2×Dic59—41no—
61C2×Dic59—82no—
62Dic118—21no—
63Dic118—21no—
64Dic118—42no—
65Dic118—41no—
66Dic118—41no—
67C8×C59—41no—
68C8×C59—41no—
69C59⋊C8—41no—
70C22×C59—22no—
71C22×C59—41no—
72C22×C59—41no—
73C22×C59#12011yescharacteristic
74C4×C59—21no—
75C4×C59—41no—
76C4×C59C2×Q1611yescharacteristic
77C4×C59#13511yescharacteristic
78C4×C59AGL(1,ℤ/8)11yescharacteristic
79C4×C59#13711yescharacteristic
80Dic59—41no—
81Dic59—42no—
82Dic59—41no—
83Dic59—81no—
84#1644—591nomaximal, Sylow
85C2×C59—42no—
86C2×C59#44911yescharacteristic
87C2×C59#45111yescharacteristic
88C2×C59#45511yescharacteristic
89#554—591no—
90#487—591no—
91#474—591no—
92#462—591no—
93#333—591no—
94Q8⋊C8—591no—
95D4⋊C8D5911yescharacteristic
96C59#164411yescharacteristic, complemented, Sylow
97C2×Q16—1181no—
98C4⋊Q8—591no—
99C4⋊Q8—591no—
100#125—1181no—
101C22⋊Q8—1182no—
102C22⋊Q8—1181no—
103C4×Q8—591no—
104C4×D4—1181no—
105C4×D4C2×D5911yescharacteristic
106C2×C4⋊C4—1181no—
107#107—1181no—
108C4⋊C8—591no—
109C4⋊C8C2×D5911yescharacteristic
110#103—1181no—
111#103—1182no—
112#102—1181no—
113C4×C8C2×D5911yescharacteristic
114C2×Q8—591no—
115C2×Q8—592no—
116C2×Q8—1181no—
117C2×D4D23611yescharacteristic
118C22×C4—21no—
119C22×C4—1182no—
120Q16—2362no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—1181no—
124C4⋊C4—591no—
125C4⋊C4—1182no—
126C4⋊C4—1182no—
127C4⋊C4—1181no—
128C4⋊C4—1181no—
129C4⋊C4—1181no—
130C4⋊C4—1181no—
131C4⋊C4—1181no—
132C4⋊C4D23611yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—1182no—
135C42—1181no—
136C42C22×D5911yescharacteristic
137C23—21no—
138Q8—1181no—
139Q8—1181no—
140Q8—2362no—
141Q8—2361no—
142Q8—2361no—
143D4—21no—
144D4Dic23612yescomplemented
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—1181no—
149C2×C4—1181no—
150C2×C4—1182no—
151C2×C4—4722no—
152C2×C4—2361no—
153C2×C4C2×D23611yescharacteristic
154C2×C4D4×D5911yescharacteristic
155C2×C4D4×D5911yescharacteristic
156C8—41no—
157C8—41no—
158C8—2361no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22#1427369711yescharacteristic
163C4—21no—
164C4—41no—
165C4—4721no—
166C4—2361no—
167C4—2361no—
168C4—2362no—
169C4C2×Dic23611yescharacteristic
170C4#1427374811yescharacteristic
171C4#1427375111yescharacteristic
172C4#1427376011yescharacteristic
173C2—42no—
174C2#1683754811yescharacteristic
175C2#1683756011yescharacteristic
176C2#1683758111yescharacteristic
subgroupG/H
centerC22#14273697
commutatorC42×C59C23
FrattiniC42C22×D59
FittingC59×D4⋊C8C2
socleC22×C59#120
hypercenterD4⋊C8D59
hypocenterC59#1644
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #16837753C2semidirect
4. #16837613C2semidirect
5. #16837150C2semidirect
6. #16837134C2semidirect
50. D4×C59Q16semidirect
96. C59#1644semidirect
144. D4Dic236semidirect