Finite Group Databasebeta

← #44537290

Subgroups — #44537290

Order 31424.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#21463327C211yesmaximal, characteristic, complemented
2C22×C22⋊D491C211yesmaximal, characteristic, complemented
3#30628333C212yesmaximal, complemented
4#30628323C212yesmaximal, complemented
5#30628310C212yesmaximal, complemented
6#30628309C212yesmaximal, complemented
7#30628307C211yesmaximal, characteristic, complemented
8#30628306C211yesmaximal, characteristic, complemented
9#30628305C211yesmaximal, characteristic, complemented
10#30628293C211yesmaximal, characteristic, complemented
11#120×C491C211yesmaximal, characteristic, complemented
12C23×D491C2211yescharacteristic
13#21463327—24no—
14C2×Dic982C2211yescharacteristic, complemented
15C2×D1964C2212yescomplemented
16C2×C22⋊D491—24no—
17C2×C22⋊D491C2212yes—
18C2×C22⋊D491C2212yes—
19C2×C22⋊D491C2211yescharacteristic
20C2×C22⋊D491C2211yescharacteristic, complemented
21C2×C22⋊D491C2212yescomplemented
22C2×C22⋊D491C2212yescomplemented
23C22×Dic491C2211yescharacteristic
24C22×Dic491C2211yescharacteristic, complemented
25C2×C4×D491C2211yescharacteristic, complemented
26Dic491⋊C4C2212yescomplemented
27C22⋊Dic491C2211yescharacteristic
28C22⋊Dic491C2212yescomplemented
29(C2×D491)⋊C4C2212yescomplemented
30(C2×D491)⋊C4C2212yescomplemented
31C4×Dic491C2212yescomplemented
32C24×C491C2211yescharacteristic
33C2×D4×C491C2211yescharacteristic
34C2×D4×C491C2212yescomplemented
35C22⋊C4×C491C2211yescharacteristic
36C22⋊C4×C491C2212yescomplemented
37C22×D491—22no—
38C22×D491—42no—
39C22×D491C2312yes—
40C22×D491C2311yescharacteristic
41Dic982—22no—
42D1964—42no—
43C22⋊D491—22no—
44C22⋊D491—24no—
45C22⋊D491—24no—
46C22⋊D491—22no—
47C22⋊D491—42no—
48C22⋊D491—42no—
49C2×Dic491—22no—
50C2×Dic491D414yescomplemented
51C2×Dic491C2312yes—
52C2×Dic491C2311yescharacteristic
53C2×Dic491C2311yescharacteristic
54C2×Dic491C2311yescharacteristic
55C4×D491—22no—
56C23×C491—22no—
57C23×C491—42no—
58C23×C491C2311yescharacteristic
59C23×C491C2312yes—
60C23×C491C2311yescharacteristic
61D4×C491—22no—
62D4×C491—42no—
63C2×C4×C491C2311yescharacteristic
64C2×C4×C491C2312yes—
65C22×C491—22no—
66C22×C491—21no—
67C22×C491—22no—
68C22×C491—22no—
69C22×C491—22no—
70C22×C491—22no—
71C22×C491—44no—
72C22×C491C2×D412yes—
73C22×C491C2411yescharacteristic
74C2×D491—21no—
75C2×D491—22no—
76C2×D491—22no—
77C2×D491—22no—
78C2×D491—22no—
79C2×D491—44no—
80C4×C491—21no—
81C4×C491—22no—
82Dic491—21no—
83Dic491—22no—
84Dic491C2×D412yes—
85Dic491C2×D412yes—
86C2×C491—22no—
87C2×C491—41no—
88C2×C491—42no—
89C2×C491C22×D411yescharacteristic
90C2×C491H(5,𝔽2)12yes—
91D491—41no—
92D491—42no—
93C491#53511yescharacteristic, complemented, Sylow
94#535—4911nomaximal, Sylow
95C2×#41—4911no—
96C22×D4—4911no—
