Finite Group Databasebeta

← #34101441

Subgroups — #34101441

Order 20096.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#23758071C211yesmaximal, characteristic
2#23757300C211yesmaximal, characteristic, complemented
3#23757269C211yesmaximal, characteristic, complemented
4#23757267C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7#410×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—21no—
10#19097801—21no—
11#19097797—21no—
12#19097796—21no—
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15C22×#11929434C2211yescharacteristic
16#19097730C2211yescharacteristic
17C4⋊#11929434—22no—
18#19097695—21no—
19#19097694—21no—
20#19097692—21no—
21#19097691—21no—
22#19097685—22no—
23C2×#41×C157C2211yescharacteristic
24#117×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26#99×C157—22no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31C2×Dic314D411yescharacteristic
32C2×D628D411yescharacteristic
33C2×C22⋊D157—21no—
34C2×C4×D157—21no—
35#14578944—22no—
36#14578942—22no—
37#14578941—22no—
38#14578939—22no—
39#14578938—21no—
40#14578938—21no—
41C2×#11929434—21no—
42C2×#11929434—21no—
43C2×#11929434—21no—
44C2×#11929434D412yes—
45C4⋊Dic157—21no—
46C22⋊Dic157—21no—
47C4×Dic157—21no—
48#41×C157—21no—
49#41×C157—21no—
50#41×C157—21no—
51#41×C157—21no—
52C2×Q8×C157D411yescharacteristic
53C2×D4×C157—21no—
54C2×D4×C157D411yescharacteristic
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57C4⋊C4×C157—21no—
58C4⋊C4×C157—21no—
59C22⋊C4×C157—21no—
60C22⋊C4×C157D412yes—
61C42×C157—21no—
62C42×C157—21no—
63C22×D157—21no—
64Dic314—21no—
65Dic314—21no—
66D628—21no—
67D628—21no—
68C22⋊D157—41no—
69C22⋊D157—41no—
70C2×Dic157—21no—
71C2×Dic157—41no—
72C4×D157—41no—
73C4×D157—41no—
74#11929434—22no—
75#11929434—41no—
76C23×C157—21no—
77C23×C157C2×D411yescharacteristic
78Q8×C157—21no—
79Q8×C157—21no—
80D4×C157—21no—
81D4×C157—21no—
82D4×C157—41no—
83D4×C157—41no—
84C2×C4×C157—22no—
85C2×C4×C157—21no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157#4111yescharacteristic
93C2×C4×C157#4111yescharacteristic
94C2×C4×C157#4111yescharacteristic
95C22×C157—41no—
96C22×C157—41no—
97C22×C157—81no—
98C22×C157#12111yescharacteristic
99C22×C157#12111yescharacteristic
100C22×C157#12111yescharacteristic
101C2×D157—41no—
102C2×D157—81no—
103C4×C157—42no—
104C4×C157—41no—
105C4×C157—81no—
106C4×C157#12011yescharacteristic
107C4×C157#12311yescharacteristic
108C4×C157#12311yescharacteristic
109C4×C157#12411yescharacteristic
110Dic157—41no—
111Dic157—81no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157—81no—
116C2×C157#39411yescharacteristic
117D157—81no—
118C157#205311yescharacteristic, complemented, Sylow
119#2053—1571nomaximal, Sylow
120#575—1571no—
121#424—1571no—
122#416—1571no—
123#416—1571no—
124#410—1571no—
125#410D15711yescharacteristic, complemented
126H(3,ℤ/4)—1571no—
127C2×#41—1571no—
128C2×#41C2×D15711yescharacteristic
129#137—3141no—
130#137—3141no—
131AGL(1,ℤ/8)—3141no—
132AGL(1,ℤ/8)—3141no—
133#135—3142no—
134C2×M16—1571no—
135C22×C8—1571no—
136#117—1571no—
137#117C2×D15711yescharacteristic
138#117C2×D15711yescharacteristic
139C4⋊C8—3142no—
140#103—3141no—
141#103—3141no—
142#102—3141no—
143#102—3141no—
144#99—22no—
145#99—3142no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—3141no—
151#41—3141no—
152#41—3141no—
153#41—3141no—
154C2×Q8—1571no—
155C2×Q8D62811yescharacteristic
156C2×D4—21no—
157C2×D4—3141no—
158C2×D4—1571no—
159C2×D4D62811yescharacteristic
160C22×C4—21no—
161C22×C4—3141no—
162C22×C4C22×D15711yescharacteristic
163Q16—3142no—
164SD16—3142no—
165SD16—3142no—
166D8—3142no—
167M16—3141no—
168M16—3141no—
169C2×C8—3141no—
170C2×C8—3141no—
171C2×C8—3141no—
172C2×C8—1572no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—3141no—
176C22⋊C4—21no—
177C22⋊C4—3141no—
178C22⋊C4C22⋊D15712yes—
179C42—21no—
180C42—21no—
181C42—3141no—
182C23—21no—
183C23—3141no—
184C23C2×C22⋊D15711yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—3141no—
188Q8—3141no—
189D4—21no—
190D4—21no—
191D4—41no—
192D4—41no—
193D4—3141no—
194D4—3141no—
195D4—6281no—
196D4—6281no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3141no—
203C2×C4—6281no—
204C2×C4—6281no—
205C2×C4—6281no—
206C2×C4D4×D15711yescharacteristic
207C2×C4D4×D15711yescharacteristic
208C2×C4C2×D62811yescharacteristic
209C2×C4#1457895311yescharacteristic
210C2×C4#1457895311yescharacteristic
211C2×C4#1457895411yescharacteristic
212C8—3142no—
213C8—6281no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—6281no—
218C22—12561no—
219C22#1909774211yescharacteristic
220C22#1909774311yescharacteristic
221C22#1909774511yescharacteristic
222C4—42no—
223C4—41no—
224C4—81no—
225C4—6281no—
226C4—12561no—
227C4#1909774111yescharacteristic
228C4#1909775311yescharacteristic
229C4#1909775511yescharacteristic
230C4#1909775711yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—12561no—
236C2#2375718911yescharacteristic
subgroupG/H
centerC4#19097757
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#410×C157C2
socleC2×C157#394
prime radical#23758071C2
hypercenter#410D157
hypocenterC157#2053
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23757300C2semidirect
3. #23757269C2semidirect
4. #23757267C2semidirect
5. #23757246C2semidirect
6. #23757029C2semidirect
7. #410×C157C2semidirect
118. C157#2053semidirect
125. #410D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237573002
#237572692
#237572672
#237572462
#237570292
#410×C1572
#2053157