Finite Group Databasebeta

← #34098693

Subgroups — #34098693

Order 20096.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#23757784C211yesmaximal, characteristic
2#23757136C211yesmaximal, characteristic, complemented
3#23757011C212yesmaximal, complemented
4#23757008C212yesmaximal, complemented
5C2×C22⋊C8×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C411yescharacteristic, complemented
7#19097754—21no—
8#19097750C411yescharacteristic, complemented
9#19097750C2211yescharacteristic
10#19097745C411yescharacteristic, complemented
11#19097745C2211yescharacteristic
12#19097740—21no—
13C4×C22⋊D157—21no—
14C4×C22⋊D157—21no—
15#19097730C411yescharacteristic, complemented
16C2×C22⋊Dic157—21no—
17#19097678—21no—
18#19097678C2212yes—
19C23×C4×C157C2211yescharacteristic
20C22×C8×C157—21no—
21C22⋊C8×C157—21no—
22C22⋊C8×C157C2212yes—
23#14578954—42no—
24C2×Dic314C2×C411yescharacteristic
25C2×D628C2×C411yescharacteristic
26C2×C22⋊D157—21no—
27C2×C22⋊D157—21no—
28C2×C22⋊D157—21no—
29C22×Dic157—22no—
30C2×C4×D157—21no—
31C4⋊Dic157C2×C411yescharacteristic
32C4⋊Dic157C2×C411yescharacteristic
33#14578933—21no—
34#14578933—21no—
35C22⋊Dic157—21no—
36C22⋊Dic157—21no—
37C22⋊Dic157—21no—
38C22⋊Dic157—42no—
39#14578931—21no—
40#14578931—21no—
41C4×Dic157—21no—
42C24×C157D411yescharacteristic
43C22×C4×C157—42no—
44C22×C4×C157C2×C411yescharacteristic
45C22×C4×C157C2×C411yescharacteristic
46C22×C4×C157D411yescharacteristic
47C22×C4×C157D411yescharacteristic
48C22×C4×C157D411yescharacteristic
49C22×C4×C157C2311yescharacteristic
50C2×C8×C157—22no—
51C2×C8×C157—42no—
52C22×D157—21no—
53Dic314—41no—
54D628—41no—
55C22⋊D157—42no—
56C22⋊D157—42no—
57C22⋊D157—42no—
58C2×Dic157—21no—
59C2×Dic157—21no—
60C2×Dic157—21no—
61C2×Dic157—42no—
62C2×Dic157—42no—
63C2×Dic157—42no—
64C4×D157—42no—
65C23×C157—22no—
66C23×C157—22no—
67C23×C157—41no—
68C23×C157C22⋊C411yescharacteristic
69C23×C157C22⋊C411yescharacteristic
70C23×C157C2×D411yescharacteristic
71C2×C4×C157—22no—
72C2×C4×C157—22no—
73C2×C4×C157—41no—
74C2×C4×C157—41no—
75C2×C4×C157—41no—
76C2×C4×C157—41no—
77C2×C4×C157C22⋊C411yescharacteristic
78C2×C4×C157C22⋊C411yescharacteristic
79C2×C4×C157C22×C411yescharacteristic
80C2×C4×C157C2×D411yescharacteristic
81C8×C157—42no—
82C22×C157—21no—
83C22×C157—21no—
84C22×C157—21no—
85C22×C157—21no—
86C22×C157—21no—
87C22×C157—21no—
88C22×C157—21no—
89C22×C157—44no—
90C22×C157#9912yes—
91C22×C157C2×C22⋊C411yescharacteristic
92C22×C157#13512yes—
93C2×D157—41no—
94C2×D157—41no—
95C2×D157—41no—
96C4×C157—21no—
97C4×C157—21no—
98C4×C157—21no—
99C4×C157—21no—
100Dic157—41no—
101Dic157—41no—
102Dic157—41no—
103Dic157—82no—
104C2×C157—22no—
105C2×C157—22no—
106C2×C157—41no—
107C2×C157C2×#9911yescharacteristic
108C2×C157#41611yescharacteristic
109C2×C157#42111yescharacteristic
110D157—81no—
111C157#151611yescharacteristic, complemented, Sylow
112#1516—1571nomaximal, Sylow
113#525—1571no—
114C2×C22⋊C8D15711yescharacteristic, complemented
115#381—1571no—
116#328—1572no—
117#327—1572no—
118C2×#41—1571no—
119C23×C4C2×D15711yescharacteristic
120C22×C8—21no—
121#123—3141no—
122C22⋊Q8—1571no—
123C22⋊Q8—1571no—
124#121—1571no—
125#121—1571no—
126#120—3141no—
127C4×D4—3141no—
128C4×D4—3141no—
129#117—1571no—
130C2×C22⋊C4—3141no—
131C22⋊C8—21no—
132C22⋊C8C2×D15712yes—
133#95—3141no—
134#95—1572no—
135C24D62811yescharacteristic
136#41—6282no—
137C2×Q8—1571no—
138C2×D4—3141no—
139C2×D4—3141no—
140C2×D4—3141no—
141C2×D4—1571no—
142C22×C4—42no—
143C22×C4—3141no—
144C22×C4—3142no—
145C22×C4C4×D15711yescharacteristic
146C22×C4C4×D15711yescharacteristic
147C22×C4C22⋊D15711yescharacteristic
148C22×C4C22⋊D15711yescharacteristic
149C22×C4D62811yescharacteristic
150C22×C4C22×D15711yescharacteristic
151C2×C8—22no—
152C2×C8—42no—
153C4⋊C4—3141no—
154C4⋊C4—3141no—
155C4⋊C4—1571no—
156C4⋊C4—1571no—
157C22⋊C4—3141no—
158C22⋊C4—3141no—
159C22⋊C4—3141no—
160C22⋊C4—3141no—
161C22⋊C4—3141no—
162C22⋊C4—6282no—
163C42—3141no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—3141no—
168C23#1457893111yescharacteristic
169C23#1457893111yescharacteristic
170C23C2×D62811yescharacteristic
171Q8—6281no—
172D4—6281no—
173D4—6282no—
174D4—6282no—
175D4—6282no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—3141no—
183C2×C4—3141no—
184C2×C4—3141no—
185C2×C4—6282no—
186C2×C4—6282no—
187C2×C4—6282no—
188C2×C4—6282no—
189C2×C4#1457893111yescharacteristic
190C2×C4#1457893111yescharacteristic
191C2×C4C2×C4×D15711yescharacteristic
192C2×C4C2×C22⋊D15711yescharacteristic
193C8—42no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—6281no—
203C22—6281no—
204C22—6281no—
205C22#1909768512yes—
206C22C2×#1457893111yescharacteristic
207C22#1909779312yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—6281no—
213C4—6281no—
214C4—6281no—
215C4—12562no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—12561no—
220C2C2×#1909768511yescharacteristic
221C2#2375727011yescharacteristic
222C2#2375728911yescharacteristic
subgroupG/H
centerC22C2×#14578931
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×C22⋊C8×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757784C2
cosocleC22×C4×C157C23
hypercenterC2×C22⋊C8D157
hypocenterC157#1516
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757136C2semidirect
3. #23757011C2semidirect
4. #23757008C2semidirect
5. C2×C22⋊C8×C157C2semidirect
6. C2×#14578954C4semidirect
8. #19097750C4semidirect
10. #19097745C4semidirect
15. #19097730C4semidirect
111. C157#1516semidirect
114. C2×C22⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
#237577842
#237571362
#237570112
#237570082
C2×C22⋊C8×C1572
#1516157