Finite Group Databasebeta

← #37571803

Subgroups — #37571803

Order 24448.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#27133317C211yesmaximal, characteristic
2#27133059C211yesmaximal, characteristic, complemented
3#27132865C211yesmaximal, characteristic, complemented
4#27132410C211yesmaximal, characteristic, complemented
5D4⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic, complemented
7#474×C191C211yesmaximal, characteristic, complemented
8C2×#15690286—21no—
9C2×#15690274—21no—
10#20259322C2211yescharacteristic
11#20259306—22no—
12#20259302—21no—
13#20259296—21no—
14C4×D764C2211yescharacteristic
15C2×C4⋊Dic191—21no—
16#20259253—22no—
17C4⋊(C191⋊C8)C2211yescharacteristic
18#20259246—21no—
19C4×C191⋊C8C2211yescharacteristic
20C4⋊Q8×C191C2211yescharacteristic
21C4×D4×C191C2211yescharacteristic
22#107×C191—21no—
23C4⋊C8×C191C2211yescharacteristic
24#102×C191—21no—
25#15690286—42no—
26#15690286—42no—
27C2×Dic382—21no—
28C2×D764D411yescharacteristic
29C2×C22⋊D191—22no—
30C22×Dic191—22no—
31C2×C4×D191—21no—
32#15690274—42no—
33C2×C191⋊C8—21no—
34C2×C191⋊C8—21no—
35C4⋊Dic191—22no—
36C4⋊Dic191—21no—
37C4⋊Dic191D411yescharacteristic
38C4⋊Dic191D412yes—
39(C2×D191)⋊C4—22no—
40(C2×D191)⋊C4—21no—
41C2×Q8×C191—21no—
42C2×D4×C191D411yescharacteristic
43C22×C4×C191—21no—
44C2×C8×C191—21no—
45C4⋊C4×C191—21no—
46C4⋊C4×C191—21no—
47C4⋊C4×C191D411yescharacteristic
48C22⋊C4×C191—21no—
49C42×C191C2311yescharacteristic
50C22×D191—21no—
51Dic382—41no—
52Dic382—41no—
53D764—21no—
54D764—21no—
55C22⋊D191—44no—
56C2×Dic191—21no—
57C2×Dic191—22no—
58C2×Dic191—44no—
59C4×D191—41no—
60C4×D191—41no—
61C191⋊C8—22no—
62C191⋊C8—41no—
63C23×C191—21no—
64Q8×C191—41no—
65Q8×C191—41no—
66D4×C191—21no—
67D4×C191SD1612yescomplemented
68C2×C4×C191—21no—
69C2×C4×C191—41no—
70C2×C4×C191—41no—
71C2×C4×C191—41no—
72C2×C4×C191C2×D411yescharacteristic
73C2×C4×C191C2×D411yescharacteristic
74C2×C4×C191C2×D411yescharacteristic
75C8×C191—41no—
76C22×C191—22no—
77C22×C191—41no—
78C22×C191—41no—
79C22×C191#12011yescharacteristic
80C2×D191—41no—
81C2×D191—41no—
82C2×D191—41no—
83C4×C191—21no—
84C4×C191—41no—
85C4×C191—81no—
86C4×C191C2×SD1611yescharacteristic
87C4×C191#13511yescharacteristic
88C4×C191AGL(1,ℤ/8)11yescharacteristic
89C4×C191AGL(1,ℤ/8)11yescharacteristic
90Dic191—41no—
91Dic191—42no—
92C2×C191—42no—
93C2×C191#44911yescharacteristic
94C2×C191#45011yescharacteristic
95C2×C191#45611yescharacteristic
96D191—81no—
97C191#165411yescharacteristic, complemented, Sylow
98#1654—1911nomaximal, Sylow
99#547—1911no—
100#498—1911no—
101#474D19111yescharacteristic, complemented
102#461—1911no—
103#332—1911no—
104D4⋊C8—1911no—
105D4⋊C8—1911no—
106C2×#41—3821no—
107C2×SD16—3821no—
108C4⋊Q8—1911no—
109C4⋊Q8C2×D19111yescharacteristic
110#123—3822no—
111C22⋊Q8—3821no—
112#121—3821no—
113C4×D4—1911no—
114C4×D4C2×D19111yescharacteristic
115C2×C4⋊C4—3821no—
116#107—21no—
117#106—3822no—
118C4⋊C8—1911no—
119C4⋊C8C2×D19111yescharacteristic
120#103—3821no—
121#102—21no—
122C4×C8—1911no—
123#41—7642no—
124#41—7642no—
125C2×Q8—21no—
126C2×Q8—3821no—
127C2×D4—3822no—
128C2×D4—1911no—
129C2×D4C22⋊D19111yescharacteristic
130C22×C4—21no—
131C22×C4—3822no—
132C22×C4—3821no—
133SD16—7642no—
134C2×C8—21no—
135C2×C8—3821no—
136C2×C8—3821no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—3822no—
140C4⋊C4—3821no—
141C4⋊C4—1911no—
142C4⋊C4—1912no—
143C4⋊C4C22⋊D19111yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—3822no—
146C22⋊C4—3821no—
147C42C22×D19111yescharacteristic
148C23—21no—
149C23—3821no—
150Q8—41no—
151Q8—41no—
152Q8—7641no—
153Q8—7641no—
154D4—21no—
155D4—3821no—
156D4—3821no—
157D4—7644no—
158D4#1569027412yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—3821no—
164C2×C4—3822no—
165C2×C4—7644no—
166C2×C4—7641no—
167C2×C4—7641no—
168C2×C4D4×D19111yescharacteristic
169C2×C4D4×D19111yescharacteristic
170C2×C4C2×C22⋊D19111yescharacteristic
171C8—41no—
172C8—3822no—
173C8—7641no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—7641no—
178C22—7641no—
179C22—7641no—
180C22#2025929011yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—7641no—
185C4—7642no—
186C4C2×#1569027411yescharacteristic
187C4#2025934111yescharacteristic
188C4#2025934611yescharacteristic
189C4#2025934711yescharacteristic
190C2—42no—
191C2—15281no—
192C2#2713280511yescharacteristic
193C2#2713281311yescharacteristic
194C2#2713284411yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#474×C191C2
socleC22×C191#120
prime radical#27133317C2
hypercenter#474D191
hypocenterC191#1654
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #27133059C2semidirect
3. #27132865C2semidirect
4. #27132410C2semidirect
5. D4⋊(C191⋊C8)C2semidirect
6. #27132390C2semidirect
7. #474×C191C2semidirect
67. D4×C191SD16semidirect
97. C191#1654semidirect
101. #474D191semidirect
158. D4#15690274semidirect

Maximal subgroups

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

subgroupindex
#271333172
#271330592
#271328652
#271324102
D4⋊(C191⋊C8)2
#271323902
#474×C1912
#1654191