Finite Group Databasebeta

← #38744682

Subgroups — #38744682

Order 25216.

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#27161880C211yesmaximal, characteristic
2#27161520C211yesmaximal, characteristic, complemented
3#27161351C211yesmaximal, characteristic, complemented
4#27160892C211yesmaximal, characteristic, complemented
5#27160887C211yesmaximal, characteristic, complemented
6#27160862C211yesmaximal, characteristic, complemented
7C2×SD32×C197C211yesmaximal, characteristic, complemented
8#20263371—21no—
9#20263357—21no—
10#20263352—21no—
11#20263350—22no—
12#20263348C2211yescharacteristic
13#20263345—21no—
14D8×D197—21no—
15#20263332C2211yescharacteristic
16#20263272—21no—
17#20263269—21no—
18#20263266C2211yescharacteristic, complemented
19#20263262C2211yescharacteristic, complemented
20#20263245—21no—
21C2×Q16×C197C2211yescharacteristic, complemented
22C2×D8×C197C2211yescharacteristic
23SD32×C197—22no—
24C2×C16×C197C2211yescharacteristic, complemented
25#15691358—21no—
26#15691358D411yescharacteristic
27#15691357—22no—
28#15691357—41no—
29C2×C22⋊D197—42no—
30D4×D197—22no—
31D4×D197—41no—
32Dic788D411yescharacteristic
33#15691348—41no—
34#15691347—21no—
35#15691346—22no—
36#15691345—41no—
37D1576D411yescharacteristic
38#15691343—22no—
39#15691342—21no—
40#15691342D411yescharacteristic
41#15691341—21no—
42C8×D197—21no—
43#15691333—21no—
44C2×Q8×C197—21no—
45C2×D4×C197—21no—
46Q16×C197—21no—
47Q16×C197—21no—
48D8×C197—21no—
49D8×C197D412yes—
50C2×C8×C197C2311yescharacteristic
51C16×C197—21no—
52C22×D197—42no—
53Dic394—21no—
54Dic394D811yescharacteristic
55D788—21no—
56D788D811yescharacteristic
57C22⋊D197—21no—
58C22⋊D197—42no—
59C22⋊D197—42no—
60C22⋊D197—41no—
61C2×Dic197—42no—
62C4×D197—21no—
63C4×D197—41no—
64#13096802—21no—
65#13096802—41no—
66C23×C197—41no—
67Q8×C197—41no—
68Q8×C197—41no—
69D4×C197—22no—
70D4×C197—41no—
71C2×C4×C197—41no—
72C2×C4×C197C2×D411yescharacteristic
73C8×C197C2×D411yescharacteristic
74C8×C197C2×D411yescharacteristic
75C22×C197—42no—
76C22×C197—161no—
77C22×C197AGL(1,ℤ/8)11yescharacteristic
78C2×D197—21no—
79C2×D197—41no—
80C2×D197—162no—
81C4×C197—81no—
82C4×C197#12011yescharacteristic
83C4×C197C2×D811yescharacteristic
84Dic197—21no—
85Dic197—41no—
86C2×C197—21no—
87C2×C197—82no—
88C2×C197#44711yescharacteristic
89D197—41no—
90D197—81no—
91C197#219411yescharacteristic, complemented, Sylow
92#2194—1971nomaximal, Sylow
93#576—1971no—
94#510—1971no—
95C2×SD32D19711yescharacteristic, complemented
96#473—1971no—
97#361—1971no—
98#359—1971no—
99#350—1971no—
100H(5,𝔽2)—3941no—
101#137—3941no—
102AGL(1,ℤ/8)—3942no—
103AGL(1,ℤ/8)—3941no—
104#135—3941no—
105#135—1971no—
106C2×Q16C2×D19711yescharacteristic, complemented
107C2×D8—3941no—
108C2×D8C2×D19711yescharacteristic
109#131—1971no—
110Q32—3941no—
111SD32—22no—
112SD32—3941no—
113M32—1971no—
114C2×C16C2×D19711yescharacteristic, complemented
115#108—1971no—
116#101—3941no—
117#41—7881no—
118#41—3942no—
119#41—3941no—
120#41—1971no—
121C2×Q8—21no—
122C2×D4—21no—
123C2×D4—7881no—
124C2×D4—7882no—
125C2×D4—3942no—
126Q16—21no—
127Q16—21no—
128Q16—7881no—
129Q16—1971no—
130SD16—7881no—
131SD16—3942no—
132SD16—3941no—
133D8—21no—
134D8—3942no—
135D8—1971no—
136D8C22⋊D19712yes—
137M16—3941no—
138M16—3941no—
139M16—1971no—
140C2×C8—3941no—
141C2×C8C22×D19711yescharacteristic
142C16—21no—
143C16—3941no—
144C23—41no—
145C23—7882no—
146Q8—41no—
147Q8—41no—
148Q8—3941no—
149Q8—1971no—
150D4—22no—
151D4—41no—
152D4—7882no—
153D4—7881no—
154D4—7882no—
155D4—3941no—
156D4—3941no—
157D4—1971no—
158C2×C4—41no—
159C2×C4—7882no—
160C2×C4—7881no—
161C2×C4—3941no—
162C2×C4D4×D19711yescharacteristic
163C8—7881no—
164C8—3941no—
165C8D4×D19711yescharacteristic
166C8C2×C22⋊D19711yescharacteristic
167C22—42no—
168C22—161no—
169C22—7881no—
170C22—31522no—
171C22—3941no—
172C22#2026335011yescharacteristic
173C4—81no—
174C4—7881no—
175C4—3941no—
176C4#2026329411yescharacteristic
177C4D8×D19711yescharacteristic
178C2—21no—
179C2—82no—
180C2—7881no—
181C2—15761no—
182C2#2716121111yescharacteristic
subgroupG/H
centerC2#27161211
commutator, cosocleC2×C8×C197C23
FrattiniC2×C8C22×D197
FittingC2×SD32×C197C2
socleC2×C197#447
prime radical#27161880C2
hypercenterC2×SD32D197
hypocenterC197#2194
LCSG ▷ C2×C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
2. #27161520C2semidirect
3. #27161351C2semidirect
4. #27160892C2semidirect
5. #27160887C2semidirect
6. #27160862C2semidirect
7. C2×SD32×C197C2semidirect
18. #20263266C22semidirect
19. #20263262C22semidirect
21. C2×Q16×C197C22semidirect
24. C2×C16×C197C22semidirect
91. C197#2194semidirect
95. C2×SD32D197semidirect
106. C2×Q16C2×D197semidirect
114. C2×C16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618802
#271615202
#271613512
#271608922
#271608872
#271608622
C2×SD32×C1972
#2194197