Finite Group Databasebeta

← #37571834

Subgroups — #37571834

Order 24448.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#27133407C211yesmaximal, characteristic
2#27133069C211yesmaximal, characteristic, complemented
3#27133025C211yesmaximal, characteristic, complemented
4#27132404C211yesmaximal, characteristic, complemented
5#27132403C211yesmaximal, characteristic, complemented
6#27132402C211yesmaximal, characteristic, complemented
7#497×C191C211yesmaximal, characteristic, complemented
8#20259314C2211yescharacteristic
9#20259310C2211yescharacteristic
10#20259306—21no—
11#20259304—21no—
12#20259299—21no—
13#20259298—21no—
14#20259295—21no—
15#20259294—21no—
16#20259255—21no—
17#20259253—21no—
18#20259245—21no—
19#20259242—21no—
20#20259233C2211yescharacteristic
21#20259233C2211yescharacteristic
22C2×Q16×C191—21no—
23C2×SD16×C191—21no—
24C4⋊Q8×C191C2211yescharacteristic
25#124×C191C2211yescharacteristic
26#97×C191C2211yescharacteristic
27C2×Dic382D411yescharacteristic
28C2×D764D411yescharacteristic
29C2×C22⋊D191—21no—
30C2×C22⋊D191—21no—
31C22×Dic191—21no—
32C2×C4×D191—21no—
33C2×C191⋊C8—21no—
34C2×C191⋊C8—21no—
35C4⋊Dic191—21no—
36C4⋊Dic191D411yescharacteristic
37C4⋊Dic191D411yescharacteristic
38Dic191⋊C4—21no—
39Dic191⋊C4—21no—
40Dic191⋊C4—21no—
41C22⋊Dic191—21no—
42(C2×D191)⋊C4—21no—
43(C2×D191)⋊C4—21no—
44(C2×D191)⋊C4—21no—
45C2×Q8×C191—21no—
46C2×Q8×C191D411yescharacteristic
47C2×D4×C191D411yescharacteristic
48Q16×C191—42no—
49SD16×C191—42no—
50C2×C8×C191—21no—
51C4⋊C4×C191—21no—
52C4⋊C4×C191—21no—
53C22⋊C4×C191—21no—
54C22⋊C4×C191—21no—
55C42×C191C2311yescharacteristic
56C22×D191—21no—
57Dic382—41no—
58D764—41no—
59C22⋊D191—81no—
60C22⋊D191—81no—
61C2×Dic191—21no—
62C2×Dic191—21no—
63C2×Dic191—21no—
64C2×Dic191—81no—
65C4×D191—81no—
66C191⋊C8—22no—
67C191⋊C8—41no—
68C23×C191—21no—
69Q8×C191—41no—
70Q8×C191—41no—
71Q8×C191—41no—
72D4×C191—41no—
73C2×C4×C191—21no—
74C2×C4×C191—41no—
75C2×C4×C191C2×D411yescharacteristic
76C2×C4×C191C2×D411yescharacteristic
77C2×C4×C191C2×D411yescharacteristic
78C8×C191—22no—
79C22×C191—41no—
80C22×C191—41no—
81C22×C191—41no—
82C22×C191#12011yescharacteristic
83C2×D191—41no—
84C2×D191—41no—
85C2×D191—41no—
86C4×C191—21no—
87C4×C191—21no—
88C4×C191—21no—
89C4×C191—41no—
90C4×C191—81no—
91Dic191—41no—
92Dic191—41no—
93Dic191—41no—
94C2×C191—81no—
95C2×C191#45311yescharacteristic
96C2×C191#45511yescharacteristic
97C2×C191#45611yescharacteristic
98D191—81no—
99C191#166111yescharacteristic, complemented, Sylow
100#1661—1911nomaximal, Sylow
101#562—1911no—
102#501—1911no—
103#497D19111yescharacteristic, complemented
104#489—1911no—
105#330—1911no—
106#329—1911no—
107#329—1911no—
108C2×Q16—21no—
109C2×SD16—21no—
110C4⋊Q8C2×D19111yescharacteristic
111#125—1911no—
112#124—1911no—
113#124C2×D19111yescharacteristic
114#123—3821no—
115#123—3821no—
116C22⋊Q8—3821no—
117C22⋊Q8—3821no—
118#121—3821no—
119#121—3821no—
120#107—3821no—
121#106—3821no—
122#103—3821no—
123#102—3821no—
124#97—1911no—
125#97—1911no—
126#97C2×D19111yescharacteristic
127C2×Q8—21no—
128C2×Q8—1911no—
129C2×Q8C22⋊D19111yescharacteristic
130C2×D4—3821no—
131C2×D4—3821no—
132C2×D4—1911no—
133C2×D4C22⋊D19111yescharacteristic
134C22×C4—3821no—
135C22×C4—3821no—
136Q16—42no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—3821no—
140C2×C8—3821no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—3821no—
144C4⋊C4—3821no—
145C4⋊C4—3821no—
146C4⋊C4—3821no—
147C4⋊C4—1911no—
148C4⋊C4—1911no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—3821no—
152C22⋊C4—3821no—
153C22⋊C4—3821no—
154C22⋊C4—3821no—
155C42C22×D19111yescharacteristic
156C23—21no—
157C23—3821no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—7641no—
162D4—41no—
163D4—15281no—
164D4—15281no—
165D4—7641no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—3821no—
169C2×C4—3821no—
170C2×C4—3821no—
171C2×C4—15281no—
172C2×C4—15281no—
173C2×C4D4×D19111yescharacteristic
174C2×C4D4×D19111yescharacteristic
175C2×C4C2×C22⋊D19111yescharacteristic
176C8—22no—
177C8—3822no—
178C8—7641no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—7641no—
183C22—7641no—
184C22—7641no—
185C22#2025929011yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—81no—
191C4—7641no—
192C4—7641no—
193C4—7641no—
194C2—81no—
195C2—15281no—
196C2#2713283011yescharacteristic
197C2#2713283911yescharacteristic
198C2#2713284411yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#497×C191C2
socleC22×C191#120
prime radical#27133407C2
hypercenter#497D191
hypocenterC191#1661
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #27133069C2semidirect
3. #27133025C2semidirect
4. #27132404C2semidirect
5. #27132403C2semidirect
6. #27132402C2semidirect
7. #497×C191C2semidirect
99. C191#1661semidirect
103. #497D191semidirect

Maximal subgroups

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

subgroupindex
#271334072
#271330692
#271330252
#271324042
#271324032
#271324022
#497×C1912
#1661191