Finite Group Databasebeta

← #34102110

Subgroups — #34102110

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#23758074C211yesmaximal, characteristic
2#23757716C211yesmaximal, characteristic, complemented
3C2×#19097715C211yesmaximal, characteristic, complemented
4#23757549C211yesmaximal, characteristic, complemented
5#23757079C211yesmaximal, characteristic, complemented
6#23757058C211yesmaximal, characteristic, complemented
7#361×C157C211yesmaximal, characteristic, complemented
8#19097816—21no—
9#19097804—21no—
10#19097797—21no—
11#19097795—22no—
12#19097793C2211yescharacteristic
13#19097790—21no—
14C2×Dic628C2211yescharacteristic, complemented
15D8×D157—21no—
16#19097777C2211yescharacteristic
17Dic1256—21no—
18#19097716—21no—
19#19097715—22no—
20C2×#14578929C2211yescharacteristic, complemented
21#19097689—21no—
22C2×D8×C157C2211yescharacteristic
23M32×C157C2211yescharacteristic, complemented
24#108×C157C2211yescharacteristic, complemented
25#14578954—21no—
26#14578954D411yescharacteristic
27#14578953—22no—
28#14578953—41no—
29C2×Dic314—21no—
30C2×C22⋊D157—42no—
31D4×D157—22no—
32D4×D157—41no—
33Dic628—21no—
34Dic628—21no—
35Dic628—41no—
36Dic628D411yescharacteristic
37#14578943—21no—
38#14578943—41no—
39#14578942—22no—
40D1256D411yescharacteristic
41#14578939—22no—
42#14578938D411yescharacteristic
43#14578937—21no—
44C8×D157—21no—
45#14578929—21no—
46C2×D4×C157—21no—
47D8×C157—21no—
48D8×C157D412yes—
49M16×C157—21no—
50C2×C8×C157C2311yescharacteristic
51C16×C157—21no—
52C22×D157—42no—
53Dic314—21no—
54Dic314—41no—
55Dic314—41no—
56Dic314D811yescharacteristic
57D628—21no—
58D628D811yescharacteristic
59C22⋊D157—21no—
60C22⋊D157—42no—
61C22⋊D157—42no—
62C22⋊D157—41no—
63C2×Dic157—42no—
64C2×Dic157—41no—
65C4×D157—21no—
66C4×D157—41no—
67#11929434—21no—
68C23×C157—41no—
69D4×C157—22no—
70D4×C157—41no—
71C2×C4×C157C2×D411yescharacteristic
72C8×C157—41no—
73C8×C157C2×D411yescharacteristic
74C8×C157C2×D411yescharacteristic
75C22×C157—42no—
76C22×C157—161no—
77C22×C157AGL(1,ℤ/8)11yescharacteristic
78C2×D157—21no—
79C2×D157—41no—
80C2×D157—162no—
81C4×C157#12011yescharacteristic
82C4×C157C2×D811yescharacteristic
83Dic157—21no—
84Dic157—41no—
85Dic157—81no—
86C2×C157—21no—
87C2×C157—82no—
88C2×C157#44711yescharacteristic
89D157—41no—
90D157—81no—
91C157#219411yescharacteristic, complemented, Sylow
92#2194—1571nomaximal, Sylow
93#576—1571no—
94#510—1571no—
95C2×SD32—1571no—
96#473—1571no—
97#361D15711yescharacteristic, complemented
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#135—1571no—
106C2×Q16—1571no—
107C2×D8—3141no—
108C2×D8C2×D15711yescharacteristic
109#131—1571no—
110Q32—3141no—
111SD32—3141no—
112SD32—3142no—
113M32C2×D15711yescharacteristic, complemented
114C2×C16—1571no—
115#108C2×D15711yescharacteristic, complemented
116#101—3141no—
117#41—3142no—
118#41—3141no—
119#41—6281no—
120#41—1571no—
121C2×Q8—3141no—
122C2×D4—21no—
123C2×D4—3142no—
124C2×D4—6281no—
125C2×D4—6282no—
126Q16—3141no—
127Q16—3141no—
128Q16—6281no—
129Q16—1571no—
130SD16—3142no—
131SD16—3141no—
132SD16—6281no—
133D8—21no—
134D8—3142no—
135D8—1571no—
136D8D62812yes—
137M16—21no—
138M16—3141no—
139M16—1571no—
140C2×C8—3141no—
141C2×C8C22×D15711yescharacteristic
142C16—21no—
143C16—3141no—
144C23—41no—
145C23—6282no—
146Q8—3141no—
147Q8—6281no—
148Q8—6281no—
149Q8—1571no—
150D4—22no—
151D4—41no—
152D4—3141no—
153D4—3141no—
154D4—6282no—
155D4—6282no—
156D4—6281no—
157D4—1571no—
158C2×C4—3141no—
159C2×C4—6282no—
160C2×C4—6281no—
161C2×C4—6281no—
162C2×C4D4×D15711yescharacteristic
163C8—41no—
164C8—3141no—
165C8D4×D15711yescharacteristic
166C8C2×D62811yescharacteristic
167C22—42no—
168C22—161no—
169C22—3141no—
170C22—6281no—
171C22—25122no—
172C22#1909779811yescharacteristic
173C4—3141no—
174C4—6281no—
175C4—12561no—
176C4#1909774111yescharacteristic
177C4D8×D15711yescharacteristic
178C2—21no—
179C2—82no—
180C2—6281no—
181C2—12561no—
182C2#2375740811yescharacteristic
subgroupG/H
centerC2#23757408
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#361×C157C2
socleC2×C157#447
prime radical#23758074C2
hypercenter#361D157
hypocenterC157#2194
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #23757716C2semidirect
3. C2×#19097715C2semidirect
4. #23757549C2semidirect
5. #23757079C2semidirect
6. #23757058C2semidirect
7. #361×C157C2semidirect
14. C2×Dic628C22semidirect
20. C2×#14578929C22semidirect
23. M32×C157C22semidirect
24. #108×C157C22semidirect
91. C157#2194semidirect
97. #361D157semidirect
113. M32C2×D157semidirect
115. #108C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580742
#237577162
C2×#190977152
#237575492
#237570792
#237570582
#361×C1572
#2194157