Finite Group Databasebeta

← #34114276

Subgroups — #34114276

Order 20096.

1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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#23758023C213yesmaximal, complemented
2#23758017C216yesmaximal, complemented
3#23758012C216yesmaximal, complemented
4#23757822C213yesmaximal, complemented
5#23757802C216yesmaximal, complemented
6#23757779C216yesmaximal, complemented
7H(3,𝔽4)×C157C211yesmaximal, characteristic, complemented
8C2×#14578955C2211yescharacteristic, complemented
9C2×#14578953C2213yes—
10C23×Dic157C2212yescomplemented
11#19097769C2213yescomplemented
12#19097767C2213yes—
13#19097764C22112yescomplemented
14#19097763C2216yescomplemented
15#19097759C2213yescomplemented
16#19097755C2216yescomplemented
17#19097753C22112yescomplemented
18#19097752C2216yes—
19#19097748C22112yescomplemented
20#19097747C2216yescomplemented
21#19097744C22112yescomplemented
22#19097743C22112yes—
23#19097742C2216yes—
24#19097740C2216yescomplemented
25Q8×Dic157C2212yescomplemented
26D4×Dic157C2216yes—
27#19097731C22112yescomplemented
28#19097727C2216yescomplemented
29C42×D157C2213yescomplemented
30#124×C157C2216yes—
31#124×C157C2213yes—
32#120×C157C2216yes—
33#14578955—42no—
34#14578953—46no—
35C2×Dic314C2313yes—
36C2×D628C2313yes—
37C2×C22⋊D157C23112yes—
38C2×C22⋊D157C2316yes—
39C22×Dic157—44no—
40C22×Dic157C2316yes—
41C22×Dic157C2316yes—
42C2×C4×D157C2313yes—
43C2×C4×D157C2316yes—
44C4⋊Dic157C2316yes—
45C4⋊Dic157C2316yes—
46#14578933C23112yes—
47#14578933C23112yes—
48C22⋊Dic157C2316yes—
49C22⋊Dic157C23112yes—
50#14578931C23112yes—
51C4×Dic157C2316yes—
52C4×Dic157C2313yes—
53C24×C157C2312yes—
54C2×Q8×C157C2312yes—
55C2×Q8×C157C2311yescharacteristic
56C2×D4×C157C2316yes—
57C2×D4×C157C2313yes—
58C22⋊C4×C157C2316yes—
59C22⋊C4×C157C23112yes—
60C42×C157C2313yes—
61C22×D157C2413yes—
62Dic314—43no—
63D628—43no—
64C22⋊D157—412no—
65C22⋊D157—46no—
66C2×Dic157—212no—
67C2×Dic157—46no—
68C2×Dic157C2416yes—
69C2×Dic157C2416yes—
70C2×Dic157C2411yescharacteristic
71C4×D157—26no—
72C4×D157—46no—
73C23×C157—42no—
74C23×C157—44no—
75C23×C157C2416yes—
76Q8×C157—41no—
77Q8×C157—42no—
78D4×C157—43no—
79D4×C157—46no—
80C2×C4×C157C2413yes—
81C2×C4×C157C2416yes—
82C22×C157—26no—
83C22×C157—212no—
84C22×C157—48no—
85C22×C157C2511yescharacteristic
86C2×D157—23no—
87C2×D157—26no—
88C4×C157—23no—
89C4×C157—26no—
90Dic157—26no—
91Dic157—26no—
92Dic157H(5,𝔽2)12yes—
93C2×C157—46no—
94C2×C157C2×H(5,𝔽2)11yescharacteristic
95C2×C157#58512yes—
96D157—43no—
97C157#352911yescharacteristic, complemented, Sylow
98#3529—1571nomaximal, Sylow
99#563—1573no—
100#562—1576no—
101H(3,𝔽4)D15711yescharacteristic, complemented
102#560—1576no—
103#532—1573no—
104#529—1576no—
105#525—1576no—
106C2×#41—1571no—
107C2×#41—1573no—
108C23×C4—1572no—
109C4⋊Q8—1573no—
110#127—1573no—
111#126—15712no—
112#125—1576no—
113#124—1573no—
114#124C2×D15716yes—
115#124C2×D15713yes—
116#123—1576no—
117#123—1576no—
118#123—15712no—
119C22⋊Q8—1576no—
120C22⋊Q8—15712no—
121#121—15712no—
122#121—1576no—
123#121—15712no—
124#120—1576no—
125#120C2×D15716yes—
126C4×Q8—1572no—
127C4×D4—1576no—
128C4×D4—15712no—
129#117—1576no—
130C2×C42—1573no—
131C24C22×D15712yes—
132#41—6286no—
133#41—6282no—
134C2×Q8—1573no—
135C2×Q8C22×D15712yes—
136C2×Q8C22×D15711yescharacteristic
137C2×D4—15712no—
138C2×D4—1576no—
139C2×D4—1573no—
140C2×D4C22×D15716yes—
141C2×D4C22×D15713yes—
142C22×C4—6284no—
143C22×C4—1576no—
144C22×C4—1573no—
145C22×C4—1576no—
146C22×C4—1576no—
147C4⋊C4—15712no—
148C4⋊C4—15712no—
149C4⋊C4—1576no—
150C4⋊C4—1576no—
151C22⋊C4—1576no—
152C22⋊C4—15712no—
153C22⋊C4—15712no—
154C22⋊C4C22×D15716yes—
155C22⋊C4C22×D157112yes—
156C42—1576no—
157C42—1573no—
158C42C22×D15713yes—
159C23—42no—
160C23—44no—
161C23—1573no—
162C23C23×D15716yes—
163Q8—41no—
164Q8—42no—
165Q8—6283no—
166D4—43no—
167D4—46no—
168D4—6283no—
169D4—6286no—
170D4—62812no—
171C2×C4—3146no—
172C2×C4—31412no—
173C2×C4—6286no—
174C2×C4—6286no—
175C2×C4—1576no—
176C2×C4—1576no—
177C2×C4—1571no—
178C2×C4C23×D15716yes—
179C2×C4C23×D15713yes—
180C22—26no—
181C22—212no—
182C22—48no—
183C22—3143no—
184C22—3146no—
185C22C24×D15711yescharacteristic
186C4—23no—
187C4—26no—
188C4—3146no—
189C4—3146no—
190C4—1572no—
191C2—46no—
192C2—6283no—
193C2H(5,𝔽2)×D15711yescharacteristic
194C2#2375810812yes—
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
FittingH(3,𝔽4)×C157C2
hypercenterH(3,𝔽4)D157
hypocenterC157#3529
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #23758023C2semidirect
2. #23758017C2semidirect
3. #23758012C2semidirect
4. #23757822C2semidirect
5. #23757802C2semidirect
6. #23757779C2semidirect
7. H(3,𝔽4)×C157C2semidirect
8. C2×#14578955C22semidirect
10. C23×Dic157C22semidirect
11. #19097769C22semidirect
13. #19097764C22semidirect
14. #19097763C22semidirect
15. #19097759C22semidirect
16. #19097755C22semidirect
17. #19097753C22semidirect
19. #19097748C22semidirect
20. #19097747C22semidirect
21. #19097744C22semidirect
24. #19097740C22semidirect
25. Q8×Dic157C22semidirect
27. #19097731C22semidirect
28. #19097727C22semidirect
29. C42×D157C22semidirect
97. C157#3529semidirect
101. H(3,𝔽4)D157semidirect

Maximal subgroups

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

subgroupindex
#237580232
#237580172
#237580122
#237578222
#237578022
#237577792
H(3,𝔽4)×C1572
#3529157