Finite Group Databasebeta

← #37581996

Subgroups — #37581996

Order 24448.

584 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259348C212yesmaximal, complemented
2C4×#15690287C211yesmaximal, characteristic, complemented
3#27132860C214yesmaximal, complemented
4#27132855C214yesmaximal, complemented
5#27132687C211yesmaximal, characteristic, complemented
6#27132668C212yesmaximal, complemented
7C2×#127×C191C211yesmaximal, characteristic, complemented
8C2×#15690287C2211yescharacteristic
9#20259348—28no—
10C2×#15690275C2214yescomplemented
11C2×#15690272C2214yescomplemented
12C2×#15690271C2212yes—
13C4×Dic382C2212yescomplemented
14C4×D764C2212yescomplemented
15C4×C22⋊D191—22no—
16#20259281C2211yescharacteristic
17#20259279—21no—
18C42×D191—21no—
19C4⋊(C191⋊C8)C2214yescomplemented
20#20259243C2218yescomplemented
21C22×D4×C191—22no—
22C22×D4×C191C2212yes—
23#127×C191—21no—
24#127×C191—21no—
25#127×C191C2214yes—
26C2×C42×C191C2211yescharacteristic
27#15690287—22no—
28#15690287D414yescomplemented
29C2×Dic382C2311yescharacteristic
30C2×D764C2311yescharacteristic
31C2×C22⋊D191—21no—
32C2×C4×D191—21no—
33C2×C4×D191—22no—
34#15690275—28no—
35#15690272—28no—
36#15690271—24no—
37C2×C191⋊C8C2314yes—
38C4⋊Dic191C2312yes—
39Dic191⋊C4—22no—
40C22⋊Dic191—21no—
41(C2×D191)⋊C4—22no—
42C4×Dic191—21no—
43C4×Dic191—22no—
44C24×C191—22no—
45C2×D4×C191—24no—
46C2×D4×C191—22no—
47C2×D4×C191—24no—
48C2×D4×C191—24no—
49C2×D4×C191—24no—
50C2×D4×C191—48no—
51C2×D4×C191C2314yes—
52C22×C4×C191D412yes—
53C22×C4×C191C2311yescharacteristic
54C42×C191D411yescharacteristic
55C42×C191D411yescharacteristic
56C42×C191C2312yes—
57C22×D191—21no—
58Dic382—21no—
59Dic382C2×D412yescomplemented
60D764—21no—
61D764C2×D412yescomplemented
62C22⋊D191—24no—
63C2×Dic191—21no—
64C2×Dic191—22no—
65C4×D191—24no—
66C4×D191—44no—
67C191⋊C8—24no—
68C23×C191—24no—
69C23×C191—48no—
70C23×C191—44no—
71C23×C191C2×D411yescharacteristic
72D4×C191—24no—
73D4×C191—24no—
74D4×C191—48no—
75D4×C191—44no—
76D4×C191—48no—
77C2×C4×C191—24no—
78C2×C4×C191C2×D412yes—
79C2×C4×C191C2×D412yes—
80C2×C4×C191C2×D411yescharacteristic
81C2×C4×C191C2×D412yes—
82C2×C4×C191#4112yes—
83C2×C4×C191C2411yescharacteristic
84C22×C191—22no—
85C22×C191—44no—
86C22×C191—48no—
87C22×C191—88no—
88C22×C191#12112yes—
89C22×C191C22×D411yescharacteristic
90C2×D191—22no—
91C2×D191—42no—
92C4×C191—22no—
93C4×C191#12112yes—
94C4×C191AGL(1,ℤ/8)14yescomplemented
95C4×C191C22×D411yescharacteristic
96C4×C191C2×#4111yescharacteristic
97Dic191—22no—
98Dic191—42no—
99C2×C191—22no—
100C2×C191—84no—
101C2×C191C2×#12111yescharacteristic
102C2×C191C2×AGL(1,ℤ/8)12yes—
103D191—42no—
104C191#303811yescharacteristic, complemented, Sylow
105#3038—1911nomaximal, Sylow
106C2×AGL(1,ℤ/8)—1912no—
