Finite Group Databasebeta

← #34099385

Subgroups — #34099385

Order 20096.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#23757676C211yesmaximal, characteristic, complemented
2#23757553C211yesmaximal, characteristic
3#23757483C211yesmaximal, characteristic, complemented
4#23757019C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic
7D4×Q8×C157C211yesmaximal, characteristic
8C2×#14578944—21no—
9#19097771C2211yescharacteristic
10#19097704—22no—
11#19097704—21no—
12C4⋊#11929434C2211yescharacteristic
13C4⋊#11929434C2211yescharacteristic
14#19097694—21no—
15#19097692—21no—
16C4×#11929434C2211yescharacteristic
17C22×Q8×C157—21no—
18C4⋊Q8×C157—21no—
19C4⋊Q8×C157C2211yescharacteristic
20C22⋊Q8×C157—21no—
21C22⋊Q8×C157—22no—
22C4×Q8×C157C2211yescharacteristic
23C4×D4×C157—21no—
24C4×D4×C157C2211yescharacteristic
25C2×Dic314—21no—
26#14578944—42no—
27C2×#11929434—21no—
28C2×#11929434—21no—
29C2×#11929434—21no—
30C4⋊Dic157—21no—
31C4⋊Dic157—21no—
32C2×Q8×C157—22no—
33C2×Q8×C157—21no—
34C2×Q8×C157—42no—
35C2×Q8×C157—42no—
36C2×Q8×C157D411yescharacteristic
37C2×D4×C157D411yescharacteristic
38C22×C4×C157—22no—
39C22×C4×C157—21no—
40C4⋊C4×C157—22no—
41C4⋊C4×C157—21no—
42C4⋊C4×C157—21no—
43C4⋊C4×C157D411yescharacteristic
44C4⋊C4×C157D411yescharacteristic
45C4⋊C4×C157D412yes—
46C22⋊C4×C157—22no—
47C22⋊C4×C157—21no—
48C42×C157—21no—
49C42×C157C2311yescharacteristic
50Dic314—41no—
51Dic314—41no—
52C2×Dic157—41no—
53#11929434—22no—
54#11929434—41no—
55#11929434—41no—
56C23×C157—21no—
57Q8×C157—21no—
58Q8×C157—44no—
59Q8×C157—41no—
60Q8×C157—41no—
61Q8×C157D812yes—
62D4×C157—21no—
63D4×C157Q1612yescomplemented
64C2×C4×C157—21no—
65C2×C4×C157—21no—
66C2×C4×C157—22no—
67C2×C4×C157—21no—
68C2×C4×C157—44no—
69C2×C4×C157—41no—
70C2×C4×C157—41no—
71C2×C4×C157C2×D411yescharacteristic
72C2×C4×C157C2×D411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C22×C157—22no—
75C22×C157—41no—
76C22×C157—41no—
77C22×C157#12011yescharacteristic
78C4×C157—22no—
79C4×C157—21no—
80C4×C157—42no—
81C4×C157—41no—
82C4×C157—41no—
83C4×C157C2×D811yescharacteristic
84C4×C157C2×Q1611yescharacteristic
85C4×C157AGL(1,ℤ/8)11yescharacteristic
86C4×C157#13711yescharacteristic
87Dic157—81no—
88C2×C157—42no—
89C2×C157#44711yescharacteristic
90C2×C157#45111yescharacteristic
91C2×C157#45611yescharacteristic
92C157#164911yescharacteristic, complemented, Sylow
93#1649—1571nomaximal, Sylow
94D4×Q8D15711yescharacteristic
95#500—1571no—
96#474—1571no—
97#462—1571no—
98#332—1571no—
99Q8⋊C8—1571no—
100D4⋊C8—1571no—
101C22×Q8—21no—
102C2×Q16—3141no—
103C4⋊Q8—21no—
104C4⋊Q8—1571no—
105C4⋊Q8C2×D15711yescharacteristic
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C4×Q8C2×D15711yescharacteristic
109C4×D4—21no—
110C4×D4C2×D15711yescharacteristic
111#107—3141no—
112#107—3142no—
113C4⋊C8—1571no—
114C4⋊C8—1571no—
115#103—3141no—
116#102—3141no—
117C4×C8—1571no—
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—3141no—
123C2×Q8C22⋊D15711yescharacteristic
124C2×D4C22⋊D15711yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127Q16—6282no—
128C2×C8—3141no—
129C2×C8—3141no—
130C2×C8—3141no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—3141no—
135C4⋊C4—3141no—
136C4⋊C4C22⋊D15711yescharacteristic
137C4⋊C4C22⋊D15711yescharacteristic
138C4⋊C4C22⋊D15712yes—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C42—21no—
142C42C22×D15711yescharacteristic
143C23—21no—
144Q8—21no—
145Q8—44no—
146Q8—41no—
147Q8—41no—
148Q8—6281no—
149Q8—6281no—
150Q8#1457893912yes—
151D4—21no—
152D4#1457894412yescomplemented
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—6281no—
161C2×C4C2×C22⋊D15711yescharacteristic
162C2×C4C2×C22⋊D15711yescharacteristic
163C2×C4C2×C22⋊D15711yescharacteristic
164C8—3142no—
165C8—6281no—
166C8—6281no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#1909774011yescharacteristic
171C4—22no—
172C4—21no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—12561no—
177C4C2×#1457893911yescharacteristic
178C4C2×#1457894411yescharacteristic
179C4#1909779711yescharacteristic
180C4#1909780211yescharacteristic
181C2—42no—
182C2#2375740711yescharacteristic
183C2#2375742911yescharacteristic
184C2#2375745611yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4×Q8×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenterD4×Q8D157
hypocenterC157#1649
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23757676C2semidirect
3. #23757483C2semidirect
4. #23757019C2semidirect
5. Q8⋊#11929434C2semidirect
63. D4×C157Q16semidirect
92. C157#1649semidirect
152. D4#14578944semidirect

Maximal subgroups

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

subgroupindex
#237576762
#237575532
#237574832
#237570192
Q8⋊#119294342
D4⋊#119294342
D4×Q8×C1572
#1649157