Finite Group Databasebeta

← #34099208

Subgroups — #34099208

Order 20096.

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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#23757647C211yesmaximal, characteristic, complemented
2#23757473C211yesmaximal, characteristic, complemented
3#23757468C211yesmaximal, characteristic
4#23757022C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic, complemented
7#554×C157C211yesmaximal, characteristic, complemented
8C2×#14578941—22no—
9C2×#14578941—21no—
10C2×#14578939—21no—
11#19097768C2211yescharacteristic
12C4⋊#11929434C2211yescharacteristic
13C4⋊#11929434C2211yescharacteristic
14#19097691—21no—
15#19097691—21no—
16C4×#11929434C2211yescharacteristic
17C4⋊Q8×C157C2211yescharacteristic
18#125×C157—21no—
19C22⋊Q8×C157—21no—
20C22⋊Q8×C157—22no—
21C4×Q8×C157C2211yescharacteristic
22C4×D4×C157—21no—
23C4×D4×C157C2211yescharacteristic
24C2×C4⋊C4×C157—21no—
25C2×D628—21no—
26C2×D628—21no—
27C2×D628—21no—
28#14578941—42no—
29#14578941—44no—
30#14578939—42no—
31C2×#11929434—21no—
32C2×#11929434—21no—
33C2×#11929434—21no—
34C2×Q8×C157D411yescharacteristic
35C2×Q8×C157D412yes—
36C2×D4×C157D411yescharacteristic
37C22×C4×C157—22no—
38C22×C4×C157—21no—
39C4⋊C4×C157—22no—
40C4⋊C4×C157—22no—
41C4⋊C4×C157—21no—
42C4⋊C4×C157—21no—
43C4⋊C4×C157—21no—
44C4⋊C4×C157D411yescharacteristic
45C4⋊C4×C157D411yescharacteristic
46C22⋊C4×C157—22no—
47C22⋊C4×C157—21no—
48C42×C157—21no—
49C42×C157C2311yescharacteristic
50C22×D157—41no—
51D628—42no—
52D628—41no—
53D628—41no—
54D628—41no—
55D628—41no—
56#11929434—22no—
57#11929434—41no—
58#11929434—41no—
59C23×C157—21no—
60Q8×C157—21no—
61Q8×C157—21no—
62Q8×C157—42no—
63D4×C157—21no—
64D4×C157SD1612yescomplemented
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—21no—
68C2×C4×C157—22no—
69C2×C4×C157—41no—
70C2×C4×C157—41no—
71C2×C4×C157—82no—
72C2×C4×C157C2×D411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157C2×D411yescharacteristic
75C22×C157—22no—
76C22×C157—41no—
77C22×C157—41no—
78C22×C157#12011yescharacteristic
79C2×D157—81no—
80C2×D157—81no—
81C2×D157—81no—
82C4×C157—21no—
83C4×C157—41no—
84C4×C157—41no—
85C4×C157—41no—
86C4×C157—42no—
87C4×C157C2×SD1611yescharacteristic
88C4×C157#13511yescharacteristic
89C4×C157AGL(1,ℤ/8)11yescharacteristic
90C4×C157#13711yescharacteristic
91C2×C157—42no—
92C2×C157#44811yescharacteristic
93C2×C157#44911yescharacteristic
94C2×C157#45311yescharacteristic
95D157—161no—
96C157#162211yescharacteristic, complemented, Sylow
97#1622—1571nomaximal, Sylow
98#554D15711yescharacteristic, complemented
99#492—1571no—
100#460—1571no—
101#459—1571no—
102#333—1571no—
103Q8⋊C8—1571no—
104D4⋊C8—1571no—
105C2×SD16—3141no—
106C2×SD16—3142no—
107C2×D8—3141no—
108C4⋊Q8C2×D15711yescharacteristic
109#127—1571no—
110#125—21no—
111C22⋊Q8—21no—
112C22⋊Q8—22no—
113C4×Q8C2×D15711yescharacteristic
114C4×D4—21no—
115C4×D4C2×D15711yescharacteristic
116C2×C4⋊C4—21no—
117C4⋊C8—1571no—
118C4⋊C8—1571no—
119#102—3141no—
120#102—3141no—
121C4×C8—1571no—
122C2×Q8C22⋊D15711yescharacteristic
123C2×Q8C22⋊D15712yes—
124C2×D4—3141no—
125C2×D4—3141no—
126C2×D4—3141no—
127C2×D4C22⋊D15711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130SD16—6284no—
131SD16—6282no—
132D8—6282no—
133C2×C8—3141no—
134C2×C8—3141no—
135C2×C8—3141no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4C22⋊D15711yescharacteristic
142C4⋊C4C22⋊D15711yescharacteristic
143C22⋊C4—22no—
144C22⋊C4—21no—
145C42—21no—
146C42C22×D15711yescharacteristic
147C23—21no—
148C23—6281no—
149Q8—21no—
150Q8—21no—
151Q8—42no—
152D4—21no—
153D4—6282no—
154D4—6281no—
155D4—6281no—
156D4—6281no—
157D4—6281no—
158D4#1457894112yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4C2×C22⋊D15711yescharacteristic
167C2×C4C2×C22⋊D15711yescharacteristic
168C2×C4C2×C22⋊D15711yescharacteristic
169C8—3142no—
170C8—6281no—
171C8—6281no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—12561no—
176C22—12561no—
177C22—12561no—
178C22#1909774011yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—42no—
184C4C2×#1457894111yescharacteristic
185C4#1909779111yescharacteristic
186C4#1909779611yescharacteristic
187C4#1909780211yescharacteristic
188C2—42no—
189C2—25121no—
190C2#2375741211yescharacteristic
191C2#2375741811yescharacteristic
192C2#2375744111yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#554×C157C2
socleC22×C157#120
prime radical#23757468C2
hypercenter#554D157
hypocenterC157#1622
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #23757647C2semidirect
2. #23757473C2semidirect
4. #23757022C2semidirect
5. Q8⋊#11929434C2semidirect
6. D4⋊#11929434C2semidirect
7. #554×C157C2semidirect
64. D4×C157SD16semidirect
96. C157#1622semidirect
98. #554D157semidirect
158. D4#14578941semidirect

Maximal subgroups

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

subgroupindex
#237576472
#237574732
#237574682
#237570222
Q8⋊#119294342
D4⋊#119294342
#554×C1572
#1622157