Finite Group Databasebeta

← #34110427

Subgroups — #34110427

Order 20096.

382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757878C211yesmaximal, characteristic
2#23757597C212yesmaximal, complemented
3#23757580C212yesmaximal, complemented
4#23757473C212yesmaximal, complemented
5#23757412C212yesmaximal, complemented
6#23757371C212yesmaximal, complemented
7C4×#14578941C212yesmaximal, complemented
8#23757334C211yesmaximal, characteristic, complemented
9#519×C157C211yesmaximal, characteristic, complemented
10C2×#14578941C2212yes—
11C2×#14578941C2212yes—
12#19097768C2211yescharacteristic
13#19097763C2211yescharacteristic
14#19097756—22no—
15#19097745C2212yes—
16#19097744—22no—
17C4×D628C2212yes—
18#19097704C2212yes—
19#19097702C2212yes—
20C4⋊#11929434C2212yes—
21#19097695C2212yes—
22#19097695C2212yes—
23#19097691C2212yes—
24#19097691C2212yes—
25C22⋊#11929434C2212yes—
26#19097682C2211yescharacteristic
27C4×#11929434C2211yescharacteristic
28C22×Q8×C157C2211yescharacteristic
29C4×Q8×C157—22no—
30C4×Q8×C157C2212yes—
31C4×Q8×C157C2212yes—
32#117×C157—22no—
33#117×C157C2211yescharacteristic
34#117×C157C2211yescharacteristic
35C2×D628—22no—
36C2×D628C2312yes—
37C2×C22⋊D157—22no—
38C2×C4×D157—22no—
39#14578941—24no—
40#14578941—24no—
41C2×#11929434C2312yes—
42C2×#11929434C2312yes—
43C4⋊Dic157C2312yes—
44#14578933—22no—
45#14578931—22no—
46#14578931—22no—
47C2×Q8×C157—24no—
48C2×Q8×C157—21no—
49C2×Q8×C157C2312yes—
50C22×C4×C157—21no—
51C22×C4×C157C2311yescharacteristic
52C4⋊C4×C157—22no—
53C4⋊C4×C157—22no—
54C4⋊C4×C157D412yes—
55C4⋊C4×C157C2312yes—
56C22⋊C4×C157—21no—
57C22⋊C4×C157D412yes—
58C42×C157—22no—
59C42×C157—22no—
60C42×C157—21no—
61C42×C157C2311yescharacteristic
62C42×C157C2311yescharacteristic
63C22×D157—22no—
64D628—22no—
65D628—22no—
66D628—82no—
67C22⋊D157—82no—
68C2×Dic157—22no—
69C4×D157—82no—
70#11929434—22no—
71#11929434—22no—
72C23×C157C2×D411yescharacteristic
73Q8×C157—24no—
74Q8×C157—22no—
75Q8×C157#4114yes—
76C2×C4×C157—24no—
77C2×C4×C157—22no—
78C2×C4×C157—22no—
79C2×C4×C157—22no—
80C2×C4×C157C2×D412yes—
81C2×C4×C157C2×D412yes—
82C2×C4×C157C2×D411yescharacteristic
83C2×C4×C157C2411yescharacteristic
84C22×C157—21no—
85C22×C157—22no—
86C22×C157C22×D411yescharacteristic
87C2×D157—42no—
88C2×D157—42no—
89C2×D157—42no—
90C4×C157—24no—
91C4×C157—21no—
92C4×C157—22no—
93C4×C157—22no—
94C4×C157—42no—
95C4×C157C2×#4112yes—
96Dic157—42no—
97C2×C157—41no—
98C2×C157#52511yescharacteristic
99C2×C157#57712yes—
100D157—82no—
101C157#312611yescharacteristic, complemented, Sylow
102#3126—1571nomaximal, Sylow
103#540—1571no—
104#519D15711yescharacteristic, complemented
105#482—1572no—
106#479—1572no—
107#460—1572no—
108#448—1572no—
109#440—1572no—
110C4×SD16—1572no—
111#433—1571no—
112C22×Q8C2×D15711yescharacteristic
113C2×SD16—1572no—
114C2×SD16—1572no—
115#127—1571no—
116#125—1571no—
117#123—3142no—
118#121—3142no—
119#121—1572no—
120C4×Q8—22no—
121C4×Q8C2×D15712yes—
122C4×Q8C2×D15712yes—
123C4×D4—1572no—
124#117—22no—
125#117C2×D15711yescharacteristic
126#117C2×D15711yescharacteristic
127#107—1572no—
128#106—1572no—
129C4⋊C8—1572no—
130#103—1572no—
131#103—1572no—
132#102—1572no—
133#102—1572no—
134C22⋊C8—1572no—
135#97—1571no—
136C4×C8—1571no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D15712yes—
140C2×D4—3142no—
141C2×D4—3142no—
142C2×D4—1572no—
143C22×C4—21no—
144C22×C4—3142no—
145C22×C4C22×D15711yescharacteristic
146SD16—3144no—
147SD16—3144no—
148C2×C8—1572no—
149C2×C8—1572no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3142no—
153C4⋊C4—1572no—
154C4⋊C4C22⋊D15712yes—
155C4⋊C4C22×D15712yes—
156C22⋊C4—21no—
157C22⋊C4—3142no—
158C22⋊C4—3142no—
159C22⋊C4C22⋊D15712yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D15711yescharacteristic
164C42C22×D15711yescharacteristic
165C23—3142no—
166C23C2×C22⋊D15711yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1457895414yes—
170D4—3142no—
171D4—3142no—
172D4—12562no—
173D4—12562no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3142no—
179C2×C4—12562no—
180C2×C4C2×C22⋊D15712yes—
181C2×C4C2×C22⋊D15712yes—
182C2×C4C2×C22⋊D15711yescharacteristic
183C2×C4C23×D15711yescharacteristic
184C8—3142no—
185C8—3142no—
186C22—22no—
187C22—21no—
188C22—6282no—
189C22—6282no—
190C22—6282no—
191C22C22×C22⋊D15711yescharacteristic
192C4—22no—
193C4—22no—
194C4—24no—
195C4—21no—
196C4—42no—
197C4—6282no—
198C4C2×#1457895412yes—
199C2—41no—
200C2—12562no—
201C2#2375778411yescharacteristic
202C2#2375808112yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#519×C157C2
socleC22×C157C22×D4
prime radical#23757878C2
hypercenter#519D157
hypocenterC157#3126
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #23757597C2semidirect
3. #23757580C2semidirect
4. #23757473C2semidirect
5. #23757412C2semidirect
6. #23757371C2semidirect
7. C4×#14578941C2semidirect
8. #23757334C2semidirect
9. #519×C157C2semidirect
101. C157#3126semidirect
104. #519D157semidirect

Maximal subgroups

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

subgroupindex
#237578782
#237575972
#237575802
#237574732
#237574122
#237573712
C4×#145789412
#237573342
#519×C1572
#3126157