Finite Group Databasebeta

← #37574492

Subgroups — #37574492

Order 24448.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#27133465C211yesmaximal, characteristic
2#27133103C211yesmaximal, characteristic, complemented
3C2×#20259267C211yesmaximal, characteristic, complemented
4#27132939C211yesmaximal, characteristic, complemented
5#27132475C211yesmaximal, characteristic, complemented
6#27132446C211yesmaximal, characteristic, complemented
7#359×C191C211yesmaximal, characteristic, complemented
8#20259368—21no—
9#20259355—21no—
10#20259349—22no—
11#20259347—21no—
12#20259345—21no—
13C2×Dic764C2211yescharacteristic, complemented
14C2×D1528C2211yescharacteristic
15D8×D191—21no—
16#20259328C2211yescharacteristic
17#20259268—21no—
18#20259267—22no—
19#20259266—21no—
20#20259262C2211yescharacteristic, complemented
21#20259240—21no—
22#135×C191C2211yescharacteristic
23C2×C16×C191C2211yescharacteristic, complemented
24#108×C191C2211yescharacteristic, complemented
25#15690288—22no—
26#15690287—41no—
27#15690287D411yescharacteristic
28C2×Dic382—21no—
29C2×D764—21no—
30C2×D764—41no—
31D4×D191—22no—
32D4×D191—42no—
33Dic764—21no—
34Dic764—21no—
35Dic764—41no—
36#15690276—41no—
37#15690274—22no—
38D1528—21no—
39D1528D412yes—
40#15690272—22no—
41#15690271D411yescharacteristic
42#15690270—21no—
43C8×D191—21no—
44C191⋊C16—21no—
45#41×C191—21no—
46Q16×C191D411yescharacteristic
47SD16×C191—21no—
48D8×C191D411yescharacteristic
49M16×C191—21no—
50C2×C8×C191C2311yescharacteristic
51C16×C191—21no—
52C22×D191—42no—
53C22×D191—41no—
54Dic382—41no—
55Dic382—41no—
56Dic382D811yescharacteristic
57D764—22no—
58D764—42no—
59D764—41no—
60D764D811yescharacteristic
61C22⋊D191—21no—
62C22⋊D191—42no—
63C2×Dic191—41no—
64C4×D191—21no—
65C4×D191—42no—
66C191⋊C8—21no—
67Q8×C191—21no—
68D4×C191—21no—
69D4×C191—41no—
70C2×C4×C191—41no—
71C2×C4×C191C2×D411yescharacteristic
72C8×C191—41no—
73C8×C191C2×D411yescharacteristic
74C8×C191C2×D411yescharacteristic
75C22×C191—41no—
76C22×C191AGL(1,ℤ/8)11yescharacteristic
77C2×D191—21no—
78C2×D191—42no—
79C2×D191—162no—
80C2×D191—161no—
81C4×C191—41no—
82C4×C191#12011yescharacteristic
83C4×C191C2×D811yescharacteristic
84Dic191—21no—
85Dic191—81no—
86C2×C191—21no—
87C2×C191—81no—
88C2×C191#44711yescharacteristic
89D191—41no—
90D191—82no—
91C191#219411yescharacteristic, complemented, Sylow
92#2194—1911nomaximal, Sylow
93#576—1911no—
94#510—1911no—
95C2×SD32—1911no—
96#473—1911no—
97#361—1911no—
98#359D19111yescharacteristic, complemented
99#350—1911no—
100H(5,𝔽2)—3821no—
101#137—3821no—
102AGL(1,ℤ/8)—3822no—
103AGL(1,ℤ/8)—3821no—
104#135—3821no—
105#135C2×D19111yescharacteristic
106C2×Q16—1911no—
107C2×D8—3821no—
108C2×D8—1911no—
109#131—1911no—
110Q32—3821no—
111SD32—3822no—
112SD32—3821no—
113M32—1911no—
114C2×C16C2×D19111yescharacteristic, complemented
115#108C2×D19111yescharacteristic, complemented
116#101—3821no—
117#41—21no—
118#41—3822no—
119#41—1911no—
120#41—7641no—
121C2×Q8—3821no—
122C2×D4—3822no—
123C2×D4—3821no—
124C2×D4—7641no—
125C2×D4—7642no—
126Q16—3821no—
127Q16—3821no—
128Q16—7641no—
129Q16D76411yescharacteristic
130SD16—21no—
131SD16—3822no—
132SD16—7641no—
133D8—3822no—
134D8—3821no—
135D8—1912no—
136D8D76411yescharacteristic
137M16—21no—
138M16—3821no—
139M16—1911no—
140C2×C8—3821no—
141C2×C8C22×D19111yescharacteristic
142C16—21no—
143C16—3821no—
144C23—7642no—
145C23—7641no—
146Q8—21no—
147Q8—1911no—
148Q8—7641no—
149Q8—7641no—
150D4—21no—
151D4—41no—
152D4—3821no—
153D4—3822no—
154D4—1911no—
155D4—7642no—
156D4—7642no—
157D4—7641no—
158C2×C4—41no—
159C2×C4—3821no—
160C2×C4—7642no—
161C2×C4—7641no—
162C2×C4D4×D19111yescharacteristic
163C8—41no—
164C8—3821no—
165C8D4×D19111yescharacteristic
166C8C2×D76411yescharacteristic
167C22—41no—
168C22—3821no—
169C22—30561no—
170C22—30562no—
171C22—7642no—
172C22#2025934911yescharacteristic
173C4—41no—
174C4—3821no—
175C4—15281no—
176C4#2025929211yescharacteristic
177C4D8×D19111yescharacteristic
178C2—21no—
179C2—81no—
180C2—15282no—
181C2—7641no—
182C2#2713279811yescharacteristic
subgroupG/H
centerC2#27132798
commutator, cosocleC2×C8×C191C23
FrattiniC2×C8C22×D191
Fitting#359×C191C2
socleC2×C191#447
prime radical#27133465C2
hypercenter#359D191
hypocenterC191#2194
LCSG ▷ C2×C8×C191 ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
2. #27133103C2semidirect
3. C2×#20259267C2semidirect
4. #27132939C2semidirect
5. #27132475C2semidirect
6. #27132446C2semidirect
7. #359×C191C2semidirect
13. C2×Dic764C22semidirect
20. #20259262C22semidirect
23. C2×C16×C191C22semidirect
24. #108×C191C22semidirect
91. C191#2194semidirect
98. #359D191semidirect
114. C2×C16C2×D191semidirect
115. #108C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334652
#271331032
C2×#202592672
#271329392
#271324752
#271324462
#359×C1912
#2194191