97#127—4912no—
98#124—4912no—
99#121—4912no—
100#121—4912no—
101#120—4911no—
102#120—4911no—
103#120—4911no—
104#120D49111yescharacteristic, complemented
105#117—4911no—
106C24—4911no—
107C24C2×D49111yescharacteristic
108#41—9824no—
109C2×Q8—4911no—
110C2×D4—9824no—
111C2×D4—4912no—
112C2×D4—4912no—
113C2×D4—4911no—
114C2×D4—4912no—
115C2×D4—4912no—
116C2×D4—4911no—
117C2×D4—4912no—
118C2×D4C2×D49111yescharacteristic
119C2×D4C2×D49112yescomplemented
120C22×C4—4911no—
121C22×C4—4911no—
122C22×C4—4911no—
123C4⋊C4—4912no—
124C22⋊C4—4911no—
125C22⋊C4—4912no—
126C22⋊C4—4912no—
127C22⋊C4—4912no—
128C22⋊C4C2×D49111yescharacteristic
129C22⋊C4C2×D49112yescomplemented
130C42—4912no—
131C23—22no—
132C23—42no—
133C23—19642no—
134C23—9822no—
135C23—4912no—
136C23—4911no—
137C23C22×D49112yes—
138C23C22×D49111yescharacteristic
139C23C22×D49111yescharacteristic
140Q8—9822no—
141D4—22no—
142D4—42no—
143D4—19642no—
144D4—19642no—
145D4—19642no—
146D4—9822no—
147D4—9824no—
148D4—9824no—
149D4—9822no—
150C2×C4—9822no—
151C2×C4—9822no—
152C2×C4—4912no—
153C2×C4—4911no—
154C2×C4—4914no—
155C2×C4—4911no—
156C2×C4—4911no—
157C2×C4C22×D49112yes—
158C2×C4C22×D49111yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—19644no—
167C22—9821no—
168C22—9822no—
169C22—9822no—
170C22—9822no—
171C22—9822no—
172C22D4×D49112yes—
173C22C23×D49111yescharacteristic
174C4—21no—
175C4—22no—
176C4—9821no—
177C4—9822no—
178C4—4912no—
179C4—4912no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—19642no—
184C2—19641no—
185C2C2×D4×D49111yescharacteristic
186C2#3062838212yes—
subgroupG/H
center, FrattiniC22C23×D491
commutator, socle, cosocleC22×C491C24
Fitting#120×C491C2
hypercenter#120D491
hypocenterC491#535
LCSG ▷ C22×C491 ▷ C491
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#21463327C2semidirect
2. C22×C22⋊D491C2semidirect
3. #30628333C2semidirect
4. #30628323C2semidirect
5. #30628310C2semidirect
6. #30628309C2semidirect
7. #30628307C2semidirect
8. #30628306C2semidirect
9. #30628305C2semidirect
10. #30628293C2semidirect
11. #120×C491C2semidirect
14. C2×Dic982C22semidirect
15. C2×D1964C22semidirect
20. C2×C22⋊D491C22semidirect
21. C2×C22⋊D491C22semidirect
22. C2×C22⋊D491C22semidirect
24. C22×Dic491C22semidirect
25. C2×C4×D491C22semidirect
26. Dic491⋊C4C22semidirect
28. C22⋊Dic491C22semidirect
29. (C2×D491)⋊C4C22semidirect
30. (C2×D491)⋊C4C22semidirect
31. C4×Dic491C22semidirect
34. C2×D4×C491C22semidirect
36. C22⋊C4×C491C22semidirect
50. C2×Dic491D4semidirect
93. C491#535semidirect
104. #120D491semidirect
119. C2×D4C2×D491semidirect
129. C22⋊C4C2×D491semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#214633272
C22×C22⋊D4912
#306283332
#306283232
#306283102
#306283092
#306283072
#306283062
#306283052
#306282932
#120×C4912
#535491