Finite Group Databasebeta

← #34099237

Subgroups — #34099237

Order 20096.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D628C211yesmaximal, characteristic, complemented
2#23757619C211yesmaximal, characteristic, complemented
3#23757465C211yesmaximal, characteristic
4#23757020C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7#460×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157—21no—
9C2×#14578939—21no—
10#19097771—21no—
11#19097771C2211yescharacteristic
12#19097751—22no—
13#19097751—21no—
14C4×D628—21no—
15C4×D628C2211yescharacteristic
16C4⋊#11929434C2211yescharacteristic
17#19097692—22no—
18#19097692—21no—
19C4×#11929434C2211yescharacteristic
20C2×SD16×C157—21no—
21#127×C157C2211yescharacteristic
22C4×Q8×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24#102×C157—21no—
25C2×Dic314—22no—
26C2×Dic314—21no—
27Q8×D157—42no—
28Q8×D157—42no—
29C2×D628D411yescharacteristic
30C2×C4×D157—22no—
31C2×C4×D157—21no—
32#14578939—42no—
33C2×#11929434—21no—
34C2×#11929434—21no—
35C4⋊Dic157—22no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38C4⋊Dic157D412yes—
39#14578931—22no—
40#14578931—21no—
41C2×Q8×C157D411yescharacteristic
42C2×D4×C157—21no—
43C2×D4×C157—21no—
44C2×D4×C157—21no—
45SD16×C157—42no—
46C2×C8×C157—21no—
47C4⋊C4×C157—21no—
48C4⋊C4×C157D411yescharacteristic
49C42×C157—21no—
50C42×C157C2311yescharacteristic
51C22×D157—21no—
52Dic314—44no—
53Dic314—41no—
54Dic314—41no—
55D628—21no—
56D628SD1612yescomplemented
57C2×Dic157—22no—
58C2×Dic157—21no—
59C4×D157—44no—
60C4×D157—41no—
61C4×D157—41no—
62#11929434—41no—
63#11929434—41no—
64C23×C157—41no—
65Q8×C157—21no—
66Q8×C157D812yescomplemented
67D4×C157—42no—
68D4×C157—41no—
69D4×C157—41no—
70D4×C157—41no—
71D4×C157—41no—
72C2×C4×C157—21no—
73C2×C4×C157—21no—
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C2×C4×C157C2×D411yescharacteristic
77C8×C157—41no—
78C22×C157—81no—
79C22×C157—81no—
80C22×C157—81no—
81C22×C157#12011yescharacteristic
82C2×D157—22no—
83C2×D157—41no—
84C2×D157—41no—
85C4×C157—22no—
86C4×C157—21no—
87C4×C157—41no—
88C4×C157C2×D811yescharacteristic
89C4×C157C2×SD1611yescharacteristic
90C4×C157AGL(1,ℤ/8)11yescharacteristic
91C4×C157#13711yescharacteristic
92Dic157—42no—
93Dic157—41no—
94C2×C157—161no—
95C2×C157#44711yescharacteristic
96C2×C157#44811yescharacteristic
97C2×C157#45411yescharacteristic
98D157—42no—
99C157#162711yescharacteristic, complemented, Sylow
100#1627—1571nomaximal, Sylow
101D4×Q8—1571no—
102#486—1571no—
103#460D15711yescharacteristic, complemented
104#459—1571no—
105#332—1571no—
106Q8⋊C8—1571no—
107D4⋊C8—1571no—
108C22×Q8—3141no—
109C2×SD16—21no—
110C2×D8—3141no—
111C4⋊Q8—3141no—
112C4⋊Q8—1571no—
113#127C2×D15711yescharacteristic
114C22⋊Q8—3142no—
115C22⋊Q8—3141no—
116C4×Q8C2×D15711yescharacteristic
117C4×D4—3141no—
118C4×D4—1571no—
119C4⋊C8—1571no—
120C4⋊C8C2×D15711yescharacteristic
121#102—21no—
122#102—3141no—
123#102—3142no—
124C4×C8—1571no—
125C2×Q8—3142no—
126C2×Q8—3141no—
127C2×Q8—6282no—
128C2×Q8—6282no—
129C2×Q8C22⋊D15711yescharacteristic
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—1571no—
134C22×C4—3142no—
135C22×C4—3141no—
136SD16—42no—
137D8—6282no—
138C2×C8—21no—
139C2×C8—3141no—
140C2×C8—3141no—
141C4⋊C4—21no—
142C4⋊C4—3142no—
143C4⋊C4—3141no—
144C4⋊C4—1571no—
145C4⋊C4—1572no—
146C4⋊C4C22⋊D15711yescharacteristic
147C22⋊C4—3142no—
148C22⋊C4—3141no—
149C42—21no—
150C42C22×D15711yescharacteristic
151C23—41no—
152C23—3141no—
153Q8—21no—
154Q8—6284no—
155Q8—6281no—
156Q8—6281no—
157Q8#1457893912yescomplemented
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—3141no—
164D4—1572no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—3142no—
168C2×C4—3141no—
169C2×C4—6284no—
170C2×C4—6281no—
171C2×C4—6281no—
172C2×C4D4×D15711yescharacteristic
173C2×C4D4×D15711yescharacteristic
174C2×C4C2×C22⋊D15711yescharacteristic
175C8—41no—
176C8—6281no—
177C8—6281no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—3142no—
182C22—6281no—
183C22—6281no—
184C22#1909773911yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—6282no—
189C4—6281no—
190C4C2×#1457893911yescharacteristic
191C4SD16×D15711yescharacteristic
192C4#1909779611yescharacteristic
193C4#1909780011yescharacteristic
194C2—161no—
195C2—6282no—
196C2#2375740611yescharacteristic
197C2#2375741011yescharacteristic
198C2#2375744411yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#460×C157C2
socleC22×C157#120
prime radical#23757465C2
hypercenter#460D157
hypocenterC157#1627
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×D628C2semidirect
2. #23757619C2semidirect
4. #23757020C2semidirect
5. Q8⋊#11929434C2semidirect
6. #23757000C2semidirect
7. #460×C157C2semidirect
56. D628SD16semidirect
66. Q8×C157D8semidirect
99. C157#1627semidirect
103. #460D157semidirect
157. Q8#14578939semidirect

Maximal subgroups

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

subgroupindex
Q8×D6282
#237576192
#237574652
#237570202
Q8⋊#119294342
#237570002
#460×C1572
#1627157