Finite Group Databasebeta

← #34099213

Subgroups — #34099213

Order 20096.

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

normal subgroups only

This group has 228 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×Dic314C211yesmaximal, characteristic, complemented
2#23757645C211yesmaximal, characteristic
3#23757470C211yesmaximal, characteristic
4#23757021C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757003C211yesmaximal, characteristic, complemented
7#460×C157C211yesmaximal, characteristic
8C2×#14578942—22no—
9C2×#14578942—21no—
10#19097771—21no—
11#19097771C2211yescharacteristic
12#19097770—21no—
13#19097770—22no—
14Q8×Dic157—21no—
15C4×Dic314—21no—
16C4×Dic314C2211yescharacteristic
17C4⋊#11929434C2211yescharacteristic
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×Dic314D411yescharacteristic
27C2×Dic314D412yes—
28#14578942—44no—
29#14578942—42no—
30C2×#11929434—21no—
31C2×#11929434—21no—
32C4⋊Dic157—22no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157D411yescharacteristic
36#14578933—22no—
37#14578933—21no—
38C4×Dic157—22no—
39C4×Dic157—21no—
40C2×Q8×C157D411yescharacteristic
41C2×D4×C157—21no—
42C2×D4×C157—21no—
43C2×D4×C157—21no—
44SD16×C157—42no—
45C2×C8×C157—21no—
46C4⋊C4×C157—21no—
47C4⋊C4×C157D411yescharacteristic
48C42×C157—21no—
49C42×C157C2311yescharacteristic
50Dic314—21no—
51Dic314—42no—
52Dic314—82no—
53Dic314SD1612yescomplemented
54C2×Dic157—21no—
55C2×Dic157—21no—
56C2×Dic157—22no—
57#11929434—22no—
58#11929434—41no—
59C23×C157—41no—
60Q8×C157—21no—
61Q8×C157SD1612yescomplemented
62D4×C157—42no—
63D4×C157—41no—
64D4×C157—41no—
65D4×C157—41no—
66D4×C157—41no—
67C2×C4×C157—21no—
68C2×C4×C157—21no—
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157C2×D411yescharacteristic
72C8×C157—41no—
73C22×C157—81no—
74C22×C157—81no—
75C22×C157—81no—
76C22×C157#12011yescharacteristic
77C4×C157—22no—
78C4×C157—21no—
79C4×C157—41no—
80C4×C157C2×SD1611yescharacteristic
81C4×C157C2×SD1611yescharacteristic
82C4×C157#13711yescharacteristic
83C4×C157#13711yescharacteristic
84Dic157—22no—
85Dic157—42no—
86Dic157—41no—
87C2×C157—161no—
88C2×C157#44811yescharacteristic
89C2×C157#44811yescharacteristic
90C2×C157#45311yescharacteristic
91C157#162311yescharacteristic, complemented, Sylow
92#1623—1571nomaximal, Sylow
93Q82—1571no—
94#492—1571no—
95#460—1571no—
96#460D15711yescharacteristic
97#333—1571no—
98Q8⋊C8—1571no—
99Q8⋊C8—1571no—
100C2×SD16—21no—
101C2×SD16—3141no—
102C2×SD16—3142no—
103C4⋊Q8—3141no—
104C4⋊Q8—3142no—
105C4⋊Q8—3141no—
106C4⋊Q8—1571no—
107#127C2×D15711yescharacteristic
108C4×Q8—3141no—
109C4×Q8—3141no—
110C4×Q8—1571no—
111C4×Q8C2×D15711yescharacteristic
112C4⋊C8—1571no—
113C4⋊C8C2×D15711yescharacteristic
114#102—21no—
115#102—3141no—
116C4×C8—1571no—
117C2×Q8—3142no—
118C2×Q8—1571no—
119C2×Q8—1572no—
120C2×Q8C22⋊D15711yescharacteristic
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124SD16—42no—
125SD16—6284no—
126SD16—6282no—
127C2×C8—21no—
128C2×C8—3141no—
129C2×C8—3141no—
130C4⋊C4—21no—
131C4⋊C4—3142no—
132C4⋊C4—3142no—
133C4⋊C4—3141no—
134C4⋊C4—3141no—
135C4⋊C4—3141no—
136C4⋊C4—1571no—
137C4⋊C4C22⋊D15711yescharacteristic
138C42—21no—
139C42—3142no—
140C42—3141no—
141C42C22×D15711yescharacteristic
142C23—41no—
143Q8—21no—
144Q8—3141no—
145Q8—6282no—
146Q8—1572no—
147Q8—12562no—
148Q8#1457894212yescomplemented
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—3141no—
157C2×C4—3142no—
158C2×C4—3141no—
159C2×C4D4×D15711yescharacteristic
160C2×C4D4×D15711yescharacteristic
161C2×C4C2×C22⋊D15711yescharacteristic
162C8—41no—
163C8—3142no—
164C8—6281no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22#1909773911yescharacteristic
169C4—22no—
170C4—21no—
171C4—41no—
172C4—3142no—
173C4—6282no—
174C4—6281no—
175C4SD16×D15711yescharacteristic
176C4C2×#1457894211yescharacteristic
177C4#1909780011yescharacteristic
178C4#1909780111yescharacteristic
179C2—161no—
180C2#2375741011yescharacteristic
181C2#2375741111yescharacteristic
182C2#2375743911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#460×C157C2
socleC22×C157#120
prime radical#127×C157C22
hypercenter#460D157
hypocenterC157#1623
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×Dic314C2semidirect
4. #23757021C2semidirect
5. Q8⋊#11929434C2semidirect
6. #23757003C2semidirect
53. Dic314SD16semidirect
61. Q8×C157SD16semidirect
91. C157#1623semidirect
148. Q8#14578942semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3142
#237576452
#237574702
#237570212
Q8⋊#119294342
#237570032
#460×C1572
#1623157