Finite Group Databasebeta

← #34102105

Subgroups — #34102105

Order 20096.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#23758078C211yesmaximal, characteristic
2#23757714C211yesmaximal, characteristic, complemented
3C2×#19097714C211yesmaximal, characteristic, complemented
4#23757085C211yesmaximal, characteristic, complemented
5#23757079C211yesmaximal, characteristic, complemented
6#23757057C211yesmaximal, characteristic, complemented
7#473×C157C211yesmaximal, characteristic, complemented
8#19097817—21no—
9#19097799—22no—
10#19097799—21no—
11#19097793—21no—
12#19097793C2211yescharacteristic
13C2×D1256—21no—
14C2×D1256C2211yescharacteristic
15#19097717—21no—
16#19097714—22no—
17#19097714—21no—
18#19097711C2211yescharacteristic, complemented
19C2×#14578929C2211yescharacteristic, complemented
20#137×C157—21no—
21C2×Q16×C157C2211yescharacteristic, complemented
22#131×C157C2211yescharacteristic
23#108×C157C2211yescharacteristic, complemented
24#101×C157—21no—
25#14578955—41no—
26#14578954—22no—
27#14578954—21no—
28C2×D628—22no—
29C2×D628—21no—
30D4×D157—42no—
31D4×D157—41no—
32Dic628D411yescharacteristic
33#14578943—22no—
34#14578943—21no—
35D1256—22no—
36D1256—21no—
37D1256D411yescharacteristic
38D1256D412yes—
39#14578929—21no—
40#14578929—21no—
41#41×C157D411yescharacteristic
42C2×Q8×C157—21no—
43Q16×C157—21no—
44Q16×C157—21no—
45Q16×C157—41no—
46SD16×C157—41no—
47M16×C157—21no—
48M16×C157—21no—
49M16×C157D411yescharacteristic
50C2×C8×C157—21no—
51C2×C8×C157C2311yescharacteristic
52C22×D157—42no—
53C22×D157—41no—
54Dic314—21no—
55D628—21no—
56D628—22no—
57D628—42no—
58D628—41no—
59C22⋊D157—42no—
60C22⋊D157—41no—
61C4×D157—42no—
62C4×D157—41no—
63Q8×C157—41no—
64Q8×C157—41no—
65Q8×C157D811yescharacteristic
66D4×C157—21no—
67D4×C157D811yescharacteristic
68C2×C4×C157—21no—
69C2×C4×C157—41no—
70C2×C4×C157C2×D411yescharacteristic
71C8×C157—21no—
72C8×C157—41no—
73C8×C157C2×D411yescharacteristic
74C8×C157C2×D411yescharacteristic
75C22×C157—21no—
76C22×C157AGL(1,ℤ/8)11yescharacteristic
77C2×D157—42no—
78C2×D157—41no—
79C2×D157—161no—
80C2×D157—162no—
81C4×C157—21no—
82C4×C157—81no—
83C4×C157#12011yescharacteristic
84C4×C157C2×D811yescharacteristic
85Dic157—41no—
86C2×C157—21no—
87C2×C157—41no—
88C2×C157#44711yescharacteristic
89D157—82no—
90D157—81no—
91C157#219411yescharacteristic, complemented, Sylow
92#2194—1571nomaximal, Sylow
93#576—1571no—
94#510—1571no—
95C2×SD32—1571no—
96#473D15711yescharacteristic, complemented
97#361—1571no—
98#359—1571no—
99#350—1571no—
100H(5,𝔽2)—3141no—
101#137—21no—
102AGL(1,ℤ/8)—3142no—
103AGL(1,ℤ/8)—3141no—
104#135—3141no—
105#135—1571no—
106C2×Q16C2×D15711yescharacteristic, complemented
107C2×D8—3141no—
108C2×D8—1571no—
109#131C2×D15711yescharacteristic
110Q32—3141no—
111SD32—3142no—
112SD32—3141no—
113M32—1571no—
114C2×C16—1571no—
115#108C2×D15711yescharacteristic, complemented
116#101—21no—
117#41—3142no—
118#41—3141no—
119#41—6281no—
120#41C22⋊D15711yescharacteristic
121C2×Q8—21no—
122C2×D4—3142no—
123C2×D4—3141no—
124C2×D4—6282no—
125C2×D4—6281no—
126Q16—21no—
127Q16—21no—
128Q16—41no—
129Q16—1571no—
130SD16—41no—
131SD16—3142no—
132SD16—3141no—
133D8—3142no—
134D8—3141no—
135D8—1571no—
136D8—1572no—
137M16—21no—
138M16—21no—
139M16C22⋊D15711yescharacteristic
140C2×C8—21no—
141C2×C8C22×D15711yescharacteristic
142C16—3141no—
143C16—3141no—
144C23—6282no—
145C23—6281no—
146Q8—41no—
147Q8—41no—
148Q8—3141no—
149Q8#1457893911yescharacteristic
150D4—21no—
151D4—3141no—
152D4—3142no—
153D4—6282no—
154D4—6282no—
155D4—6281no—
156D4—6281no—
157D4#1457893911yescharacteristic
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—6282no—
161C2×C4—6281no—
162C2×C4C2×C22⋊D15711yescharacteristic
163C8—21no—
164C8—41no—
165C8D4×D15711yescharacteristic
166C8D4×D15711yescharacteristic
167C22—21no—
168C22—6282no—
169C22—6281no—
170C22—25121no—
171C22—25122no—
172C22#1909779611yescharacteristic
173C4—21no—
174C4—81no—
175C4—6281no—
176C4#1909773911yescharacteristic
177C4C2×#1457893911yescharacteristic
178C2—21no—
179C2—41no—
180C2—12562no—
181C2—12561no—
182C2#2375740611yescharacteristic
subgroupG/H
centerC2#23757406
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#473×C157C2
socleC2×C157#447
prime radical#23758078C2
hypercenter#473D157
hypocenterC157#2194
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #23757714C2semidirect
3. C2×#19097714C2semidirect
4. #23757085C2semidirect
5. #23757079C2semidirect
6. #23757057C2semidirect
7. #473×C157C2semidirect
18. #19097711C22semidirect
19. C2×#14578929C22semidirect
21. C2×Q16×C157C22semidirect
23. #108×C157C22semidirect
91. C157#2194semidirect
96. #473D157semidirect
106. C2×Q16C2×D157semidirect
115. #108C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580782
#237577142
C2×#190977142
#237570852
#237570792
#237570572
#473×C1572
#2194157