Finite Group Databasebeta

← #34111597

Subgroups — #34111597

Order 20096.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#23757894C212yesmaximal, complemented
2#23757569C212yesmaximal, complemented
3#23757554C212yesmaximal, complemented
4#23757519C212yesmaximal, complemented
5#23757514C211yesmaximal, characteristic, complemented
6#23757509C211yesmaximal, characteristic
7#23757496C212yesmaximal, complemented
8#23757489C212yesmaximal, complemented
9#424×C157C211yesmaximal, characteristic, complemented
10C2×Dic628C2211yescharacteristic
11C2×#14578943C2212yes—
12C2×D1256C2211yescharacteristic
13#19097771C2212yes—
14#19097766—22no—
15#19097759C2212yes—
16#19097757—22no—
17#19097753—22no—
18#19097750C2212yes—
19#19097749—22no—
20#19097745C2212yes—
21C4×Dic314C2212yes—
22C4×D628C2212yes—
23#19097705C2211yescharacteristic
24#19097705C2212yes—
25#19097705C2211yescharacteristic
26#19097696C2212yes—
27#19097696C2212yes—
28#19097693C2212yes—
29#19097693C2212yes—
30C2×M16×C157C2211yescharacteristic
31C22×C8×C157C2211yescharacteristic
32#117×C157C2211yescharacteristic
33C4⋊C8×C157C2212yes—
34C4⋊C8×C157C2212yes—
35C2×Dic314—22no—
36C2×Dic314C2312yes—
37C2×D628C2312yes—
38C2×C22⋊D157—22no—
39C2×C4×D157—22no—
40Dic628—41no—
41#14578943—42no—
42D1256—41no—
43C4⋊Dic157—22no—
44C4⋊Dic157C2312yes—
45C4⋊Dic157C2312yes—
46#14578933—22no—
47#14578933—22no—
48C22⋊Dic157—22no—
49#14578931—22no—
50#14578931—22no—
51#14578931—22no—
52#14578931—22no—
53C4×Dic157—22no—
54C22×C4×C157C2311yescharacteristic
55M16×C157—41no—
56C2×C8×C157—41no—
57C2×C8×C157C2311yescharacteristic
58C2×C8×C157C2311yescharacteristic
59C2×C8×C157C2312yes—
60C4⋊C4×C157D411yescharacteristic
61C4⋊C4×C157D411yescharacteristic
62C22⋊C4×C157D412yes—
63C42×C157C2312yes—
64C22×D157—22no—
65Dic314—22no—
66Dic314—22no—
67Dic314—82no—
68D628—22no—
69D628—22no—
70C22⋊D157—82no—
71C2×Dic157—22no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C4×D157—82no—
75C23×C157C2×D411yescharacteristic
76C2×C4×C157—21no—
77C2×C4×C157—21no—
78C2×C4×C157C2×D412yes—
79C2×C4×C157C2×D412yes—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2411yescharacteristic
82C8×C157—21no—
83C8×C157—21no—
84C8×C157—22no—
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—22no—
94C4×C157—22no—
95C4×C157H(5,𝔽2)11yescharacteristic
96C4×C1572−1+411yescharacteristic
97Dic157—42no—
98Dic157—42no—
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#467—1571no—
112#466—1571no—
113#464—1572no—
114#463—1572no—
115#424D15711yescharacteristic, complemented
116C2×Q16—1571no—
117C2×SD16—1572no—
118C2×D8—1571no—
119C2×M16C2×D15711yescharacteristic
120C22×C8C2×D15711yescharacteristic
121C4⋊Q8—1572no—
122#126—3142no—
123#124—3142no—
124#124—1572no—
125#123—3142no—
126C22⋊Q8—3142no—
127C22⋊Q8—1572no—
128#121—1572no—
129C4×Q8—1572no—
130C4×D4—1572no—
131#117C2×D15711yescharacteristic
132#107—1571no—
133#107—1571no—
134#107—1572no—
135C4⋊C8C2×D15712yes—
136C4⋊C8C2×D15712yes—
137#103—1572no—
138#103—1572no—
139#102—1572no—
140#102—1572no—
141C2×Q8—3142no—
142C2×Q8—1572no—
143C2×D4—3142no—
144C2×D4—1572no—
145C22×C4—3142no—
146C22×C4C22×D15711yescharacteristic
147Q16—6281no—
148SD16—6282no—
149D8—6281no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D15711yescharacteristic
153C2×C8C22×D15712yes—
154C2×C8C22×D15711yescharacteristic
155C4⋊C4—3142no—
156C4⋊C4—3142no—
157C4⋊C4—3142no—
158C4⋊C4—1572no—
159C4⋊C4—1572no—
160C4⋊C4D62811yescharacteristic
161C4⋊C4D62811yescharacteristic
162C22⋊C4—3142no—
163C22⋊C4—3142no—
164C22⋊C4—3142no—
165C22⋊C4—3142no—
166C22⋊C4—3142no—
167C22⋊C4D62812yes—
168C42—3142no—
169C42C22×D15712yes—
170C23—3142no—
171C23C2×D62811yescharacteristic
172Q8—3142no—
173Q8—3142no—
174Q8—12562no—
175D4—3142no—
176D4—3142no—
177D4—12562no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—3142no—
181C2×C4—3142no—
182C2×C4—3142no—
183C2×C4—12562no—
184C2×C4C2×D62812yes—
185C2×C4C2×D62812yes—
186C2×C4C2×D62811yescharacteristic
187C2×C4C23×D15711yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—6282no—
195C22—6282no—
196C22—6282no—
197C22C22×D62811yescharacteristic
198C4—22no—
199C4—21no—
200C4—22no—
201C4—6282no—
202C4—6282no—
203C4—6282no—
204C4#1909781611yescharacteristic
205C4#1909781911yescharacteristic
206C2—41no—
207C2—12562no—
208C2#2375785411yescharacteristic
209C2#2375807811yescharacteristic
210C2#2375809011yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#424×C157C2
socleC22×C157C22×D4
prime radical#23757509C2
hypercenter#424D157
hypocenterC157#3262
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
1. #23757894C2semidirect
2. #23757569C2semidirect
3. #23757554C2semidirect
4. #23757519C2semidirect
5. #23757514C2semidirect
7. #23757496C2semidirect
8. #23757489C2semidirect
9. #424×C157C2semidirect
105. C157#3262semidirect
115. #424D157semidirect

Maximal subgroups

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

subgroupindex
#237578942
#237575692
#237575542
#237575192
#237575142
#237575092
#237574962
#237574892
#424×C1572
#3262157