107C2×#127D19111yescharacteristic, complemented
108C4×#41—1911no—
109#460—1914no—
110#459—1914no—
111#423—1911no—
112#418—1912no—
113C2×#41—1911no—
114C22×D4—22no—
115C22×D4C2×D19112yes—
116AGL(1,ℤ/8)—3828no—
117C2×SD16—1914no—
118C2×D8—1914no—
119C2×M16—1912no—
120#127—21no—
121#127—21no—
122#127C2×D19114yes—
123C4×Q8—1912no—
124C4×D4—3822no—
125C4×D4—1912no—
126#117—3821no—
127#117—1911no—
128C2×C42—3821no—
129C2×C42C2×D19111yescharacteristic
130C4⋊C8—1914no—
131#102—1918no—
132C24—22no—
133#41—3822no—
134#41—1914no—
135C2×Q8—1911no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—24no—
140C2×D4—24no—
141C2×D4—48no—
142C2×D4—3821no—
143C2×D4—1911no—
144C2×D4C22×D19114yes—
145C22×C4—3821no—
146C22×C4—3822no—
147C22×C4C22⋊D19112yes—
148C22×C4C22×D19111yescharacteristic
149SD16—3828no—
150D8—3828no—
151M16—3824no—
152C2×C8—1914no—
153C4⋊C4—3822no—
154C4⋊C4—1912no—
155C22⋊C4—3821no—
156C22⋊C4—3822no—
157C42—3821no—
158C42—3822no—
159C42C22⋊D19111yescharacteristic
160C42C22⋊D19111yescharacteristic
161C42C22×D19112yes—
162C23—24no—
163C23—48no—
164C23—44no—
165C23—3821no—
166C23C2×C22⋊D19111yescharacteristic
167Q8—3821no—
168Q8—1912no—
169D4—24no—
170D4—24no—
171D4—48no—
172D4—48no—
173D4—44no—
174D4—3824no—
175D4—3821no—
176D4—1912no—
177C2×C4—24no—
178C2×C4—3821no—
179C2×C4—3824no—
180C2×C4—3822no—
181C2×C4—7644no—
182C2×C4D4×D19112yes—
183C2×C4C2×C22⋊D19112yes—
184C2×C4C2×C22⋊D19112yes—
185C2×C4C2×C22⋊D19111yescharacteristic
186C2×C4#1569028612yes—
187C2×C4C23×D19111yescharacteristic
188C8—3824no—
189C22—22no—
190C22—48no—
191C22—44no—
192C22—88no—
193C22—3822no—
194C22—7642no—
195C22#2025929312yes—
196C22C22×C22⋊D19111yescharacteristic
197C4—22no—
198C4—3822no—
199C4—7642no—
200C4#2025929312yes—
201C4#2025934814yescomplemented
202C4C2×D4×D19111yescharacteristic
203C4C2×#1569028611yescharacteristic
204C2—22no—
205C2—84no—
206C2—7642no—
207C2C2×#2025929311yescharacteristic
208C2C2×#2025934812yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#127×C191C2
socleC22×C191C22×D4
hypercenterC2×#127D191
hypocenterC191#3038
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. C4×#15690287C2semidirect
3. #27132860C2semidirect
4. #27132855C2semidirect
5. #27132687C2semidirect
6. #27132668C2semidirect
7. C2×#127×C191C2semidirect
10. C2×#15690275C22semidirect
11. C2×#15690272C22semidirect
13. C4×Dic382C22semidirect
14. C4×D764C22semidirect
19. C4⋊(C191⋊C8)C22semidirect
20. #20259243C22semidirect
28. #15690287D4semidirect
59. Dic382C2×D4semidirect
61. D764C2×D4semidirect
94. C4×C191AGL(1,ℤ/8)semidirect
104. C191#3038semidirect
107. C2×#127D191semidirect
201. C4#20259348semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
C4×#156902872
#271328602
#271328552
#271326872
#271326682
C2×#127×C1912
#3038191