Finite Group Databasebeta

← #34102104

Subgroups — #34102104

Order 20096.

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#23758075C211yesmaximal, characteristic
2C2×#19097714C211yesmaximal, characteristic, complemented
3#23757545C211yesmaximal, characteristic, complemented
4#23757087C211yesmaximal, characteristic, complemented
5#23757080C211yesmaximal, characteristic, complemented
6#23757058C211yesmaximal, characteristic, complemented
7#510×C157C211yesmaximal, characteristic, complemented
8#19097817—21no—
9#19097802—21no—
10#19097798—22no—
11#19097796—21no—
12#19097794—21no—
13C2×D1256C2211yescharacteristic
14D8×D157—21no—
15#19097777C2211yescharacteristic
16#19097714—22no—
17C2×#14578929C2211yescharacteristic, complemented
18#19097707C2211yescharacteristic, complemented
19#19097690—21no—
20#135×C157C2211yescharacteristic
21C2×Q16×C157C2211yescharacteristic, complemented
22Q32×C157—21no—
23SD32×C157—21no—
24M32×C157C2211yescharacteristic, complemented
25#14578955—22no—
26#14578954—41no—
27#14578954D411yescharacteristic
28C2×D628—21no—
29C2×D628—41no—
30D4×D157—22no—
31D4×D157—42no—
32#14578944—41no—
33#14578941—22no—
34#14578941—41no—
35D1256—21no—
36D1256D412yes—
37#14578939—22no—
38#14578938—21no—
39#14578938D411yescharacteristic
40#14578937—21no—
41C8×D157—21no—
42#14578929—21no—
43#41×C157—21no—
44C2×Q8×C157—21no—
45Q16×C157—21no—
46Q16×C157—21no—
47Q16×C157D411yescharacteristic
48SD16×C157—21no—
49D8×C157D411yescharacteristic
50C2×C8×C157C2311yescharacteristic
51C16×C157—21no—
52C22×D157—42no—
53C22×D157—41no—
54Dic314D811yescharacteristic
55D628—22no—
56D628—42no—
57D628—41no—
58D628D811yescharacteristic
59C22⋊D157—21no—
60C22⋊D157—42no—
61C4×D157—21no—
62C4×D157—42no—
63#11929434—21no—
64#11929434—41no—
65Q8×C157—21no—
66Q8×C157—41no—
67Q8×C157—41no—
68D4×C157—21no—
69D4×C157—41no—
70C2×C4×C157—41no—
71C2×C4×C157—41no—
72C2×C4×C157C2×D411yescharacteristic
73C8×C157C2×D411yescharacteristic
74C8×C157C2×D411yescharacteristic
75C22×C157—41no—
76C22×C157AGL(1,ℤ/8)11yescharacteristic
77C2×D157—21no—
78C2×D157—42no—
79C2×D157—162no—
80C2×D157—161no—
81C4×C157—41no—
82C4×C157—81no—
83C4×C157#12011yescharacteristic
84C4×C157C2×D811yescharacteristic
85Dic157—21no—
86C2×C157—21no—
87C2×C157—81no—
88C2×C157#44711yescharacteristic
89D157—41no—
90D157—82no—
91C157#219411yescharacteristic, complemented, Sylow
92#2194—1571nomaximal, Sylow
93#576—1571no—
94#510D15711yescharacteristic, complemented
95C2×SD32—1571no—
96#473—1571no—
97#361—1571no—
98#359—1571no—
99#350—1571no—
100H(5,𝔽2)—3141no—
101#137—3141no—
102AGL(1,ℤ/8)—3142no—
103AGL(1,ℤ/8)—3141no—
104#135—3141no—
105#135C2×D15711yescharacteristic
106C2×Q16C2×D15711yescharacteristic, complemented
107C2×D8—3141no—
108C2×D8—1571no—
109#131—1571no—
110Q32—21no—
111SD32—21no—
112SD32—3142no—
113M32C2×D15711yescharacteristic, complemented
114C2×C16—1571no—
115#108—1571no—
116#101—3141no—
117#41—21no—
118#41—3142no—
119#41—6281no—
120#41—1571no—
121C2×Q8—21no—
122C2×D4—3142no—
123C2×D4—3141no—
124C2×D4—6281no—
125C2×D4—6282no—
126Q16—21no—
127Q16—21no—
128Q16—6281no—
129Q16C22⋊D15711yescharacteristic
130SD16—21no—
131SD16—3142no—
132SD16—6281no—
133D8—3142no—
134D8—3141no—
135D8—1572no—
136D8C22⋊D15711yescharacteristic
137M16—3141no—
138M16—3141no—
139M16—1571no—
140C2×C8—3141no—
141C2×C8C22×D15711yescharacteristic
142C16—21no—
143C16—3141no—
144C23—6282no—
145C23—6281no—
146Q8—21no—
147Q8—41no—
148Q8—41no—
149Q8—1571no—
150D4—21no—
151D4—41no—
152D4—3141no—
153D4—3142no—
154D4—6282no—
155D4—6282no—
156D4—6281no—
157D4—1571no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—3141no—
161C2×C4—6282no—
162C2×C4D4×D15711yescharacteristic
163C8—3141no—
164C8—6281no—
165C8D4×D15711yescharacteristic
166C8C2×C22⋊D15711yescharacteristic
167C22—41no—
168C22—3141no—
169C22—6282no—
170C22—25121no—
171C22—25122no—
172C22#1909779511yescharacteristic
173C4—41no—
174C4—81no—
175C4—3141no—
176C4#1909773911yescharacteristic
177C4D8×D15711yescharacteristic
178C2—21no—
179C2—81no—
180C2—6281no—
181C2—12562no—
182C2#2375740511yescharacteristic
subgroupG/H
centerC2#23757405
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#510×C157C2
socleC2×C157#447
prime radical#23758075C2
hypercenter#510D157
hypocenterC157#2194
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#19097714C2semidirect
3. #23757545C2semidirect
4. #23757087C2semidirect
5. #23757080C2semidirect
6. #23757058C2semidirect
7. #510×C157C2semidirect
17. C2×#14578929C22semidirect
18. #19097707C22semidirect
21. C2×Q16×C157C22semidirect
24. M32×C157C22semidirect
91. C157#2194semidirect
94. #510D157semidirect
106. C2×Q16C2×D157semidirect
113. M32C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580752
C2×#190977142
#237575452
#237570872
#237570802
#237570582
#510×C1572
#2194157