Finite Group Databasebeta

← C2×#27132998 #37582425

Subgroups — C2×#27132998 #37582425

Order 24448.

460 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 165 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×#20259301C211yesmaximal, characteristic
2#27132998C218yesmaximal, direct product factor
3C2×#20259255C212yesmaximal, complemented
4C2×#20259245C212yesmaximal, complemented
5C2×C22⋊(C191⋊C8)C211yesmaximal, characteristic
6C22×C4⋊C4×C191C211yesmaximal, characteristic
7C22×Dic382C2211yescharacteristic
8C22×C191⋊C8C2212yes—
9#20259301—22no—
10#20259301C2214yes—
11C2×C4⋊Dic191C2211yescharacteristic
12C2×Dic191⋊C4—21no—
13C2×C22⋊Dic191—21no—
14#20259255C2218yescomplemented
15#20259245C2218yescomplemented
16C22⋊(C191⋊C8)C2214yes—
17C23×C4×C191—21no—
18C23×C4×C191C2211yescharacteristic
19C2×C4⋊C4×C191—21no—
20C2×C4⋊C4×C191—22no—
21C2×C4⋊C4×C191C2212yes—
22C2×C4⋊C4×C191C2214yes—
23C2×Dic382—21no—
24C2×Dic382—24no—
25C2×Dic382C2312yes—
26C22×Dic191—21no—
27C22×Dic191—21no—
28C2×C191⋊C8—24no—
29C2×C191⋊C8C2314yes—
30C4⋊Dic191—21no—
31C4⋊Dic191C2312yes—
32Dic191⋊C4—24no—
33C22⋊Dic191—24no—
34C24×C191D411yescharacteristic
35C22×C4×C191—24no—
36C22×C4×C191—24no—
37C22×C4×C191—22no—
38C22×C4×C191—44no—
39C22×C4×C191D411yescharacteristic
40C22×C4×C191D412yes—
41C22×C4×C191C2312yes—
42C22×C4×C191C2311yescharacteristic
43C4⋊C4×C191—24no—
44C4⋊C4×C191—22no—
45C4⋊C4×C191C2314yes—
46Dic382—24no—
47Dic382—42no—
48C2×Dic191—22no—
49C2×Dic191—22no—
50C2×Dic191—42no—
51C2×Dic191—42no—
52C191⋊C8—24no—
53C23×C191—22no—
54C23×C191—22no—
55C23×C191Q1614yes—
56C23×C191C2×D412yes—
57C23×C191C2×D411yescharacteristic
58C2×C4×C191—22no—
59C2×C4×C191—24no—
60C2×C4×C191—24no—
61C2×C4×C191—24no—
62C2×C4×C191—48no—
63C2×C4×C191—44no—
64C2×C4×C191C2×D412yes—
65C2×C4×C191C2×D411yescharacteristic
66C2×C4×C191#4114yes—
67C2×C4×C191C2411yescharacteristic
68C22×C191—22no—
69C22×C191—22no—
70C22×C191—28no—
71C22×C191#12312yes—
72C22×C191C2×Q1614yes—
73C22×C191C2×Q1612yes—
74C22×C191AGL(1,ℤ/8)12yes—
75C22×C191C22×D411yescharacteristic
76C4×C191—22no—
77C4×C191—44no—
78C4×C191#12312yes—
79C4×C191C2×#4112yes—
80Dic191—42no—
81Dic191—42no—
82C2×C191—24no—
83C2×C191#48414yescomplemented
84C2×C191C2×#12311yescharacteristic
85C2×C191C22×Q1611yescharacteristic
86C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
87C191C2×#48411yescharacteristic, complemented, Sylow
88C2×#484—1911nomaximal, Sylow
89C2×C22⋊Q8—1911no—
90C22×C4⋊C4D19111yescharacteristic
91#484—1918no—
92C2×#107—1912no—
93C2×#103—1912no—
94C2×C22⋊C8—1911no—
95C22×Q8—1911no—
96C23×C4—21no—
97C23×C4C2×D19111yescharacteristic
98C22×C8—1912no—
99C22⋊Q8—3822no—
100C22⋊Q8—1914no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—3821no—
104C2×C4⋊C4—1911no—
105C2×C4⋊C4C2×D19112yes—
106C2×C4⋊C4C2×D19114yes—
107C2×C22⋊C4—3821no—
108#107—1918no—
109#103—1918no—
110C22⋊C8—1914no—
111C24C22⋊D19111yescharacteristic
112C2×Q8—3821no—
113C2×Q8—3824no—
114C2×Q8—1912no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—3821no—
120C22×C4—3821no—
121C22×C4C22⋊D19112yes—
122C22×C4C22⋊D19111yescharacteristic
123C22×C4C22×D19112yes—
124C22×C4C22×D19111yescharacteristic
125C2×C8—3824no—
126C2×C8—1914no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—3824no—
130C4⋊C4—3821no—
131C4⋊C4—1912no—
132C4⋊C4C22×D19114yes—
133C22⋊C4—3824no—
134C23—22no—
135C23—22no—
136C23#1569027714yes—
137C23C2×C22⋊D19112yes—
138C23C2×C22⋊D19111yescharacteristic
139Q8—3824no—
140Q8—7642no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—48no—
146C2×C4—44no—
147C2×C4—3822no—
148C2×C4—3822no—
149C2×C4—7642no—
150C2×C4—7642no—
151C2×C4C2×C22⋊D19112yes—
152C2×C4C2×C22⋊D19111yescharacteristic
153C2×C4#1569028714yes—
154C2×C4C23×D19111yescharacteristic
155C8—3824no—
156C22—22no—
157C22—28no—
158C22—22no—
159C22#2025930512yes—
160C22C2×#1569027714yes—
161C22C2×#1569027712yes—
162C22#2025934812yes—
163C22C22×C22⋊D19111yescharacteristic
164C4—22no—
165C4—44no—
166C4—7642no—
167C4—7642no—
168C4#2025930512yes—
169C4C2×#1569028712yes—
170C2—24no—
171C2#2713299814yesdirect product factor
172C2C2×#2025930511yescharacteristic
173C2C22×#1569027711yescharacteristic
174C2C2×#2025934811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×C4⋊C4×C191C2
socleC23×C191C2×D4
prime radicalC24×C191D4
hypercenterC22×C4⋊C4D191
hypocenterC191C2×#484
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #27132998C2direct (N × H)
3. C2×#20259255C2semidirect
4. C2×#20259245C2semidirect
14. #20259255C22semidirect
15. #20259245C22semidirect
83. C2×C191#484semidirect
87. C191C2×#484semidirect
171. C2#27132998direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202593012
#271329982
C2×#202592552
C2×#202592452
C2×C22⋊(C191⋊C8)2
C22×C4⋊C4×C1912
C2×#484191