Finite Group Databasebeta

← #34102155

Subgroups — #34102155

Order 20096.

216 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 41 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#23758022C211yesmaximal, characteristic, complemented
2#23757456C211yesmaximal, characteristic, complemented
3#23757446C211yesmaximal, characteristic
4#23757072C211yesmaximal, characteristic, complemented
5#23757071C211yesmaximal, characteristic, complemented
6#23757069C211yesmaximal, characteristic, complemented
7#456×C157C211yesmaximal, characteristic, complemented
8#19097801—21no—
9#19097796—21no—
10#19097769C2211yescharacteristic
11#19097766—21no—
12#19097765—21no—
13#19097761—21no—
14#19097760C2211yescharacteristic
15#19097756—21no—
16#19097749—21no—
17#19097747—21no—
18#19097698—21no—
19#19097698—21no—
20#19097690C2211yescharacteristic
21#19097690C2211yescharacteristic
222−1+4×C157C2211yescharacteristic
23#137×C157—21no—
24C4⋊Q8×C157C2211yescharacteristic
25#104×C157—21no—
26#101×C157C2211yescharacteristic
27C2×Dic314D411yescharacteristic
28C2×D628D411yescharacteristic
29C2×C4×D157—21no—
30#14578944—41no—
31#14578942—41no—
32#14578941—41no—
33#14578939—41no—
34#14578938—21no—
35#14578938—21no—
36C4⋊Dic157—21no—
37#14578933—21no—
38#14578933—21no—
39#14578933—21no—
40#14578933—21no—
41#14578933—21no—
42#14578931—21no—
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46#14578931—21no—
47C4×Dic157D411yescharacteristic
48C4×Dic157D411yescharacteristic
49#41×C157—21no—
50#41×C157—21no—
51#41×C157—21no—
52#41×C157—41no—
53C2×Q8×C157—41no—
54C2×Q8×C157D411yescharacteristic
55C2×Q8×C157C2311yescharacteristic
56Q16×C157—41no—
57SD16×C157—41no—
58M16×C157—21no—
59C4⋊C4×C157—21no—
60C4⋊C4×C157—21no—
61C42×C157D411yescharacteristic
62C22×D157—21no—
63Dic314—41no—
64D628—41no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C4×D157—81no—
69#11929434—41no—
70#11929434—41no—
71Q8×C157—21no—
72Q8×C157—21no—
73Q8×C157—21no—
74Q8×C157—41no—
75Q8×C157—41no—
76D4×C157—21no—
77D4×C157—21no—
78D4×C157—21no—
79D4×C157—41no—
80C2×C4×C157—21no—
81C2×C4×C157—21no—
82C2×C4×C157—41no—
83C2×C4×C157—41no—
84C2×C4×C157—41no—
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C8×C157—41no—
89C22×C157—41no—
90C22×C157#12011yescharacteristic
91C2×D157—42no—
92C2×D157—41no—
93C4×C157—21no—
94C4×C157—21no—
95C4×C157—21no—
96C4×C157—41no—
97C4×C157—41no—
98C4×C157—41no—
99Dic157—41no—
100Dic157—41no—
101Dic157—41no—
102C2×C157—21no—
103C2×C157—81no—
104C2×C157B(4,𝔽2)11yescharacteristic
105D157—81no—
106C157#220311yescharacteristic, complemented, Sylow
107#2203—1571nomaximal, Sylow
108#563—1571no—
109#456—1571no—
110#456D15711yescharacteristic, complemented
111#454—1571no—
112#356—1571no—
113#356—1571no—
114#355—1571no—
1152−1+4C2×D15711yescharacteristic
116#137—21no—
117#137—3141no—
118AGL(1,ℤ/8)—3141no—
119C4⋊Q8—1571no—
120C4⋊Q8C2×D15711yescharacteristic
121#126—3141no—
122#126—3141no—
123#125—3141no—
124#124—1571no—
125#123—3141no—
126C22⋊Q8—3141no—
127C22⋊Q8—3141no—
128#104—21no—
129#104—3141no—
130#104—3141no—
131#101—1571no—
132#101—1571no—
133#101C2×D15711yescharacteristic
134#41—21no—
135#41—21no—
136#41—21no—
137#41—41no—
138C2×Q8—41no—
139C2×Q8—1571no—
140C2×Q8C22⋊D15711yescharacteristic
141C2×Q8C22×D15711yescharacteristic
142C2×D4—1571no—
143C22×C4—3141no—
144Q16—41no—
145Q16—6281no—
146SD16—41no—
147SD16—6281no—
148SD16—6281no—
149D8—6281no—
150M16—21no—
151M16—3141no—
152M16—3141no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—3141no—
156C4⋊C4—3141no—
157C4⋊C4—3141no—
158C4⋊C4—3141no—
159C4⋊C4—3141no—
160C4⋊C4—3141no—
161C22⋊C4—3141no—
162C22⋊C4—3141no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—3141no—
166C42—1571no—
167C42—1571no—
168C42C22⋊D15711yescharacteristic
169C23—3141no—
170Q8—21no—
171Q8—21no—
172Q8—21no—
173Q8—41no—
174Q8—41no—
175Q8—6281no—
176D4—21no—
177D4—21no—
178D4—21no—
179D4—41no—
180D4—6281no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3141no—
187C2×C4—3141no—
188C2×C4—3141no—
189C2×C4—12561no—
190C2×C4D4×D15711yescharacteristic
191C2×C4D4×D15711yescharacteristic
192C2×C4C2×C22⋊D15711yescharacteristic
193C8—41no—
194C8—6281no—
195C8—6281no—
196C22—41no—
197C22—6282no—
198C22—6281no—
199C22#1909773911yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—41no—
206C4—6281no—
207C4—6281no—
208C4—6281no—
209C2—21no—
210C2—81no—
211C2—12561no—
212C2#2375745911yescharacteristic
subgroupG/H
centerC2#23757459
commutator, cosocleC2×Q8×C157C23
FrattiniC2×Q8C22×D157
Fitting#456×C157C2
socleC2×C157B(4,𝔽2)
prime radical#23757446C2
hypercenter#456D157
hypocenterC157#2203
LCSG ▷ C2×Q8×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23758022C2semidirect
2. #23757456C2semidirect
4. #23757072C2semidirect
5. #23757071C2semidirect
6. #23757069C2semidirect
7. #456×C157C2semidirect
106. C157#2203semidirect
110. #456D157semidirect

Maximal subgroups

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

subgroupindex
#237580222
#237574562
#237574462
#237570722
#237570712
#237570692
#456×C1572
#2203157