Finite Group Databasebeta

← #34111583

Subgroups — #34111583

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#23757885C211yesmaximal, characteristic, complemented
2#23757878C211yesmaximal, characteristic
3#23757581C212yesmaximal, complemented
4#23757574C212yesmaximal, complemented
5#23757519C212yesmaximal, complemented
6#23757502C212yesmaximal, complemented
7#23757479C212yesmaximal, complemented
8#23757474C212yesmaximal, complemented
9#424×C157C211yesmaximal, characteristic, complemented
10C2×#14578943C2212yes—
11C2×#14578943C2212yes—
12#19097771C2211yescharacteristic
13#19097768C2211yescharacteristic
14#19097764—22no—
15#19097763C2211yescharacteristic
16#19097763C2211yescharacteristic
17#19097761—22no—
18#19097756—22no—
19#19097750C2212yes—
20#19097745C2212yes—
21#19097744—22no—
22C4×Dic314C2212yes—
23C4×D628C2212yes—
24#19097705C2212yes—
25#19097703C2212yes—
26#19097696C2212yes—
27#19097696C2212yes—
28#19097693C2212yes—
29#19097693C2212yes—
30C2×M16×C157C2211yescharacteristic
31C22×C8×C157C2211yescharacteristic
32#117×C157C2211yescharacteristic
33C4⋊C8×C157C2212yes—
34C4⋊C8×C157C2212yes—
35C2×Dic314C2312yes—
36C2×D628—22no—
37C2×D628C2312yes—
38C2×C22⋊D157—22no—
39C2×C4×D157—22no—
40#14578943—42no—
41#14578943—42no—
42C4⋊Dic157—22no—
43C4⋊Dic157C2312yes—
44C4⋊Dic157C2312yes—
45#14578933—22no—
46#14578933—22no—
47#14578933—22no—
48#14578933—22no—
49C22⋊Dic157—22no—
50#14578931—22no—
51#14578931—22no—
52C4×Dic157—22no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—41no—
55C2×C8×C157—41no—
56C2×C8×C157C2312yes—
57C2×C8×C157C2312yes—
58C4⋊C4×C157D412yes—
59C22⋊C4×C157D412yes—
60C42×C157C2311yescharacteristic
61C42×C157C2311yescharacteristic
62C22×D157—22no—
63Dic314—22no—
64Dic314—22no—
65D628—22no—
66D628—22no—
67D628—82no—
68C22⋊D157—82no—
69C2×Dic157—22no—
70C2×Dic157—22no—
71C2×Dic157—22no—
72C4×D157—82no—
73C23×C157C2×D411yescharacteristic
74C2×C4×C157—21no—
75C2×C4×C157—21no—
76C2×C4×C157C2×D412yes—
77C2×C4×C157C2×D412yes—
78C2×C4×C157C2×D411yescharacteristic
79C2×C4×C157C2411yescharacteristic
80C8×C157—22no—
81C8×C157—22no—
82C22×C157—21no—
83C22×C157—22no—
84C22×C157C22×D411yescharacteristic
85C2×D157—42no—
86C2×D157—42no—
87C2×D157—42no—
88C4×C157—21no—
89C4×C157—22no—
90C4×C157—22no—
91C4×C157H(5,𝔽2)11yescharacteristic
92C4×C1572−1+411yescharacteristic
93Dic157—42no—
94Dic157—42no—
95Dic157—42no—
96C2×C157—41no—
97C2×C157#53711yescharacteristic
98C2×C157#57712yes—
99D157—82no—
100C157#326011yescharacteristic, complemented, Sylow
101#3260—1571nomaximal, Sylow
102#541—1571no—
103#540—1571no—
104#479—1572no—
105#478—1572no—
106#468—1572no—
107#465—1572no—
108#461—1572no—
109#460—1572no—
110#424D15711yescharacteristic, complemented
111C2×SD16—1572no—
112C2×SD16—1572no—
113C2×M16C2×D15711yescharacteristic
114C22×C8C2×D15711yescharacteristic
115C4⋊Q8—1571no—
116#127—1571no—
117#126—3142no—
118#125—3142no—
119#125—1571no—
120#125—1571no—
121#123—3142no—
122C22⋊Q8—1572no—
123#121—3142no—
124#121—1572no—
125C4×Q8—1572no—
126C4×D4—1572no—
127#117C2×D15711yescharacteristic
128#107—1572no—
129#106—1572no—
130C4⋊C8C2×D15712yes—
131C4⋊C8C2×D15712yes—
132#103—1572no—
133#103—1572no—
134#102—1572no—
135#102—1572no—
136C2×Q8—1572no—
137C2×D4—3142no—
138C2×D4—3142no—
139C2×D4—1572no—
140C22×C4—3142no—
141C22×C4C22×D15711yescharacteristic
142SD16—6282no—
143SD16—6282no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D15712yes—
147C2×C8C22×D15712yes—
148C4⋊C4—3142no—
149C4⋊C4—3142no—
150C4⋊C4—3142no—
151C4⋊C4—3142no—
152C4⋊C4—3142no—
153C4⋊C4—1572no—
154C4⋊C4—1572no—
155C4⋊C4D62812yes—
156C22⋊C4—3142no—
157C22⋊C4—3142no—
158C22⋊C4—3142no—
159C22⋊C4D62812yes—
160C42—3142no—
161C42C22×D15711yescharacteristic
162C42C22×D15711yescharacteristic
163C23—3142no—
164C23C2×D62811yescharacteristic
165Q8—3142no—
166Q8—3142no—
167D4—3142no—
168D4—3142no—
169D4—12562no—
170D4—12562no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—3142no—
174C2×C4—3142no—
175C2×C4—3142no—
176C2×C4—12562no—
177C2×C4C2×D62812yes—
178C2×C4C2×D62812yes—
179C2×C4C2×D62811yescharacteristic
180C2×C4C23×D15711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—6282no—
186C22—6282no—
187C22—6282no—
188C22C22×D62811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—6282no—
193C4—6282no—
194C4—6282no—
195C4#1909781611yescharacteristic
196C4#1909781911yescharacteristic
197C2—41no—
198C2—12562no—
199C2#2375785411yescharacteristic
200C2#2375808512yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#424×C157C2
socleC22×C157C22×D4
prime radical#23757878C2
hypercenter#424D157
hypocenterC157#3260
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. #23757885C2semidirect
3. #23757581C2semidirect
4. #23757574C2semidirect
5. #23757519C2semidirect
6. #23757502C2semidirect
7. #23757479C2semidirect
8. #23757474C2semidirect
9. #424×C157C2semidirect
100. C157#3260semidirect
110. #424D157semidirect

Maximal subgroups

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

subgroupindex
#237578852
#237578782
#237575812
#237575742
#237575192
#237575022
#237574792
#237574742
#424×C1572
#3260157