Finite Group Databasebeta

← #34099091

Subgroups — #34099091

Order 20096.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#23757898C211yesmaximal, characteristic
2#23757429C211yesmaximal, characteristic, complemented
3#23757184C211yesmaximal, characteristic
4#23757011C211yesmaximal, characteristic, complemented
5#23757010C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic
7#451×C157C211yesmaximal, characteristic, complemented
8C2×#14578944—21no—
9#19097751—21no—
10#19097750C2211yescharacteristic
11#19097749—21no—
12#19097749—21no—
13#19097748—21no—
14#19097747—21no—
15C2×#14578931—21no—
16C2×#14578931C2211yescharacteristic
17C22⋊C4×D157—21no—
18#19097695—21no—
19C22⋊#11929434C2211yescharacteristic
20#19097678—21no—
21#19097678C2211yescharacteristic
22C22×Q8×C157C2211yescharacteristic
23C2×Q16×C157—21no—
24C22⋊Q8×C157C2211yescharacteristic
25#103×C157—21no—
26C22⋊C8×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28C2×Dic314—21no—
29C2×Dic314D411yescharacteristic
30C22×Dic157—21no—
31C22×Dic157D411yescharacteristic
32C2×C4×D157—21no—
33C2×C4×D157—21no—
34#14578944—42no—
35C2×#11929434—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38#14578933—21no—
39#14578933—21no—
40#14578933—21no—
41C22⋊Dic157—21no—
42#14578931—21no—
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46#14578931—42no—
47C2×Q8×C157—21no—
48C2×Q8×C157—21no—
49C2×Q8×C157—42no—
50C2×Q8×C157D411yescharacteristic
51C22×C4×C157—21no—
52C22×C4×C157C2311yescharacteristic
53Q16×C157—42no—
54C2×C8×C157—21no—
55C4⋊C4×C157—21no—
56C4⋊C4×C157D411yescharacteristic
57C22⋊C4×C157—21no—
58C22×D157—21no—
59C22×D157—21no—
60C22×D157—41no—
61C22×D157—41no—
62C22×D157Q1612yescomplemented
63Dic314—41no—
64Dic314—81no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C4×D157—81no—
73C4×D157—81no—
74#11929434—41no—
75C23×C157C2×D411yescharacteristic
76Q8×C157—22no—
77Q8×C157—22no—
78Q8×C157—41no—
79Q8×C157—41no—
80Q8×C157—41no—
81C2×C4×C157—22no—
82C2×C4×C157—21no—
83C2×C4×C157—21no—
84C2×C4×C157—41no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
87C2×C4×C157C2×D411yescharacteristic
88C2×C4×C157C2×D411yescharacteristic
89C8×C157—41no—
90C22×C157—21no—
91C22×C157—21no—
92C22×C157#12011yescharacteristic
93C22×C157C2×Q1611yescharacteristic
94C22×C157#13711yescharacteristic
95C2×D157—22no—
96C2×D157—41no—
97C2×D157—41no—
98C2×D157—42no—
99C2×D157—81no—
100C4×C157—21no—
101C4×C157—21no—
102C4×C157—41no—
103C4×C157—42no—
104C4×C157—41no—
105Dic157—41no—
106Dic157—41no—
107Dic157—41no—
108Dic157—81no—
109C2×C157—21no—
110C2×C157—21no—
111C2×C157#45111yescharacteristic
112C2×C157#45511yescharacteristic
113C2×C157B(4,𝔽2)11yescharacteristic
114D157—42no—
115C157#160211yescharacteristic, complemented, Sylow
116#1602—1571nomaximal, Sylow
117#543—1571no—
118#451—1571no—
119#451D15711yescharacteristic, complemented
120#393—1571no—
121#328—1571no—
122#328—1571no—
123#323—1571no—
124C22×Q8C2×D15711yescharacteristic
125C2×Q16—21no—
126C2×Q16—3141no—
127C22⋊Q8—3141no—
128C22⋊Q8—3141no—
129C22⋊Q8—3141no—
130C22⋊Q8—3141no—
131C22⋊Q8—3141no—
132C22⋊Q8—1571no—
133C22⋊Q8C2×D15711yescharacteristic
134C2×C22⋊C4—3141no—
135C2×C22⋊C4—3141no—
136C2×C22⋊C4—1571no—
137#103—21no—
138#103—3141no—
139C22⋊C8—1571no—
140C22⋊C8C2×D15711yescharacteristic
141#95—3141no—
142#95—1571no—
143C24—1571no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—42no—
147C2×Q8—3141no—
148C2×Q8—1571no—
149C2×Q8C22⋊D15711yescharacteristic
150C22×C4—21no—
151C22×C4—3141no—
152C22×C4—3141no—
153C22×C4—3141no—
154C22×C4—1571no—
155C22×C4C22×D15711yescharacteristic
156Q16—42no—
157Q16—6282no—
158C2×C8—21no—
159C2×C8—3141no—
160C4⋊C4—21no—
161C4⋊C4—3141no—
162C4⋊C4—3141no—
163C4⋊C4—3141no—
164C4⋊C4—3141no—
165C4⋊C4—1571no—
166C4⋊C4C22⋊D15711yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—3141no—
169C22⋊C4—3141no—
170C22⋊C4—3141no—
171C22⋊C4—3141no—
172C22⋊C4—3141no—
173C22⋊C4—6282no—
174C23—3141no—
175C23—3141no—
176C23—6281no—
177C23—6281no—
178C23—1572no—
179C23D4×D15711yescharacteristic
180Q8—22no—
181Q8—22no—
182Q8—41no—
183Q8—41no—
184Q8—41no—
185Q8—6281no—
186Q8—12561no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—3141no—
194C2×C4—3141no—
195C2×C4—3141no—
196C2×C4—6281no—
197C2×C4—6281no—
198C2×C4—6281no—
199C2×C4—6281no—
200C2×C4—12561no—
201C2×C4—12561no—
202C2×C4D4×D15711yescharacteristic
203C2×C4C2×C22⋊D15711yescharacteristic
204C8—41no—
205C8—6281no—
206C22—21no—
207C22—21no—
208C22—3142no—
209C22—6282no—
210C22—6281no—
211C22—6281no—
212C22—12561no—
213C22#1909773911yescharacteristic
214C22Q16×D15711yescharacteristic
215C22#1909780311yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—6281no—
222C4—6281no—
223C4—6281no—
224C4—12561no—
225C2—21no—
226C2—21no—
227C2—6282no—
228C2#2375742711yescharacteristic
229C2#2375744911yescharacteristic
230C2#2375745911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#451×C157C2
socleC22×C157#120
prime radicalC23×D157D4
hypercenter#451D157
hypocenterC157#1602
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #23757429C2semidirect
4. #23757011C2semidirect
5. #23757010C2semidirect
7. #451×C157C2semidirect
62. C22×D157Q16semidirect
115. C157#1602semidirect
119. #451D157semidirect

Maximal subgroups

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

subgroupindex
#237578982
#237574292
#237571842
#237570112
#237570102
#237569962
#451×C1572
#1602157