Finite Group Databasebeta

← #37571653

Subgroups — #37571653

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#27133373C211yesmaximal, characteristic
2#27133042C211yesmaximal, characteristic, complemented
3#27132950C211yesmaximal, characteristic
4#27132411C211yesmaximal, characteristic, complemented
5Q8⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic
7#477×C191C211yesmaximal, characteristic, complemented
8C2×#15690277—22no—
9#20259322C2211yescharacteristic
10#20259314—21no—
11#20259300—22no—
12#20259298—21no—
13C4×D764—21no—
14C4×D764C2211yescharacteristic
15C4⋊C4×D191—21no—
16#20259253—21no—
17C4⋊(C191⋊C8)C2211yescharacteristic
18#20259242—21no—
19C4×C191⋊C8C2211yescharacteristic
20C4⋊Q8×C191C2211yescharacteristic
21C4×Q8×C191C2211yescharacteristic
22#107×C191—21no—
23C4⋊C8×C191C2211yescharacteristic
24#103×C191—21no—
25C2×Dic382D412yes—
26C2×D764D411yescharacteristic
27C2×C4×D191—22no—
28C2×C4×D191—21no—
29#15690277—44no—
30C2×C191⋊C8—21no—
31C2×C191⋊C8—21no—
32C4⋊Dic191—21no—
33C4⋊Dic191—21no—
34C4⋊Dic191D411yescharacteristic
35Dic191⋊C4—22no—
36Dic191⋊C4—22no—
37(C2×D191)⋊C4—22no—
38(C2×D191)⋊C4—21no—
39C2×Q8×C191—21no—
40C2×Q8×C191D411yescharacteristic
41C2×C8×C191—21no—
42C4⋊C4×C191—21no—
43C4⋊C4×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191D411yescharacteristic
46C42×C191—21no—
47C42×C191C2311yescharacteristic
48C22×D191—21no—
49Dic382—42no—
50D764—21no—
51D764Q1612yescomplemented
52C2×Dic191—21no—
53C2×Dic191—22no—
54C4×D191—41no—
55C4×D191—41no—
56C4×D191—82no—
57C191⋊C8—22no—
58C191⋊C8—41no—
59Q8×C191—21no—
60Q8×C191—21no—
61Q8×C191—42no—
62C2×C4×C191—21no—
63C2×C4×C191—21no—
64C2×C4×C191—41no—
65C2×C4×C191C2×D411yescharacteristic
66C2×C4×C191C2×D411yescharacteristic
67C2×C4×C191C2×D411yescharacteristic
68C8×C191—41no—
69C22×C191#12011yescharacteristic
70C2×D191—22no—
71C2×D191—41no—
72C2×D191—41no—
73C4×C191—21no—
74C4×C191—41no—
75C4×C191—41no—
76C4×C191—81no—
77C4×C191C2×Q1611yescharacteristic
78C4×C191#13511yescharacteristic
79C4×C191#13711yescharacteristic
80C4×C191#13711yescharacteristic
81Dic191—41no—
82Dic191—42no—
83C2×C191#45111yescharacteristic
84C2×C191#45211yescharacteristic
85C2×C191#45311yescharacteristic
86D191—42no—
87C191#163211yescharacteristic, complemented, Sylow
88#1632—1911nomaximal, Sylow
89#554—1911no—
90#494—1911no—
91#477D19111yescharacteristic, complemented
92#476—1911no—
93#333—1911no—
94Q8⋊C8—1911no—
95D4⋊C8—1911no—
96C2×Q16—3822no—
97C4⋊Q8—1911no—
98C4⋊Q8C2×D19111yescharacteristic
99#125—3821no—
100C22⋊Q8—3822no—
101C22⋊Q8—3821no—
102C4×Q8C2×D19111yescharacteristic
103C4×D4—3821no—
104C4×D4—1911no—
105C2×C4⋊C4—3821no—
106#107—21no—
107#106—3821no—
108C4⋊C8—1911no—
109C4⋊C8C2×D19111yescharacteristic
110#103—21no—
111#102—3821no—
112C4×C8—1911no—
113C2×Q8—21no—
114C2×Q8—1912no—
115C2×Q8C22⋊D19111yescharacteristic
116C2×D4—1911no—
117C22×C4—3822no—
118C22×C4—3821no—
119Q16—7644no—
120C2×C8—21no—
121C2×C8—3821no—
122C2×C8—3821no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—3822no—
127C4⋊C4—3822no—
128C4⋊C4—3821no—
129C4⋊C4—3821no—
130C4⋊C4—1911no—
131C4⋊C4C22⋊D19111yescharacteristic
132C22⋊C4—3822no—
133C22⋊C4—3821no—
134C42—21no—
135C42C22×D19111yescharacteristic
136C23—3821no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—7642no—
141D4—3821no—
142D4—1912no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—3821no—
147C2×C4—3822no—
148C2×C4—15282no—
149C2×C4—7641no—
150C2×C4—7641no—
151C2×C4D4×D19111yescharacteristic
152C2×C4D4×D19111yescharacteristic
153C2×C4C2×C22⋊D19111yescharacteristic
154C8—41no—
155C8—3822no—
156C8—7641no—
157C22—3822no—
158C22—7641no—
159C22—7641no—
160C22#2025929011yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—7641no—
166C4—7642no—
167C4Q16×D19111yescharacteristic
168C4#2025934211yescharacteristic
169C4#2025935311yescharacteristic
170C4#2025935411yescharacteristic
171C2—7642no—
172C2#2713281711yescharacteristic
173C2#2713282311yescharacteristic
174C2#2713282911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#477×C191C2
socleC22×C191#120
prime radicalC2×D764D4
hypercenter#477D191
hypocenterC191#1632
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #27133042C2semidirect
4. #27132411C2semidirect
5. Q8⋊(C191⋊C8)C2semidirect
7. #477×C191C2semidirect
51. D764Q16semidirect
87. C191#1632semidirect
91. #477D191semidirect

Maximal subgroups

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

subgroupindex
#271333732
#271330422
#271329502
#271324112
Q8⋊(C191⋊C8)2
#271323902
#477×C1912
#1632191