Finite Group Databasebeta

← #34099162

Subgroups — #34099162

Order 20096.

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

normal subgroups only

This group has 236 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#23757609C211yesmaximal, characteristic, complemented
3#23757606C211yesmaximal, characteristic, complemented
4#23757210C211yesmaximal, characteristic
5#23757009C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic
7#328×C157C211yesmaximal, characteristic, complemented
8#19097751—21no—
9#19097750C2211yescharacteristic
10#19097749—21no—
11#19097749—21no—
12#19097748—21no—
13#19097747—21no—
14C2×C4⋊Dic157—21no—
15C2×C4⋊Dic157C2211yescharacteristic
16C2×#14578931—21no—
17C2×#14578931C2211yescharacteristic
18C22⋊C4×D157—21no—
19#19097705—21no—
20#19097704—21no—
21#19097696—21no—
22#19097695—21no—
23C22⋊#11929434C2211yescharacteristic
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28C2×Dic314—21no—
29C2×Dic314D411yescharacteristic
30C22×Dic157—21no—
31C22×Dic157D411yescharacteristic
32C2×C4×D157—21no—
33C2×C4×D157—21no—
34C2×#11929434—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157—21no—
38C4⋊Dic157—42no—
39C4⋊Dic157D411yescharacteristic
40#14578933—21no—
41#14578933—21no—
42#14578933—21no—
43C22⋊Dic157—21no—
44#14578931—21no—
45#14578931—21no—
46#14578931—21no—
47#14578931—21no—
48#14578931—42no—
49C2×Q8×C157D411yescharacteristic
50C22×C4×C157—21no—
51C22×C4×C157C2311yescharacteristic
52C2×C8×C157—21no—
53C4⋊C4×C157—21no—
54C4⋊C4×C157D411yescharacteristic
55C22⋊C4×C157—21no—
56C22×D157—21no—
57C22×D157—21no—
58C22×D157—41no—
59C22×D157—41no—
60C22×D157Q1612yes—
61Dic314—41no—
62Dic314—81no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C2×Dic157—41no—
67C2×Dic157—41no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C4×D157—81no—
71C4×D157—81no—
72#11929434—41no—
73C23×C157C2×D411yescharacteristic
74Q8×C157—41no—
75C2×C4×C157—22no—
76C2×C4×C157—21no—
77C2×C4×C157—21no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157—41no—
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C8×C157—41no—
84C22×C157—21no—
85C22×C157—21no—
86C22×C157#12011yescharacteristic
87C22×C157C2×Q1611yescharacteristic
88C22×C157#13711yescharacteristic
89C2×D157—22no—
90C2×D157—41no—
91C2×D157—41no—
92C2×D157—42no—
93C2×D157—81no—
94C4×C157—21no—
95C4×C157—21no—
96C4×C157—42no—
97C4×C157—41no—
98C4×C157—41no—
99Dic157—41no—
100Dic157—41no—
101Dic157—41no—
102Dic157—81no—
103C2×C157—21no—
104C2×C157—21no—
105C2×C157#45111yescharacteristic
106C2×C157#45311yescharacteristic
107C2×C157#45811yescharacteristic
108D157—42no—
109C157#161411yescharacteristic, complemented, Sylow
110#1614—1571nomaximal, Sylow
111#543—1571no—
112#484—1571no—
113#484—1571no—
114#399—1571no—
115#328—1571no—
116#328D15711yescharacteristic, complemented
117#323—1571no—
118C22⋊Q8—3141no—
119C22⋊Q8—3141no—
120C22⋊Q8—3141no—
121C22⋊Q8—3141no—
122C22⋊Q8—3141no—
123C22⋊Q8—1571no—
124C22⋊Q8C2×D15711yescharacteristic
125C2×C4⋊C4—3141no—
126C2×C4⋊C4—1571no—
127C2×C22⋊C4—3141no—
128C2×C22⋊C4—3141no—
129C2×C22⋊C4—1571no—
130#107—3141no—
131#107—3141no—
132#103—3141no—
133#103—3141no—
134C22⋊C8—1571no—
135C22⋊C8C2×D15711yescharacteristic
136#95C2×D15711yescharacteristic
137C24—1571no—
138C2×Q8—3141no—
139C2×Q8—1571no—
140C2×Q8D62811yescharacteristic
141C22×C4—21no—
142C22×C4—3141no—
143C22×C4—3141no—
144C22×C4—3141no—
145C22×C4—1571no—
146C22×C4C22×D15711yescharacteristic
147C2×C8—21no—
148C2×C8—3141no—
149C4⋊C4—21no—
150C4⋊C4—3141no—
151C4⋊C4—3141no—
152C4⋊C4—3141no—
153C4⋊C4—3141no—
154C4⋊C4—3141no—
155C4⋊C4—3141no—
156C4⋊C4—6282no—
157C4⋊C4—1571no—
158C4⋊C4D62811yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—3141no—
161C22⋊C4—3141no—
162C22⋊C4—3141no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—6282no—
166C23—3141no—
167C23—3141no—
168C23—6281no—
169C23—6281no—
170C23—1572no—
171C23D4×D15711yescharacteristic
172Q8—41no—
173Q8—6281no—
174Q8—12561no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—3141no—
182C2×C4—3141no—
183C2×C4—3141no—
184C2×C4—6281no—
185C2×C4—6281no—
186C2×C4—6281no—
187C2×C4—6281no—
188C2×C4—12561no—
189C2×C4—12561no—
190C2×C4D4×D15711yescharacteristic
191C2×C4C2×D62811yescharacteristic
192C8—41no—
193C8—6281no—
194C22—21no—
195C22—21no—
196C22—3142no—
197C22—6282no—
198C22—6281no—
199C22—6281no—
200C22—12561no—
201C22#1909774111yescharacteristic
202C22Q16×D15711yescharacteristic
203C22#1909780011yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—6281no—
210C4—6281no—
211C4—6281no—
212C4—12561no—
213C2—21no—
214C2—21no—
215C2—6282no—
216C2#2375743011yescharacteristic
217C2#2375744211yescharacteristic
218C2#2375746411yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157#120
prime radicalC23×D157D4
hypercenter#328D157
hypocenterC157#1614
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
2. #23757609C2semidirect
3. #23757606C2semidirect
5. #23757009C2semidirect
7. #328×C157C2semidirect
109. C157#1614semidirect
116. #328D157semidirect

Maximal subgroups

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

subgroupindex
#237578982
#237576092
#237576062
#237572102
#237570092
#237569962
#328×C1572
#1614157