Finite Group Databasebeta

← #34110259

Subgroups — #34110259

Order 20096.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#23757823C211yesmaximal, characteristic
2C2×C4×Dic314C211yesmaximal, characteristic, complemented
3#23757587C212yesmaximal, complemented
4#23757569C212yesmaximal, complemented
5#23757474C212yesmaximal, complemented
6#23757414C212yesmaximal, complemented
7#23757375C214yesmaximal, complemented
8#432×C157C211yesmaximal, characteristic, complemented
9C2×#14578954—21no—
10C22×Dic314C2211yescharacteristic
11C2×#14578943C2214yes—
12#19097771C2211yescharacteristic
13#19097768C2211yescharacteristic
14#19097759—21no—
15#19097745—21no—
16#19097745C2212yes—
17C4×Dic314—21no—
18C4×Dic314—21no—
19C4×Dic314C2212yes—
20C4×Dic314C2212yes—
21C4×D628—21no—
22C4×D628C2212yes—
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933—21no—
25C2×C4×Dic157—21no—
26#19097705C2214yes—
27#19097696C2214yes—
28#19097693C2214yes—
29C2×C42×C157C2211yescharacteristic
30C4⋊C8×C157C2212yes—
31C22⋊C8×C157C2212yes—
32#97×C157C2212yes—
33#14578954—44no—
34C2×Dic314—24no—
35C2×Dic314—21no—
36C2×Dic314—21no—
37C2×Dic314C2312yes—
38C2×D628—21no—
39C2×D628—21no—
40C2×D628C2312yes—
41C2×C22⋊D157—22no—
42C22×Dic157—21no—
43C22×Dic157—21no—
44C2×C4×D157—22no—
45#14578943—28no—
46C4⋊Dic157—21no—
47C4⋊Dic157—21no—
48C4⋊Dic157C2312yes—
49C4⋊Dic157C2312yes—
50#14578933—22no—
51#14578933—22no—
52#14578931—22no—
53#14578931—22no—
54C4×Dic157—22no—
55C4×Dic157—22no—
56C22×C4×C157D411yescharacteristic
57C22×C4×C157D411yescharacteristic
58C22×C4×C157C2311yescharacteristic
59C2×C8×C157C2314yes—
60C42×C157D411yescharacteristic
61C42×C157D411yescharacteristic
62C42×C157C2311yescharacteristic
63C42×C157C2311yescharacteristic
64C22×D157—22no—
65Dic314—24no—
66Dic314—22no—
67Dic314—42no—
68Dic314#4114yescomplemented
69D628—24no—
70D628—42no—
71D628—42no—
72C22⋊D157—44no—
73C2×Dic157—24no—
74C2×Dic157—22no—
75C2×Dic157—22no—
76C2×Dic157—22no—
77C2×Dic157—42no—
78C4×D157—44no—
79C23×C157C2×D411yescharacteristic
80C2×C4×C157—22no—
81C2×C4×C157—22no—
82C2×C4×C157—22no—
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C2×C4×C157C2411yescharacteristic
89C8×C157—24no—
90C22×C157—21no—
91C22×C157—21no—
92C22×C157AGL(1,ℤ/8)12yes—
93C22×C157C22×D411yescharacteristic
94C2×D157—42no—
95C2×D157—42no—
96C2×D157—42no—
97C4×C157—21no—
98C4×C157—21no—
99C4×C157—21no—
100C4×C157—21no—
101C4×C157#13712yes—
102C4×C157C2×#4112yes—
103Dic157—24no—
104Dic157—42no—
105Dic157—42no—
106C2×C157—22no—
107C2×C157#52511yescharacteristic
108C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C157C2×#13711yescharacteristic
110D157—82no—
111C157#311211yescharacteristic, complemented, Sylow
112#3112—1571nomaximal, Sylow
113#532—1571no—
114C2×C4×Q8—1571no—
115#480—1572no—
116#477—1572no—
117#460—1572no—
118#448—1572no—
119#440—1574no—
120#432D15711yescharacteristic, complemented
121C2×#41—3141no—
122C22×Q8—1571no—
123C2×SD16—1574no—
124C4⋊Q8—1571no—
125#127—1571no—
126#124—3141no—
127#121—3141no—
128#121—1572no—
129C4×Q8—3141no—
130C4×Q8—3141no—
131C4×Q8—1572no—
132C4×Q8—1572no—
133C4×D4—3141no—
134C4×D4—1572no—
135C2×C4⋊C4—3141no—
136C2×C4⋊C4—1571no—
137C2×C42—3141no—
138C2×C42C2×D15711yescharacteristic
139#107—1574no—
140C4⋊C8C2×D15712yes—
141#103—1574no—
142#102—1574no—
143C22⋊C8C2×D15712yes—
144#97C2×D15712yes—
145#41—6284no—
146C2×Q8—3144no—
147C2×Q8—3141no—
148C2×Q8—3141no—
149C2×Q8—1572no—
150C2×D4—3142no—
151C2×D4—3141no—
152C2×D4—3141no—
153C2×D4—1572no—
154C22×C4—3141no—
155C22×C4—3141no—
156C22×C4—3142no—
157C22×C4D62811yescharacteristic
158C22×C4D62811yescharacteristic
159C22×C4C22×D15711yescharacteristic
160SD16—3148no—
161C2×C8C22×D15714yes—
162C4⋊C4—3142no—
163C4⋊C4—3142no—
164C4⋊C4—3141no—
165C4⋊C4—3141no—
166C4⋊C4—1572no—
167C4⋊C4—1572no—
168C22⋊C4—3142no—
169C22⋊C4—3142no—
170C42—3142no—
171C42—3142no—
172C42D62811yescharacteristic
173C42D62811yescharacteristic
174C42C22×D15711yescharacteristic
175C42C22×D15711yescharacteristic
176C23—3142no—
177C23C2×D62811yescharacteristic
178Q8—3144no—
179Q8—3142no—
180Q8—6282no—
181Q8—1574no—
182D4—3144no—
183D4—6282no—
184D4—6284no—
185D4—6282no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—3144no—
190C2×C4—3142no—
191C2×C4—3142no—
192C2×C4—3142no—
193C2×C4—6284no—
194C2×C4—6282no—
195C2×C4C2×D62811yescharacteristic
196C2×C4C2×D62811yescharacteristic
197C2×C4C2×D62811yescharacteristic
198C2×C4C2×D62811yescharacteristic
199C2×C4C2×D62811yescharacteristic
200C2×C4C23×D15711yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—6282no—
205C22—6282no—
206C22—6282no—
207C22#1909779912yes—
208C22C22×D62811yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—3144no—
214C4—6282no—
215C4—6282no—
216C4#1909780412yes—
217C4#41×D15712yes—
218C2—22no—
219C2—12562no—
220C2#2375778511yescharacteristic
221C2C2×#1909779911yescharacteristic
222C2C2×#1909780411yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#432×C157C2
socleC22×C157C22×D4
prime radical#23757823C2
hypercenter#432D157
hypocenterC157#3112
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic314C2semidirect
3. #23757587C2semidirect
4. #23757569C2semidirect
5. #23757474C2semidirect
6. #23757414C2semidirect
7. #23757375C2semidirect
8. #432×C157C2semidirect
68. Dic314#41semidirect
111. C157#3112semidirect
120. #432D157semidirect

Maximal subgroups

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

subgroupindex
#237578232
C2×C4×Dic3142
#237575872
#237575692
#237574742
#237574142
#237573752
#432×C1572
#3112157