Finite Group Databasebeta

← #34099074

Subgroups — #34099074

Order 20096.

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

normal subgroups only

This group has 266 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#23757902C211yesmaximal, characteristic, complemented
2#23757424C211yesmaximal, characteristic
3#23757187C211yesmaximal, characteristic
4#23757011C211yesmaximal, characteristic, complemented
5#23757010C211yesmaximal, characteristic, complemented
6#23756998C211yesmaximal, characteristic, complemented
7#450×C157C211yesmaximal, characteristic
8C2×#14578942—21no—
9#19097770—21no—
10#19097769—21no—
11#19097761—21no—
12#19097761—21no—
13#19097750C2211yescharacteristic
14#19097748—21no—
15#19097727—21no—
16C2×#14578933C2211yescharacteristic
17C2×C22⋊Dic157—21no—
18C2×C22⋊Dic157—21no—
19#19097692—21no—
20C22⋊#11929434C2211yescharacteristic
21#19097678C2211yescharacteristic
22C22×D4×C157C2211yescharacteristic
23C2×SD16×C157—21no—
24C22⋊Q8×C157C2211yescharacteristic
25#102×C157—21no—
26C22⋊C8×C157C2211yescharacteristic
27C2×Dic314—21no—
28C2×Dic314D411yescharacteristic
29C22×Dic157—21no—
30C22×Dic157D411yescharacteristic
31C22×Dic157D411yescharacteristic
32#14578942—42no—
33C2×#11929434—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157D411yescharacteristic
36#14578933—21no—
37#14578933—21no—
38#14578933—21no—
39#14578933—21no—
40#14578933—21no—
41#14578933—21no—
42#14578933—21no—
43C22⋊Dic157—21no—
44C22⋊Dic157—42no—
45C22⋊Dic157—42no—
46C4×Dic157—21no—
47C4×Dic157—21no—
48C24×C157—21no—
49C2×Q8×C157D411yescharacteristic
50C2×D4×C157—21no—
51C2×D4×C157—21no—
52C2×D4×C157—42no—
53C22×C4×C157C2311yescharacteristic
54SD16×C157—42no—
55C2×C8×C157—21no—
56C4⋊C4×C157—21no—
57C4⋊C4×C157D411yescharacteristic
58C22⋊C4×C157—21no—
59Dic314—41no—
60Dic314—81no—
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C2×Dic157—41no—
67C2×Dic157—41no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157SD1612yescomplemented
71#11929434—41no—
72C23×C157—22no—
73C23×C157—41no—
74C23×C157—41no—
75C23×C157—41no—
76C23×C157—41no—
77C23×C157—41no—
78C23×C157—41no—
79C23×C157C2×D411yescharacteristic
80Q8×C157—41no—
81D4×C157—22no—
82D4×C157—22no—
83D4×C157—41no—
84D4×C157—41no—
85C2×C4×C157—21no—
86C2×C4×C157—21no—
87C2×C4×C157—41no—
88C2×C4×C157C2×D411yescharacteristic
89C2×C4×C157C2×D411yescharacteristic
90C8×C157—41no—
91C22×C157—21no—
92C22×C157—21no—
93C22×C157—42no—
94C22×C157—42no—
95C22×C157—42no—
96C22×C157—81no—
97C22×C157—81no—
98C22×C157—81no—
99C22×C157—81no—
100C22×C157#12011yescharacteristic
101C22×C157C2×SD1611yescharacteristic
102C22×C157#13711yescharacteristic
103C4×C157—21no—
104C4×C157—21no—
105C4×C157—41no—
106C4×C157—41no—
107Dic157—22no—
108Dic157—41no—
109Dic157—41no—
110Dic157—41no—
111Dic157—81no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—82no—
115C2×C157#44811yescharacteristic
116C2×C157#45411yescharacteristic
117C2×C157B(4,𝔽2)11yescharacteristic
118C157#159911yescharacteristic, complemented, Sylow
119#1599—1571nomaximal, Sylow
120#544—1571no—
121#450—1571no—
122#450D15711yescharacteristic
123#394—1571no—
124#328—1571no—
125#328—1571no—
126#324—1571no—
127C22×D4C2×D15711yescharacteristic
128C2×SD16—21no—
129C2×SD16—3141no—
130C4⋊Q8—3141no—
131C4⋊Q8—3141no—
132#125—3141no—
133#125—3141no—
134C22⋊Q8—3141no—
135C22⋊Q8—1571no—
136C22⋊Q8C2×D15711yescharacteristic
137#117—3141no—
138C2×C4⋊C4—1571no—
139C2×C22⋊C4—3141no—
140C2×C22⋊C4—3141no—
141#102—21no—
142#102—3141no—
143C22⋊C8—1571no—
144C22⋊C8C2×D15711yescharacteristic
145#95—1571no—
146C24—21no—
147C2×Q8—3141no—
148C2×Q8—1571no—
149C2×Q8C22⋊D15711yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C22×C4—3141no—
154C22×C4—1571no—
155C22×C4—1571no—
156C22×C4C22×D15711yescharacteristic
157SD16—42no—
158SD16—6282no—
159C2×C8—21no—
160C2×C8—3141no—
161C4⋊C4—21no—
162C4⋊C4—3141no—
163C4⋊C4—3141no—
164C4⋊C4—3141no—
165C4⋊C4—3141no—
166C4⋊C4—3141no—
167C4⋊C4—3141no—
168C4⋊C4—3141no—
169C4⋊C4—3141no—
170C4⋊C4—1571no—
171C4⋊C4C22⋊D15711yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—3141no—
174C22⋊C4—6282no—
175C22⋊C4—6282no—
176C42—3141no—
177C42—3141no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D15711yescharacteristic
186Q8—41no—
187Q8—6281no—
188Q8—12561no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—3141no—
197C2×C4—3141no—
198C2×C4—3141no—
199C2×C4—3141no—
200C2×C4—3141no—
201C2×C4—6281no—
202C2×C4—6281no—
203C2×C4—6281no—
204C2×C4—6281no—
205C2×C4—1572no—
206C2×C4D4×D15711yescharacteristic
207C2×C4C2×C22⋊D15711yescharacteristic
208C8—41no—
209C8—6281no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#1909773911yescharacteristic
220C22SD16×D15711yescharacteristic
221C22#1909780011yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—3142no—
227C4—6281no—
228C4—6281no—
229C4—6281no—
230C4—12561no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#2375741011yescharacteristic
235C2#2375744411yescharacteristic
236C2#2375745911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#450×C157C2
socleC22×C157#120
prime radicalC22×D4×C157C22
hypercenter#450D157
hypocenterC157#1599
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757902C2semidirect
4. #23757011C2semidirect
5. #23757010C2semidirect
6. #23756998C2semidirect
70. C2×Dic157SD16semidirect
118. C157#1599semidirect

Maximal subgroups

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

subgroupindex
#237579022
#237574242
#237571872
#237570112
#237570102
#237569982
#450×C1572
#1599157