Finite Group Databasebeta

← #34101306

Subgroups — #34101306

Order 20096.

244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 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#23757826C211yesmaximal, characteristic, complemented
2C2×#19097698C211yesmaximal, characteristic
3#23757247C211yesmaximal, characteristic
4#23757044C211yesmaximal, characteristic, complemented
5#23757034C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic
8C2×#14578938C2211yescharacteristic
9#19097770—21no—
10#19097762—21no—
11#19097748—22no—
12C4×Dic314—22no—
13#19097730C2211yescharacteristic
14#19097730C2211yescharacteristic
15C2×C4×Dic157C2211yescharacteristic
16#19097698—21no—
17#19097698—21no—
18#19097686—22no—
19C2×#41×C157C2211yescharacteristic
20C2×M16×C157C2211yescharacteristic
21#117×C157C2211yescharacteristic
22#104×C157—22no—
23C2×Dic314—22no—
24C22×Dic157—21no—
25#14578938—21no—
26#14578938—21no—
27C2×#11929434—21no—
28C4⋊Dic157—22no—
29C4⋊Dic157—21no—
30C4⋊Dic157—21no—
31#14578933—22no—
32#14578933—22no—
33C22⋊Dic157—21no—
34C22⋊Dic157D412yes—
35C4×Dic157—21no—
36C4×Dic157—21no—
37C4×Dic157—21no—
38C4×Dic157D411yescharacteristic
39C4×Dic157D411yescharacteristic
40#41×C157—21no—
41#41×C157—21no—
42#41×C157—21no—
43#41×C157—21no—
44C2×Q8×C157Q811yescharacteristic, complemented
45C2×D4×C157—21no—
46C2×D4×C157Q811yescharacteristic, complemented
47C22×C4×C157—21no—
48C22×C4×C157C2311yescharacteristic
49M16×C157—21no—
50M16×C157—21no—
51C2×C8×C157—21no—
52C4⋊C4×C157—21no—
53C22⋊C4×C157—21no—
54C42×C157D412yes—
55Dic314—44no—
56C2×Dic157—21no—
57C2×Dic157—21no—
58C2×Dic157—22no—
59C2×Dic157—42no—
60C2×Dic157—41no—
61#11929434—41no—
62C23×C157—21no—
63C23×C157C2×D411yescharacteristic
64Q8×C157—21no—
65Q8×C157—21no—
66D4×C157—21no—
67D4×C157—21no—
68D4×C157—41no—
69D4×C157—41no—
70C2×C4×C157—22no—
71C2×C4×C157—21no—
72C2×C4×C157—41no—
73C2×C4×C157—41no—
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C2×C4×C157C2×Q811yescharacteristic
77C2×C4×C157#4111yescharacteristic
78C2×C4×C157#4111yescharacteristic
79C2×C4×C157#4111yescharacteristic
80C8×C157—41no—
81C22×C157—41no—
82C22×C157—41no—
83C22×C157—81no—
84C22×C157#12011yescharacteristic
85C22×C157C22⋊Q811yescharacteristic
86C22×C157#12411yescharacteristic
87C4×C157—42no—
88C4×C157—41no—
89C4×C157C22⋊Q811yescharacteristic
90C4×C157C22⋊Q811yescharacteristic
91C4×C157#12411yescharacteristic
92C4×C157#12411yescharacteristic
93Dic157—22no—
94Dic157—41no—
95Dic157—42no—
96Dic157—81no—
97C2×C157—21no—
98C2×C157—21no—
99C2×C157—21no—
100C2×C157—81no—
101C2×C157#39311yescharacteristic
102C157#203111yescharacteristic, complemented, Sylow
103#2031—1571nomaximal, Sylow
104#533—1571no—
105#421D15711yescharacteristic
106C2×#104—1571no—
107#410—1571no—
108#344—1571no—
109#339—1571no—
110H(3,ℤ/4)—1571no—
111C2×#41C2×D15711yescharacteristic
112C2×M16—1571no—
113C2×M16C2×D15711yescharacteristic
114C4⋊Q8—3141no—
115#125—3141no—
116C22⋊Q8—3142no—
117C4×Q8—3142no—
118#117—1571no—
119#117—1571no—
120#117C2×D15711yescharacteristic
121C2×C42—1571no—
122#104—22no—
123#104—3141no—
124#104—3141no—
125#99—3142no—
126#41—21no—
127#41—21no—
128#41—21no—
129#41—21no—
130C2×Q8—3142no—
131C2×Q8Dic31411yescharacteristic, complemented
132C2×D4—21no—
133C2×D4Dic31411yescharacteristic, complemented
134C22×C4—21no—
135C22×C4—3141no—
136C22×C4C22×D15711yescharacteristic
137M16—21no—
138M16—21no—
139M16—3141no—
140M16—3141no—
141C2×C8—21no—
142C2×C8—3141no—
143C4⋊C4—21no—
144C4⋊C4—3142no—
145C4⋊C4—3142no—
146C4⋊C4—3142no—
147C4⋊C4—3141no—
148C4⋊C4—3141no—
149C22⋊C4—21no—
150C22⋊C4—3141no—
151C22⋊C4—1572no—
152C42—3141no—
153C42—3141no—
154C42—3141no—
155C42—1571no—
156C42—1571no—
157C42C22⋊D15712yes—
158C23—21no—
159C23D4×D15711yescharacteristic
160Q8—21no—
161Q8—21no—
162Q8—6284no—
163D4—21no—
164D4—21no—
165D4—41no—
166D4—41no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—3141no—
172C2×C4—3141no—
173C2×C4—3142no—
174C2×C4—6282no—
175C2×C4—6281no—
176C2×C4D4×D15711yescharacteristic
177C2×C4C2×C22⋊D15711yescharacteristic
178C2×C4C2×Dic31411yescharacteristic
179C2×C4#1457895311yescharacteristic
180C2×C4#1457895311yescharacteristic
181C2×C4#1457895411yescharacteristic
182C8—41no—
183C8—6281no—
184C22—41no—
185C22—41no—
186C22—81no—
187C22#1909773911yescharacteristic
188C22#1909774811yescharacteristic
189C22#1909775711yescharacteristic
190C4—42no—
191C4—41no—
192C4—3142no—
193C4—6281no—
194C4—6282no—
195C4—12561no—
196C4#1909774811yescharacteristic
197C4#1909775011yescharacteristic
198C4#1909775711yescharacteristic
199C4#1909775811yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—81no—
204C2#2375718511yescharacteristic
subgroupG/H
centerC4#19097758
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#393
prime radicalC2×#41×C157C22
hypercenter#421D157
hypocenterC157#2031
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #23757826C2semidirect
4. #23757044C2semidirect
5. #23757034C2semidirect
6. #23757029C2semidirect
44. C2×Q8×C157Q8semidirect
46. C2×D4×C157Q8semidirect
102. C157#2031semidirect
131. C2×Q8Dic314semidirect
133. C2×D4Dic314semidirect

Maximal subgroups

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

subgroupindex
#237578262
C2×#190976982
#237572472
#237570442
#237570342
#237570292
#421×C1572
#2031157