Finite Group Databasebeta

← #34102124

Subgroups — #34102124

Order 20096.

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

normal subgroups only

This group has 258 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#23757954C211yesmaximal, characteristic, complemented
2#23757463C211yesmaximal, characteristic, complemented
3#23757451C211yesmaximal, characteristic, complemented
4#23757067C211yesmaximal, characteristic, complemented
5#23757062C211yesmaximal, characteristic, complemented
6#23757060C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C157C211yesmaximal, characteristic
8#19097801—21no—
9#19097764—21no—
10#19097758C2211yescharacteristic
11#19097752—21no—
12#19097752—21no—
13#19097752C2211yescharacteristic
14#19097748—21no—
15D4×Dic157—21no—
16#19097727—21no—
17C2×C22⋊Dic157—21no—
18#19097698—21no—
19#19097688C2211yescharacteristic
20#19097686—21no—
21#19097686C2211yescharacteristic
22H(5,𝔽2)×C157C2211yescharacteristic
23#120×C157—21no—
24#120×C157C2211yescharacteristic
25#99×C157—21no—
26#99×C157C2211yescharacteristic
27C2×Dic314D411yescharacteristic, complemented
28C22×Dic157—21no—
29C22×Dic157—21no—
30C22×Dic157D411yescharacteristic, complemented
31#14578944—41no—
32#14578942—41no—
33#14578938—21no—
34C4⋊Dic157—21no—
35#14578933—21no—
36#14578933—21no—
37#14578933—21no—
38#14578933—41no—
39C22⋊Dic157—21no—
40C22⋊Dic157—21no—
41C22⋊Dic157—21no—
42C22⋊Dic157—21no—
43C22⋊Dic157—21no—
44C22⋊Dic157—21no—
45C22⋊Dic157—41no—
46C22⋊Dic157D411yescharacteristic, complemented
47C4×Dic157—21no—
48C4×Dic157D411yescharacteristic, complemented
49C24×C157D411yescharacteristic
50#41×C157—21no—
51#41×C157—41no—
52C2×D4×C157—21no—
53C2×D4×C157—21no—
54C2×D4×C157—21no—
55C2×D4×C157—41no—
56C2×D4×C157—41no—
57C2×D4×C157C2311yescharacteristic
58C22⋊C4×C157—21no—
59C22⋊C4×C157—21no—
60C22⋊C4×C157—41no—
61C22⋊C4×C157D411yescharacteristic
62Dic314—41no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—42no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157—81no—
71#11929434—41no—
72C23×C157—21no—
73C23×C157—21no—
74C23×C157—21no—
75C23×C157—41no—
76C23×C157—42no—
77C23×C157C2×D411yescharacteristic
78C23×C157C2×D411yescharacteristic
79Q8×C157—21no—
80D4×C157—21no—
81D4×C157—41no—
82D4×C157—42no—
83D4×C157—41no—
84D4×C157—41no—
85D4×C157—41no—
86D4×C157—82no—
87C2×C4×C157—21no—
88C2×C4×C157—21no—
89C2×C4×C157—21no—
90C2×C4×C157—41no—
91C2×C4×C157—41no—
92C2×C4×C157C2×D411yescharacteristic
93C22×C157—22no—
94C22×C157—21no—
95C22×C157—21no—
96C22×C157—41no—
97C22×C157—41no—
98C22×C157—41no—
99C22×C157—42no—
100C22×C157—41no—
101C22×C157—41no—
102C22×C157—81no—
103C22×C157—81no—
104C22×C157—81no—
105C22×C157#12011yescharacteristic
106C4×C157—21no—
107C4×C157—41no—
108C4×C157—41no—
109C4×C157—81no—
110Dic157—41no—
111Dic157—41no—
112Dic157—41no—
113Dic157—41no—
114Dic157—81no—
115C2×C157—21no—
116C2×C157—41no—
117C2×C157—41no—
118C2×C157—42no—
119C2×C157—81no—
120C2×C157B(4,𝔽2)11yescharacteristic
121C157#219711yescharacteristic, complemented, Sylow
122#2197—1571nomaximal, Sylow
123#551—1571no—
124#458—1571no—
125B(4,𝔽2)D15711yescharacteristic
126#455—1571no—
127#354—1571no—
128#352—1571no—
129AΣL(1,𝔽16)—1571no—
130H(5,𝔽2)C2×D15711yescharacteristic
131#137—3141no—
132#126—3141no—
133#124—1571no—
134#123—3141no—
135#123—3141no—
136#123—1571no—
137C22⋊Q8—3141no—
138#120—21no—
139#120C2×D15711yescharacteristic
140C4×D4—3141no—
141#117—3141no—
142C2×C22⋊C4—3141no—
143#104—3141no—
144#100—1571no—
145#99—21no—
146#99—3141no—
147#99—1571no—
148#99C2×D15711yescharacteristic
149C24C22⋊D15711yescharacteristic
150#41—21no—
151#41—41no—
152C2×Q8—1571no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C22×D15711yescharacteristic
159C22×C4—3141no—
160C22×C4—3141no—
161C22×C4—1571no—
162Q16—6281no—
163SD16—6281no—
164M16—3141no—
165C4⋊C4—3141no—
166C4⋊C4—3141no—
167C4⋊C4—3141no—
168C4⋊C4—3141no—
169C4⋊C4—6281no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—41no—
173C22⋊C4—3141no—
174C22⋊C4—3141no—
175C22⋊C4—3141no—
176C22⋊C4—3141no—
177C22⋊C4—3141no—
178C22⋊C4—3141no—
179C22⋊C4—6281no—
180C22⋊C4—1571no—
181C22⋊C4C22⋊D15711yescharacteristic
182C42—3141no—
183C42—1571no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23D4×D15711yescharacteristic
190C23C2×C22⋊D15711yescharacteristic
191Q8—21no—
192Q8—6281no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—41no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—3141no—
206C2×C4—3141no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—6282no—
210C2×C4—6281no—
211C2×C4—6281no—
212C2×C4—12561no—
213C2×C4D4×D15711yescharacteristic
214C8—6281no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#1909773911yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—81no—
232C4—6281no—
233C4—6281no—
234C4—6281no—
235C4—6281no—
236C4—12561no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—81no—
242C2#2375745911yescharacteristic
subgroupG/H
centerC2#23757459
commutator, cosocleC2×D4×C157C23
FrattiniC2×D4C22×D157
Fitting, prime radicalB(4,𝔽2)×C157C2
socleC2×C157B(4,𝔽2)
hypercenterB(4,𝔽2)D157
hypocenterC157#2197
LCSG ▷ C2×D4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23757954C2semidirect
2. #23757463C2semidirect
3. #23757451C2semidirect
4. #23757067C2semidirect
5. #23757062C2semidirect
6. #23757060C2semidirect
27. C2×Dic314D4semidirect
30. C22×Dic157D4semidirect
46. C22⋊Dic157D4semidirect
48. C4×Dic157D4semidirect
121. C157#2197semidirect

Maximal subgroups

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

subgroupindex
#237579542
#237574632
#237574512
#237570672
#237570622
#237570602
B(4,𝔽2)×C1572
#2197157