Finite Group Databasebeta

← #34099077

Subgroups — #34099077

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#23757426C211yesmaximal, characteristic
3#23757423C211yesmaximal, characteristic
4#23757191C211yesmaximal, characteristic
5#23757009C211yesmaximal, characteristic, complemented
6#23756998C211yesmaximal, characteristic, complemented
7#328×C157C211yesmaximal, characteristic, complemented
8C22×D628C2211yescharacteristic
9C2×#14578943—21no—
10C2×#14578941—21no—
11#19097770—21no—
12#19097769—21no—
13#19097761—21no—
14#19097761—21no—
15#19097750C2211yescharacteristic
16#19097748—21no—
17#19097727—21no—
18C2×#14578933C2211yescharacteristic
19C2×#14578931—21no—
20C2×#14578931—21no—
21#19097693—21no—
22#19097691—21no—
23C22⋊#11929434C2211yescharacteristic
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157—21no—
28C2×Dic314—21no—
29C2×Dic314D411yescharacteristic
30C2×D628—21no—
31C2×D628—21no—
32C2×D628—42no—
33C22×Dic157D411yescharacteristic
34C22×Dic157D411yescharacteristic
35#14578943—42no—
36#14578941—42no—
37C2×#11929434—21no—
38C4⋊Dic157—21no—
39C4⋊Dic157D411yescharacteristic
40#14578933—21no—
41#14578933—21no—
42#14578933—21no—
43#14578933—21no—
44#14578933—21no—
45#14578933—21no—
46#14578933—21no—
47C22⋊Dic157—21no—
48#14578931—42no—
49#14578931—42no—
50C4×Dic157—21no—
51C4×Dic157—21no—
52C2×Q8×C157D411yescharacteristic
53C22×C4×C157—21no—
54C22×C4×C157C2311yescharacteristic
55C2×C8×C157—21no—
56C4⋊C4×C157—21no—
57C4⋊C4×C157D411yescharacteristic
58C22⋊C4×C157—21no—
59C22×D157—22no—
60C22×D157—41no—
61C22×D157—41no—
62C22×D157—41no—
63C22×D157—41no—
64C22×D157—41no—
65C22×D157—41no—
66Dic314—41no—
67Dic314—81no—
68D628—22no—
69D628—22no—
70D628—41no—
71D628—41no—
72C2×Dic157—21no—
73C2×Dic157—21no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—41no—
78C2×Dic157SD1612yescomplemented
79#11929434—41no—
80C23×C157C2×D411yescharacteristic
81Q8×C157—41no—
82C2×C4×C157—21no—
83C2×C4×C157—21no—
84C2×C4×C157—41no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157C2×D411yescharacteristic
89C2×C4×C157C2×D411yescharacteristic
90C8×C157—41no—
91C22×C157—21no—
92C22×C157—21no—
93C22×C157#12011yescharacteristic
94C22×C157C2×SD1611yescharacteristic
95C22×C157#13711yescharacteristic
96C2×D157—42no—
97C2×D157—42no—
98C2×D157—42no—
99C2×D157—81no—
100C2×D157—81no—
101C2×D157—81no—
102C2×D157—81no—
103C4×C157—21no—
104C4×C157—21no—
105C4×C157—41no—
106C4×C157—41no—
107C4×C157—81no—
108Dic157—22no—
109Dic157—41no—
110Dic157—41no—
111Dic157—41no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157#44811yescharacteristic
115C2×C157#45411yescharacteristic
116C2×C157B(4,𝔽2)11yescharacteristic
117D157—82no—
118C157#159911yescharacteristic, complemented, Sylow
119#1599—1571nomaximal, Sylow
120#544—1571no—
121#450—1571no—
122#450—1571no—
123#394—1571no—
124#328—1571no—
125#328D15711yescharacteristic, complemented
126#324—1571no—
127C22×D4—1571no—
128C2×SD16—3141no—
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—3141no—
142#102—3141no—
143C22⋊C8—1571no—
144C22⋊C8C2×D15711yescharacteristic
145#95C2×D15711yescharacteristic
146C24—3141no—
147C2×Q8—3141no—
148C2×Q8—1571no—
149C2×Q8D62811yescharacteristic
150C2×D4—3141no—
151C2×D4—3141no—
152C2×D4—6282no—
153C22×C4—21no—
154C22×C4—1571no—
155C22×C4—1571no—
156C22×C4C22×D15711yescharacteristic
157SD16—6282no—
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⋊C4D62811yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—3141no—
174C22⋊C4—6282no—
175C22⋊C4—6282no—
176C42—3141no—
177C42—3141no—
178C23—3142no—
179C23—6281no—
180C23—6281no—
181C23—6281no—
182C23—6281no—
183C23—6281no—
184C23—6281no—
185C23D4×D15711yescharacteristic
186Q8—41no—
187Q8—6281no—
188Q8—12561no—
189D4—3142no—
190D4—3142no—
191D4—6281no—
192D4—6281no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3141no—
200C2×C4—3141no—
201C2×C4—3141no—
202C2×C4—3141no—
203C2×C4—3141no—
204C2×C4—6281no—
205C2×C4—1572no—
206C2×C4D4×D15711yescharacteristic
207C2×C4C2×D62811yescharacteristic
208C8—41no—
209C8—6281no—
210C22—21no—
211C22—21no—
212C22—6282no—
213C22—6282no—
214C22—6282no—
215C22—12561no—
216C22—12561no—
217C22—12561no—
218C22—12561no—
219C22#1909774111yescharacteristic
220C22SD16×D15711yescharacteristic
221C22#1909780311yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—3142no—
228C4—6281no—
229C4—6281no—
230C4—6281no—
231C2—21no—
232C2—21no—
233C2—12562no—
234C2#2375741311yescharacteristic
235C2#2375744711yescharacteristic
236C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157#120
prime radicalC22×D628C22
hypercenter#328D157
hypocenterC157#1599
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #23757902C2semidirect
5. #23757009C2semidirect
6. #23756998C2semidirect
7. #328×C157C2semidirect
78. C2×Dic157SD16semidirect
118. C157#1599semidirect
125. #328D157semidirect

Maximal subgroups

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

subgroupindex
#237579022
#237574262
#237574232
#237571912
#237570092
#237569982
#328×C1572
#1599157