Finite Group Databasebeta

← #37571719

Subgroups — #37571719

Order 24448.

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

normal subgroups only

This group has 250 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#27133335C211yesmaximal, characteristic
2#27133011C211yesmaximal, characteristic, complemented
3#27132865C211yesmaximal, characteristic, complemented
4#27132406C211yesmaximal, characteristic, complemented
5Q8⋊(C191⋊C8)C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic, complemented
7#475×C191C211yesmaximal, characteristic, complemented
8C2×#15690288—21no—
9C2×#15690274—21no—
10#20259319—21no—
11#20259319C2211yescharacteristic
12#20259297—22no—
13#20259297—21no—
14C4×D764—21no—
15C4×D764C2211yescharacteristic
16C4⋊(C191⋊C8)C2211yescharacteristic
17#20259246—21no—
18#20259242—22no—
19C4×C191⋊C8C2211yescharacteristic
20C4⋊Q8×C191C2211yescharacteristic
21C4×Q8×C191C2211yescharacteristic
22#106×C191—21no—
23C4⋊C8×C191C2211yescharacteristic
24#103×C191—21no—
25#15690288—42no—
26#15690288—42no—
27C2×D764—22no—
28C2×D764—22no—
29C2×D764—21no—
30C2×D764—21no—
31C2×D764D411yescharacteristic
32C2×D764D412yes—
33C2×C4×D191—22no—
34C2×C4×D191—21no—
35#15690274—42no—
36C2×C191⋊C8—21no—
37C2×C191⋊C8—21no—
38C4⋊Dic191D411yescharacteristic
39(C2×D191)⋊C4—22no—
40(C2×D191)⋊C4—21no—
41C2×Q8×C191—21no—
42C2×Q8×C191D411yescharacteristic
43C2×C8×C191—21no—
44C4⋊C4×C191—21no—
45C4⋊C4×C191—21no—
46C4⋊C4×C191—21no—
47C4⋊C4×C191D411yescharacteristic
48C42×C191—21no—
49C42×C191C2311yescharacteristic
50C22×D191—21no—
51C22×D191—22no—
52D764—21no—
53D764—21no—
54D764—42no—
55D764—44no—
56D764—41no—
57D764—41no—
58D764—41no—
59D764—41no—
60D764—82no—
61C2×Dic191—21no—
62C4×D191—44no—
63C4×D191—41no—
64C4×D191—41no—
65C191⋊C8—41no—
66C191⋊C8—41no—
67Q8×C191—21no—
68Q8×C191—41no—
69Q8×C191—41no—
70Q8×C191SD1612yescomplemented
71C2×C4×C191—21no—
72C2×C4×C191—21no—
73C2×C4×C191—41no—
74C2×C4×C191C2×D411yescharacteristic
75C2×C4×C191C2×D411yescharacteristic
76C2×C4×C191C2×D411yescharacteristic
77C8×C191—41no—
78C22×C191#12011yescharacteristic
79C2×D191—41no—
80C2×D191—42no—
81C2×D191—42no—
82C2×D191—41no—
83C2×D191—42no—
84C2×D191—41no—
85C4×C191—22no—
86C4×C191—21no—
87C4×C191—41no—
88C4×C191—81no—
89C4×C191C2×SD1611yescharacteristic
90C4×C191#13511yescharacteristic
91C4×C191AGL(1,ℤ/8)11yescharacteristic
92C4×C191AGL(1,ℤ/8)11yescharacteristic
93Dic191—41no—
94C2×C191#44911yescharacteristic
95C2×C191#45011yescharacteristic
96C2×C191#45511yescharacteristic
97D191—81no—
98D191—82no—
99C191#164211yescharacteristic, complemented, Sylow
100#1642—1911nomaximal, Sylow
101#550—1911no—
102#486—1911no—
103#475D19111yescharacteristic, complemented
104#461—1911no—
105#331—1911no—
106Q8⋊C8—1911no—
107D4⋊C8—1911no—
108C2×#41—3821no—
109C2×SD16—3821no—
110C4⋊Q8C2×D19111yescharacteristic
111#127—3821no—
112#127—1911no—
113#121—3822no—
114#121—3821no—
115C4×Q8C2×D19111yescharacteristic
116C4×D4—3821no—
117C4×D4—1911no—
118#106—21no—
119C4⋊C8—1911no—
120C4⋊C8C2×D19111yescharacteristic
121#103—21no—
122#103—3821no—
123#102—3822no—
124C4×C8—1911no—
125#41—7642no—
126#41—7642no—
127C2×Q8—21no—
128C2×Q8C22⋊D19111yescharacteristic
129C2×D4—3822no—
130C2×D4—3822no—
131C2×D4—3821no—
132C2×D4—3821no—
133C2×D4—1911no—
134C2×D4—1912no—
135C22×C4—3822no—
136C22×C4—3821no—
137SD16—7642no—
138C2×C8—21no—
139C2×C8—3821no—
140C2×C8—3821no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1911no—
145C4⋊C4C22⋊D19111yescharacteristic
146C22⋊C4—3822no—
147C22⋊C4—3821no—
148C42—21no—
149C42C22×D19111yescharacteristic
150C23—3821no—
151C23—3822no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#1569027412yescomplemented
156D4—3821no—
157D4—3821no—
158D4—15282no—
159D4—7642no—
160D4—7644no—
161D4—7641no—
162D4—7641no—
163D4—7641no—
164D4—7641no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—3821no—
169C2×C4—7644no—
170C2×C4—7641no—
171C2×C4—7641no—
172C2×C4D4×D19111yescharacteristic
173C2×C4D4×D19111yescharacteristic
174C2×C4C2×C22⋊D19111yescharacteristic
175C8—41no—
176C8—7641no—
177C8—7641no—
178C22—7641no—
179C22—7642no—
180C22—7642no—
181C22—7641no—
182C22—7641no—
183C22—7642no—
184C22#2025929011yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—7641no—
190C4C2×#1569027411yescharacteristic
191C4#2025934311yescharacteristic
192C4#2025934711yescharacteristic
193C4#2025934911yescharacteristic
194C2—15281no—
195C2—15282no—
196C2#2713280711yescharacteristic
197C2#2713281311yescharacteristic
198C2#2713283911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#475×C191C2
socleC22×C191#120
prime radical#27133335C2
hypercenter#475D191
hypocenterC191#1642
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #27133011C2semidirect
3. #27132865C2semidirect
4. #27132406C2semidirect
5. Q8⋊(C191⋊C8)C2semidirect
6. #27132390C2semidirect
7. #475×C191C2semidirect
70. Q8×C191SD16semidirect
99. C191#1642semidirect
103. #475D191semidirect
155. Q8#15690274semidirect

Maximal subgroups

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

subgroupindex
#271333352
#271330112
#271328652
#271324062
Q8⋊(C191⋊C8)2
#271323902
#475×C1912
#1642191