Finite Group Databasebeta

← #34109667

Subgroups — #34109667

Order 20096.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#19097804C211yesmaximal, characteristic
2C22×Dic628C211yesmaximal, characteristic, complemented
3#23757849C211yesmaximal, characteristic
4#23757750C211yesmaximal, characteristic, complemented
5#23757496C214yesmaximal, complemented
6#23757484C214yesmaximal, complemented
7#23757283C211yesmaximal, characteristic, complemented
8#23757270C211yesmaximal, characteristic
9#424×C157C211yesmaximal, characteristic, complemented
10C2×#14578954C2211yescharacteristic
11C22×Dic314C2211yescharacteristic
12C22×Dic314C2211yescharacteristic
13#19097804—24no—
14C2×Dic628—24no—
15C2×Dic628C2212yes—
16C2×Dic628C2214yes—
17C2×#14578943C2212yes—
18#19097771C2212yes—
19#19097759C2212yes—
20#19097755—22no—
21#19097751—22no—
22C4×Dic314C2214yes—
23#19097736—22no—
24#19097730C2211yescharacteristic
25#19097727—22no—
26#19097696C2212yes—
27#19097696C2214yes—
28#19097693C2212yes—
29C2×M16×C157C2211yescharacteristic
30C22×C8×C157C2211yescharacteristic
31#117×C157C2211yescharacteristic
32C4⋊C8×C157C2214yes—
33#14578954—22no—
34#14578954—22no—
35C2×Dic314—21no—
36C2×Dic314—21no—
37C2×Dic314—22no—
38C2×Dic314—22no—
39C2×Dic314—22no—
40C2×Dic314D414yes—
41C2×Dic314C2312yes—
42C2×Dic314C2311yescharacteristic
43C2×Dic314C2312yes—
44C2×D628C2311yescharacteristic
45C2×C22⋊D157—21no—
46C22×Dic157—21no—
47C22×Dic157—21no—
48C2×C4×D157—21no—
49Dic628—28no—
50Dic628—24no—
51#14578943—24no—
52C4⋊Dic157—24no—
53C4⋊Dic157C2312yes—
54#14578933—24no—
55C22⋊Dic157—21no—
56#14578931—24no—
57C4×Dic157—24no—
58C4×Dic157—21no—
59C22×C4×C157C2311yescharacteristic
60M16×C157—22no—
61C2×C8×C157—22no—
62C2×C8×C157C2312yes—
63C2×C8×C157C2312yes—
64C4⋊C4×C157D412yes—
65C22⋊C4×C157D412yes—
66C42×C157C2312yes—
67C22×D157—21no—
68Dic314—24no—
69Dic314—22no—
70Dic314—22no—
71Dic314—21no—
72Dic314—22no—
73Dic314—21no—
74Dic314—42no—
75Dic314C2×D414yes—
76D628—21no—
77D628—21no—
78C22⋊D157—42no—
79C2×Dic157—24no—
80C2×Dic157—22no—
81C2×Dic157—21no—
82C2×Dic157—22no—
83C2×Dic157—22no—
84C2×Dic157—42no—
85C4×D157—42no—
86C23×C157C2×D411yescharacteristic
87C2×C4×C157C2×D412yes—
88C2×C4×C157C2×D414yes—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157#4112yes—
91C2×C4×C157C2411yescharacteristic
92C8×C157—22no—
93C8×C157—22no—
94C22×C157—22no—
95C22×C157#12112yes—
96C22×C157C22×D411yescharacteristic
97C2×D157—41no—
98C2×D157—42no—
99C4×C157—24no—
100C4×C157#12112yes—
101C4×C157C22×D411yescharacteristic
102C4×C157C2×#4111yescharacteristic
103Dic157—24no—
104Dic157—41no—
105Dic157—42no—
106Dic157—42no—
107C2×C157—22no—
108C2×C157C2×#12111yescharacteristic
109C2×C157#57812yes—
110D157—81no—
111C157#304411yescharacteristic, complemented, Sylow
112#3044—1571nomaximal, Sylow
113C2×#137—1571no—
114C22×Q16—1571no—
115#536—1571no—
116#519—1571no—
117#464—1574no—
118#462—1574no—
119#424D15711yescharacteristic, complemented
120#419—1571no—
121#416—1571no—
122C2×#41—1571no—
123C22×Q8—1571no—
124C22×Q8—1571no—
125#137—3144no—
126C2×Q16—3144no—
127C2×Q16—1574no—
128C2×Q16—1572no—
129C2×SD16—1572no—
130C2×M16C2×D15711yescharacteristic
131C22×C8C2×D15711yescharacteristic
132C4⋊Q8—1572no—
133#124—1572no—
134#123—3142no—
135C22⋊Q8—3142no—
136C4×Q8—3142no—
137C4×Q8—1574no—
138#117—3142no—
139#117—1571no—
140#117C2×D15711yescharacteristic
141C4⋊C8C2×D15714yes—
142#103—1572no—
143#103—1574no—
144#102—1572no—
145#41—3142no—
146#41—3142no—
147C2×Q8—3141no—
148C2×Q8—3141no—
149C2×Q8—3142no—
150C2×Q8—3142no—
151C2×Q8—3142no—
152C2×Q8—1574no—
153C2×Q8—1572no—
154C2×Q8—1571no—
155C2×Q8—1572no—
156C2×D4—3141no—
157C2×D4—1571no—
158C22×C4—3141no—
159C22×C4—3141no—
160C22×C4—3141no—
161C22×C4C22×D15711yescharacteristic
162Q16—3148no—
163Q16—3144no—
164SD16—3144no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D15712yes—
168C2×C8C22×D15712yes—
169C4⋊C4—3144no—
170C4⋊C4—3144no—
171C4⋊C4—1572no—
172C4⋊C4D62812yes—
173C22⋊C4—3141no—
174C22⋊C4—3144no—
175C22⋊C4D62812yes—
176C42—3144no—
177C42—3141no—
178C42C22×D15712yes—
179C23—3141no—
180C23C2×D62811yescharacteristic
181Q8—3144no—
182Q8—3142no—
183Q8—3142no—
184Q8—3141no—
185Q8—3142no—
186Q8—3141no—
187Q8—6282no—
188Q8—1574no—
189D4—3141no—
190D4—3141no—
191D4—6282no—
192C2×C4—3144no—
193C2×C4—3142no—
194C2×C4—3142no—
195C2×C4—3142no—
196C2×C4—3141no—
197C2×C4—6282no—
198C2×C4—6282no—
199C2×C4D4×D15712yes—
200C2×C4C2×D62814yes—
201C2×C4C2×D62811yescharacteristic
202C2×C4#1457895512yes—
203C2×C4C23×D15711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—6282no—
208C22—6281no—
209C22#1909774612yes—
210C22C22×D62811yescharacteristic
211C4—24no—
212C4—3144no—
213C4—6282no—
214C4—6281no—
215C4—6282no—
216C4#1909774612yes—
217C4C2×D4×D15711yescharacteristic
218C4C2×#1457895511yescharacteristic
219C2—22no—
220C2—12561no—
221C2C2×#1909774611yescharacteristic
222C2#2375809012yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#424×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578954C22
hypercenter#424D157
hypocenterC157#3044
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic628C2semidirect
4. #23757750C2semidirect
5. #23757496C2semidirect
6. #23757484C2semidirect
7. #23757283C2semidirect
9. #424×C157C2semidirect
111. C157#3044semidirect
119. #424D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C22×Dic6282
#237578492
#237577502
#237574962
#237574842
#237572832
#237572702
#424×C1572
#3044157