Finite Group Databasebeta

← #34111598

Subgroups — #34111598

Order 20096.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#23757896C212yesmaximal, complemented
2#23757566C212yesmaximal, complemented
3#23757552C212yesmaximal, complemented
4#23757521C212yesmaximal, complemented
5#23757509C211yesmaximal, characteristic
6#23757497C212yesmaximal, complemented
7#23757491C212yesmaximal, complemented
8#23757299C211yesmaximal, characteristic, complemented
9#467×C157C211yesmaximal, characteristic, complemented
10C2×#14578941C2212yes—
11C2×D1256C2211yescharacteristic
12C2×#14578938C2211yescharacteristic
13#19097770C2212yes—
14#19097766—22no—
15#19097760C2212yes—
16#19097757—22no—
17#19097753—22no—
18#19097751—22no—
19#19097745C2212yes—
20Q8×Dic157C2212yes—
21#19097733C2212yes—
22#19097730C2211yescharacteristic
23#19097705C2211yescharacteristic
24#19097704C2212yes—
25#19097700C2212yes—
26#19097700C2212yes—
27#19097695C2212yes—
28#19097693C2212yes—
29#19097691C2212yes—
30C2×Q16×C157C2211yescharacteristic
31C22×C8×C157C2211yescharacteristic
32C22⋊Q8×C157C2212yes—
33#107×C157C2211yescharacteristic
34#103×C157C2212yes—
35C2×Dic314—22no—
36C2×D628C2312yes—
37C2×C22⋊D157—22no—
38C2×C4×D157—22no—
39#14578941—42no—
40D1256—41no—
41#14578938—41no—
42C2×#11929434C2312yes—
43C4⋊Dic157—22no—
44C4⋊Dic157D411yescharacteristic
45C4⋊Dic157D411yescharacteristic
46C4⋊Dic157C2312yes—
47#14578933—22no—
48C22⋊Dic157D412yes—
49#14578931—22no—
50#14578931—22no—
51#14578931—22no—
52#14578931—22no—
53C4×Dic157—22no—
54C4×Dic157C2312yes—
55C2×Q8×C157C2312yes—
56C22×C4×C157C2311yescharacteristic
57Q16×C157—41no—
58C2×C8×C157—41no—
59C2×C8×C157C2311yescharacteristic
60C2×C8×C157C2311yescharacteristic
61C4⋊C4×C157—22no—
62C4⋊C4×C157C2312yes—
63C22⋊C4×C157—22no—
64C22×D157—22no—
65Dic314—82no—
66D628—22no—
67D628—22no—
68C22⋊D157—82no—
69C2×Dic157—22no—
70C2×Dic157C2×D412yes—
71C2×Dic157C2×D412yes—
72C4×D157—82no—
73#11929434—22no—
74C23×C157C2×D411yescharacteristic
75Q8×C157—22no—
76Q8×C157—22no—
77C2×C4×C157—21no—
78C2×C4×C157—21no—
79C2×C4×C157—22no—
80C2×C4×C157—22no—
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2411yescharacteristic
83C8×C157—21no—
84C8×C157—21no—
85C22×C157—21no—
86C22×C157—21no—
87C22×C157—21no—
88C22×C157C22×D411yescharacteristic
89C2×D157—42no—
90C2×D157—42no—
91C2×D157—42no—
92C4×C157—21no—
93C4×C157—42no—
94C4×C157—42no—
95C4×C157H(5,𝔽2)11yescharacteristic
96C4×C1572−1+411yescharacteristic
97Dic157—22no—
98Dic157—22no—
99Dic157—42no—
100C2×C157—41no—
101C2×C157#53711yescharacteristic
102C2×C157#57611yescharacteristic
103C2×C157#57811yescharacteristic
104D157—82no—
105C157#326211yescharacteristic, complemented, Sylow
106#3262—1571nomaximal, Sylow
107#542—1572no—
108#477—1572no—
109#474—1572no—
110#468—1572no—
111#467D15711yescharacteristic, complemented
112#466—1571no—
113#464—1572no—
114#463—1572no—
115#424—1571no—
116C2×Q16C2×D15711yescharacteristic
117C2×SD16—1572no—
118C2×D8—1571no—
119C2×M16—1571no—
120C22×C8C2×D15711yescharacteristic
121C4⋊Q8—1572no—
122#126—3142no—
123#124—3142no—
124#124—1572no—
125#123—3142no—
126C22⋊Q8—3142no—
127C22⋊Q8C2×D15712yes—
128#121—1572no—
129C4×Q8—1572no—
130C4×D4—1572no—
131#117—1571no—
132#107—1571no—
133#107—1572no—
134#107C2×D15711yescharacteristic
135C4⋊C8—1572no—
136C4⋊C8—1572no—
137#103—1572no—
138#103C2×D15712yes—
139#102—1572no—
140#102—1572no—
141C2×Q8—3142no—
142C2×Q8C22×D15712yes—
143C2×D4—3142no—
144C2×D4—1572no—
145C22×C4—3142no—
146C22×C4C22×D15711yescharacteristic
147Q16—41no—
148SD16—6282no—
149D8—6281no—
150M16—6281no—
151C2×C8—41no—
152C2×C8—1572no—
153C2×C8C22×D15711yescharacteristic
154C2×C8C22×D15711yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—3142no—
157C4⋊C4—3142no—
158C4⋊C4—1572no—
159C4⋊C4—1571no—
160C4⋊C4—1571no—
161C4⋊C4C22×D15712yes—
162C22⋊C4—22no—
163C22⋊C4—3142no—
164C22⋊C4—3142no—
165C22⋊C4—3142no—
166C22⋊C4—3142no—
167C22⋊C4—1572no—
168C42—3142no—
169C42—1572no—
170C23—3142no—
171C23D4×D15711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—12562no—
175D4—3142no—
176D4—3142no—
177D4—12562no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—3142no—
183C2×C4—1572no—
184C2×C4—1572no—
185C2×C4—12562no—
186C2×C4D4×D15711yescharacteristic
187C2×C4C23×D15711yescharacteristic
188C8—21no—
189C8—21no—
190C8—3142no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—6282no—
195C22—6282no—
196C22—6282no—
197C22C2×D4×D15711yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—3142no—
202C4—3142no—
203C4—6282no—
204C4#1909781611yescharacteristic
205C4#1909781811yescharacteristic
206C2—41no—
207C2—12562no—
208C2#2375785011yescharacteristic
209C2#2375807511yescharacteristic
210C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#467×C157C2
socleC22×C157C22×D4
prime radical#23757509C2
hypercenter#467D157
hypocenterC157#3262
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #23757896C2semidirect
2. #23757566C2semidirect
3. #23757552C2semidirect
4. #23757521C2semidirect
6. #23757497C2semidirect
7. #23757491C2semidirect
8. #23757299C2semidirect
9. #467×C157C2semidirect
105. C157#3262semidirect
111. #467D157semidirect

Maximal subgroups

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

subgroupindex
#237578962
#237575662
#237575522
#237575212
#237575092
#237574972
#237574912
#237572992
#467×C1572
#3262157