Finite Group Databasebeta

← #37571651

Subgroups — #37571651

Order 24448.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#27133374C211yesmaximal, characteristic
2#27132955C211yesmaximal, characteristic, complemented
3#27132950C211yesmaximal, characteristic
4#27132412C211yesmaximal, characteristic, complemented
5#27132395C211yesmaximal, characteristic, complemented
6#27132392C211yesmaximal, characteristic
7#494×C191C211yesmaximal, characteristic, complemented
8#20259313—21no—
9#20259302—21no—
10#20259298—22no—
11C4×Dic382C2211yescharacteristic
12C4×D764C2211yescharacteristic
13#20259284—21no—
14C4⋊C4×D191—21no—
15#20259255—21no—
16#20259253—21no—
17C4⋊(C191⋊C8)C2211yescharacteristic
18C4⋊(C191⋊C8)C2211yescharacteristic
19#20259245—21no—
20#20259242—21no—
21C2×Q16×C191—22no—
22C4⋊Q8×C191C2211yescharacteristic
23C4⋊Q8×C191C2211yescharacteristic
24C4×C8×C191C2211yescharacteristic
25C2×Dic382D411yescharacteristic
26C2×D764D411yescharacteristic
27C2×C4×D191—22no—
28C2×C4×D191—21no—
29C2×C191⋊C8—21no—
30C2×C191⋊C8—21no—
31C4⋊Dic191—22no—
32C4⋊Dic191D411yescharacteristic
33C4⋊Dic191D411yescharacteristic
34Dic191⋊C4—22no—
35Dic191⋊C4—21no—
36(C2×D191)⋊C4—22no—
37(C2×D191)⋊C4—21no—
38C4×Dic191—21no—
39C2×Q8×C191—21no—
40C2×Q8×C191D412yes—
41Q16×C191—44no—
42C2×C8×C191—21no—
43C4⋊C4×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191—21no—
46C4⋊C4×C191—21no—
47C42×C191C2311yescharacteristic
48C22×D191—21no—
49Dic382—21no—
50Dic382—21no—
51D764—21no—
52D764Q1612yescomplemented
53C2×Dic191—21no—
54C2×Dic191—21no—
55C2×Dic191—21no—
56C4×D191—41no—
57C4×D191—41no—
58C4×D191—82no—
59C191⋊C8—41no—
60C191⋊C8—41no—
61Q8×C191—42no—
62Q8×C191—42no—
63C2×C4×C191—22no—
64C2×C4×C191—41no—
65C2×C4×C191C2×D411yescharacteristic
66C2×C4×C191C2×D411yescharacteristic
67C2×C4×C191C2×D411yescharacteristic
68C8×C191—22no—
69C22×C191#12011yescharacteristic
70C2×D191—22no—
71C2×D191—41no—
72C2×D191—41no—
73C4×C191—21no—
74C4×C191—42no—
75C4×C191—81no—
76C4×C191C2×Q1611yescharacteristic
77C4×C191#13511yescharacteristic
78C4×C191#13711yescharacteristic
79C4×C191#13711yescharacteristic
80Dic191—41no—
81Dic191—41no—
82Dic191—41no—
83C2×C191#45111yescharacteristic
84C2×C191#45211yescharacteristic
85C2×C191#45311yescharacteristic
86D191—42no—
87C191#163211yescharacteristic, complemented, Sylow
88#1632—1911nomaximal, Sylow
89#554—1911no—
90#494D19111yescharacteristic, complemented
91#477—1911no—
92#476—1911no—
93#333—1911no—
94Q8⋊C8—1911no—
95D4⋊C8—1911no—
96C2×Q16—22no—
97C4⋊Q8C2×D19111yescharacteristic
98C4⋊Q8C2×D19111yescharacteristic
99#125—3821no—
100C22⋊Q8—3821no—
101C22⋊Q8—3822no—
102C4×Q8—1911no—
103C4×D4—3821no—
104C4×D4—1911no—
105C2×C4⋊C4—3821no—
106#107—3821no—
107#106—3821no—
108C4⋊C8—1911no—
109C4⋊C8—1911no—
110#103—3821no—
111#102—3821no—
112C4×C8C2×D19111yescharacteristic
113C2×Q8—21no—
114C2×Q8—1911no—
115C2×Q8C22⋊D19112yes—
116C2×D4—1911no—
117C22×C4—3822no—
118C22×C4—3821no—
119Q16—44no—
120C2×C8—21no—
121C2×C8—3821no—
122C2×C8—3821no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—3822no—
128C4⋊C4—3822no—
129C4⋊C4—3821no—
130C4⋊C4—1911no—
131C4⋊C4—1911no—
132C22⋊C4—3822no—
133C22⋊C4—3821no—
134C42—3821no—
135C42C22×D19111yescharacteristic
136C23—3821no—
137Q8—42no—
138Q8—42no—
139Q8—3821no—
140Q8—3821no—
141D4—3821no—
142D4—1912no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—3821no—
146C2×C4—3821no—
147C2×C4—3821no—
148C2×C4—15282no—
149C2×C4—7641no—
150C2×C4—7641no—
151C2×C4D4×D19111yescharacteristic
152C2×C4D4×D19111yescharacteristic
153C2×C4C2×C22⋊D19111yescharacteristic
154C8—22no—
155C8—7641no—
156C8—7641no—
157C22—3822no—
158C22—7641no—
159C22—7641no—
160C22#2025929011yescharacteristic
161C4—21no—
162C4—42no—
163C4—81no—
164C4—7641no—
165C4—7641no—
166C4—7641no—
167C4Q16×D19111yescharacteristic
168C4#2025934511yescharacteristic
169C4#2025935411yescharacteristic
170C4#2025935411yescharacteristic
171C2—7642no—
172C2#2713281711yescharacteristic
173C2#2713282411yescharacteristic
174C2#2713283011yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#494×C191C2
socleC22×C191#120
prime radicalC2×D764D4
hypercenter#494D191
hypocenterC191#1632
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #27132955C2semidirect
4. #27132412C2semidirect
5. #27132395C2semidirect
7. #494×C191C2semidirect
52. D764Q16semidirect
87. C191#1632semidirect
90. #494D191semidirect

Maximal subgroups

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

subgroupindex
#271333742
#271329552
#271329502
#271324122
#271323952
#271323922
#494×C1912
#1632191