Finite Group Databasebeta

← #34100670

Subgroups — #34100670

Order 20096.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#23757844C211yesmaximal, characteristic
2#23757750C211yesmaximal, characteristic, complemented
3#23757290C211yesmaximal, characteristic
4C2×#19097689C211yesmaximal, characteristic, complemented
5#23757030C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic
8C22×Dic314C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097769C412yescomplemented
11#19097758C412yescomplemented
12#19097752—22no—
13#19097750—22no—
14C4×Dic314—22no—
15#19097736—22no—
16#19097730C2211yescharacteristic
17#19097730C2211yescharacteristic
18#19097727—22no—
19#19097698—22no—
20#19097689—22no—
21C2×#41×C157C2211yescharacteristic
22C2×M16×C157C2211yescharacteristic
23#117×C157C2211yescharacteristic
24#104×C157—22no—
25C2×Dic314—21no—
26C2×Dic314—21no—
27C2×Dic314—21no—
28C2×Dic314—21no—
29C2×Dic314C2×C412yes—
30C2×Dic314D412yes—
31C22×Dic157D412yes—
32#14578938—21no—
33#14578938—21no—
34C2×#11929434—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37#14578933—24no—
38#14578933—24no—
39C22⋊Dic157—24no—
40C22⋊Dic157—21no—
41C22⋊Dic157—21no—
42C4×Dic157—24no—
43C4×Dic157C2×C412yes—
44C4×Dic157D412yes—
45#41×C157—21no—
46#41×C157—21no—
47#41×C157—21no—
48#41×C157—21no—
49C2×Q8×C157C2×C411yescharacteristic
50C2×D4×C157—21no—
51C2×D4×C157C2×C411yescharacteristic
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—21no—
55M16×C157—21no—
56C2×C8×C157—21no—
57C4⋊C4×C157—21no—
58C22⋊C4×C157—21no—
59C42×C157D412yes—
60Dic314—22no—
61Dic314—22no—
62Dic314—22no—
63Dic314—22no—
64C2×Dic157—22no—
65C2×Dic157—22no—
66C2×Dic157—22no—
67C2×Dic157—22no—
68C2×Dic157C22⋊C414yes—
69#11929434—41no—
70C23×C157—21no—
71C23×C157C2×D411yescharacteristic
72Q8×C157—21no—
73Q8×C157—21no—
74D4×C157—21no—
75D4×C157—21no—
76D4×C157—41no—
77D4×C157—41no—
78C2×C4×C157—21no—
79C2×C4×C157—22no—
80C2×C4×C157—41no—
81C2×C4×C157—41no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157#4111yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C8×C157—41no—
89C22×C157—41no—
90C22×C157—42no—
91C22×C157—41no—
92C22×C157C2×C22⋊C411yescharacteristic
93C22×C157#12011yescharacteristic
94C22×C157#12311yescharacteristic
95C4×C157—42no—
96C4×C157—41no—
97C4×C157C4×D411yescharacteristic
98C4×C157C4×D411yescharacteristic
99C4×C157#12111yescharacteristic
100C4×C157#12111yescharacteristic
101Dic157—24no—
102Dic157—42no—
103Dic157—42no—
104C2×C157—21no—
105C2×C157—21no—
106C2×C157—21no—
107C2×C157—81no—
108C2×C157#38611yescharacteristic
109C157#191211yescharacteristic, complemented, Sylow
110#1912—1571nomaximal, Sylow
111#536—1571no—
112#519—1571no—
113#421—1571no—
114#421D15711yescharacteristic
115C2×#101—1571no—
116H(3,ℤ/4)—1571no—
117H(3,ℤ/4)—1571no—
118C2×#41C2×D15711yescharacteristic
119C22×Q8—1571no—
120C2×M16—1571no—
121C2×M16C2×D15711yescharacteristic
122C4⋊Q8—1572no—
123#124—1572no—
124#123—3142no—
125C22⋊Q8—3142no—
126C4×Q8—3142no—
127C4×Q8—3142no—
128#117—3142no—
129#117—1571no—
130#117—1571no—
131#117C2×D15711yescharacteristic
132#104—22no—
133#104—3142no—
134#101—3142no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—3141no—
140C2×Q8—3141no—
141C2×Q8—3141no—
142C2×Q8—3141no—
143C2×Q8—1572no—
144C2×Q8—1572no—
145C2×Q8C4×D15711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D15711yescharacteristic
148C22×C4—21no—
149C22×C4—1572no—
150C22×C4C22×D15711yescharacteristic
151M16—21no—
152M16—21no—
153M16—3141no—
154M16—3141no—
155C2×C8—21no—
156C2×C8—3141no—
157C4⋊C4—21no—
158C4⋊C4—3144no—
159C4⋊C4—3144no—
160C4⋊C4—3141no—
161C4⋊C4—3141no—
162C22⋊C4—21no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—3144no—
166C42—3144no—
167C42—1572no—
168C42—1572no—
169C42C22⋊D15712yes—
170C23—21no—
171C23D4×D15711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—3142no—
175Q8—3142no—
176Q8—3142no—
177Q8—3142no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3142no—
187C2×C4—3142no—
188C2×C4—3142no—
189C2×C4—3142no—
190C2×C4—1574no—
191C2×C4C2×C4×D15711yescharacteristic
192C2×C4D4×D15711yescharacteristic
193C2×C4D4×D15711yescharacteristic
194C2×C4C2×C22⋊D15711yescharacteristic
195C2×C4#1457895311yescharacteristic
196C2×C4#1457895411yescharacteristic
197C8—41no—
198C8—6281no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D15711yescharacteristic
203C22#1909773911yescharacteristic
204C22#1909775311yescharacteristic
205C4—42no—
206C4—41no—
207C4—3144no—
208C4—6282no—
209C4—6282no—
210C4#1909773111yescharacteristic
211C4C4×C22⋊D15711yescharacteristic
212C4#1909774211yescharacteristic
213C4#1909774411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2375714911yescharacteristic
subgroupG/H
centerC2#23757149
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#386
prime radicalC2×#41×C157C22
cosocleC22×C4×C157C23
hypercenter#421D157
hypocenterC157#1912
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #23757750C2semidirect
4. C2×#19097689C2semidirect
5. #23757030C2semidirect
6. #23757029C2semidirect
10. #19097769C4semidirect
11. #19097758C4semidirect
109. C157#1912semidirect

Maximal subgroups

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

subgroupindex
#237578442
#237577502
#237572902
C2×#190976892
#237570302
#237570292
#421×C1572
#1912157