Finite Group Databasebeta

← #34111577

Subgroups — #34111577

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#23757878C211yesmaximal, characteristic
2#23757580C212yesmaximal, complemented
3#23757570C212yesmaximal, complemented
4#23757521C212yesmaximal, complemented
5#23757504C212yesmaximal, complemented
6#23757475C212yesmaximal, complemented
7#23757473C212yesmaximal, complemented
8#23757300C211yesmaximal, characteristic, complemented
9#541×C157C211yesmaximal, characteristic, complemented
10C2×#14578941C2212yes—
11C2×#14578941C2212yes—
12C2×#14578938C2211yescharacteristic
13C22×#11929434C2211yescharacteristic
14#19097768C2211yescharacteristic
15#19097763C2211yescharacteristic
16#19097756—22no—
17#19097745C2212yes—
18#19097744—22no—
19C4×D628C2212yes—
20#19097704C2212yes—
21#19097702C2212yes—
22C4⋊#11929434C2212yes—
23C4⋊#11929434C2212yes—
24#19097695C2212yes—
25#19097695C2212yes—
26#19097691C2212yes—
27#19097691C2212yes—
28C4⋊Q8×C157C2211yescharacteristic
29#126×C157—22no—
30#125×C157—22no—
31#125×C157C2211yescharacteristic
32C22⋊Q8×C157C2212yes—
33C4×Q8×C157C2212yes—
34#117×C157C2211yescharacteristic
35C2×D628—22no—
36C2×D628C2312yes—
37C2×C22⋊D157—22no—
38C2×C4×D157—22no—
39#14578941—42no—
40#14578941—42no—
41#14578938—41no—
42C2×#11929434—41no—
43C2×#11929434C2312yes—
44C2×#11929434C2312yes—
45C4⋊Dic157C2312yes—
46#14578933—22no—
47#14578931—22no—
48#14578931—22no—
49C2×Q8×C157C2312yes—
50C22×C4×C157C2311yescharacteristic
51C4⋊C4×C157—22no—
52C4⋊C4×C157—22no—
53C4⋊C4×C157—22no—
54C4⋊C4×C157—22no—
55C4⋊C4×C157D412yes—
56C4⋊C4×C157C2312yes—
57C22⋊C4×C157—22no—
58C22⋊C4×C157D412yes—
59C42×C157—22no—
60C42×C157C2311yescharacteristic
61C42×C157C2311yescharacteristic
62C22×D157—22no—
63D628—22no—
64D628—22no—
65D628—82no—
66C22⋊D157—82no—
67C2×Dic157—22no—
68C4×D157—82no—
69#11929434—22no—
70#11929434—22no—
71C23×C157C2×D411yescharacteristic
72Q8×C157—22no—
73Q8×C157—22no—
74C2×C4×C157—21no—
75C2×C4×C157—21no—
76C2×C4×C157—22no—
77C2×C4×C157—22no—
78C2×C4×C157C2×D412yes—
79C2×C4×C157C2×D412yes—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2411yescharacteristic
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×C157—42no—
92C4×C157—42no—
93C4×C157H(5,𝔽2)11yescharacteristic
94C4×C1572−1+411yescharacteristic
95Dic157—42no—
96C2×C157—41no—
97C2×C157#53711yescharacteristic
98C2×C157#57712yes—
99D157—82no—
100C157#326011yescharacteristic, complemented, Sylow
101#3260—1571nomaximal, Sylow
102#541D15711yescharacteristic, complemented
103#540—1571no—
104#479—1572no—
105#478—1572no—
106#468—1572no—
107#465—1572no—
108#461—1572no—
109#460—1572no—
110#424—1571no—
111C2×SD16—1572no—
112C2×SD16—1572no—
113C2×M16—1571no—
114C22×C8—1571no—
115C4⋊Q8C2×D15711yescharacteristic
116#127—1571no—
117#126—22no—
118#125—22no—
119#125—1571no—
120#125C2×D15711yescharacteristic
121#123—3142no—
122C22⋊Q8C2×D15712yes—
123#121—3142no—
124#121—1572no—
125C4×Q8C2×D15712yes—
126C4×D4—1572no—
127#117C2×D15711yescharacteristic
128#107—1572no—
129#106—1572no—
130C4⋊C8—1572no—
131C4⋊C8—1572no—
132#103—1572no—
133#103—1572no—
134#102—1572no—
135#102—1572no—
136C2×Q8C22×D15712yes—
137C2×D4—3142no—
138C2×D4—3142no—
139C2×D4—1572no—
140C22×C4—3142no—
141C22×C4C22×D15711yescharacteristic
142SD16—6282no—
143SD16—6282no—
144M16—6281no—
145C2×C8—6281no—
146C2×C8—1572no—
147C2×C8—1572no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—3142no—
153C4⋊C4—1572no—
154C4⋊C4C22⋊D15712yes—
155C4⋊C4C22×D15712yes—
156C22⋊C4—22no—
157C22⋊C4—3142no—
158C22⋊C4—3142no—
159C22⋊C4C22⋊D15712yes—
160C42—22no—
161C42C22×D15711yescharacteristic
162C42C22×D15711yescharacteristic
163C23—3142no—
164C23C2×C22⋊D15711yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—3142no—
168D4—3142no—
169D4—12562no—
170D4—12562no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—3142no—
176C2×C4—12562no—
177C2×C4C2×C22⋊D15712yes—
178C2×C4C2×C22⋊D15712yes—
179C2×C4C2×C22⋊D15711yescharacteristic
180C2×C4C23×D15711yescharacteristic
181C8—3142no—
182C8—3142no—
183C22—22no—
184C22—21no—
185C22—6282no—
186C22—6282no—
187C22—6282no—
188C22C22×C22⋊D15711yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—6282no—
195C4#1909781711yescharacteristic
196C4#1909781811yescharacteristic
197C2—41no—
198C2—12562no—
199C2#2375785111yescharacteristic
200C2#2375808112yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#541×C157C2
socleC22×C157C22×D4
prime radical#23757878C2
hypercenter#541D157
hypocenterC157#3260
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #23757580C2semidirect
3. #23757570C2semidirect
4. #23757521C2semidirect
5. #23757504C2semidirect
6. #23757475C2semidirect
7. #23757473C2semidirect
8. #23757300C2semidirect
9. #541×C157C2semidirect
100. C157#3260semidirect
102. #541D157semidirect

Maximal subgroups

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

subgroupindex
#237578782
#237575802
#237575702
#237575212
#237575042
#237574752
#237574732
#237573002
#541×C1572
#3260157