Finite Group Databasebeta

← #34100497

Subgroups — #34100497

Order 20096.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#23758073C211yesmaximal, characteristic
2#23758039C211yesmaximal, characteristic, complemented
3#23757291C211yesmaximal, characteristic, complemented
4#23757267C211yesmaximal, characteristic, complemented
5#23757241C211yesmaximal, characteristic, complemented
6#23757037C211yesmaximal, characteristic, complemented
7#344×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578954C2211yescharacteristic
10#19097804—21no—
11#19097804C412yescomplemented
12#19097799—21no—
13#19097799C412yescomplemented
14#19097793—22no—
15#19097779—22no—
16#19097777—22no—
17C2×#14578938C2211yescharacteristic
18M16×D157—22no—
19C22×#11929434C2211yescharacteristic
20#19097699—22no—
21#19097694—21no—
22#19097691—21no—
23#19097683—22no—
24C2×M16×C157C2211yescharacteristic
25C2×M16×C157C2211yescharacteristic
26#117×C157C2211yescharacteristic
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33#14578954C2×C412yes—
34#14578954D412yes—
35C2×Dic314C2×C411yescharacteristic
36C2×Dic314D411yescharacteristic
37C2×D628C2×C411yescharacteristic
38C2×D628D411yescharacteristic
39C2×C22⋊D157—21no—
40C2×C22⋊D157—21no—
41C2×C4×D157—21no—
42C2×C4×D157—21no—
43Dic628—22no—
44#14578943—22no—
45#14578943—22no—
46D1256—22no—
47#14578938—21no—
48#14578938D412yes—
49#14578937—24no—
50C2×#11929434—21no—
51C2×#11929434—21no—
52C2×#11929434—21no—
53C2×#11929434—21no—
54C8×D157—24no—
55C22×C4×C157C2311yescharacteristic
56M16×C157—21no—
57M16×C157—21no—
58M16×C157—21no—
59M16×C157C2×C412yes—
60C2×C8×C157—21no—
61C2×C8×C157D412yes—
62C4⋊C4×C157—21no—
63C22⋊C4×C157—21no—
64C42×C157—21no—
65C22×D157—21no—
66C22×D157—21no—
67Dic314—21no—
68Dic314—21no—
69Dic314—21no—
70Dic314C22⋊C412yes—
71D628—21no—
72D628—21no—
73D628—21no—
74D628C22⋊C412yes—
75C22⋊D157—22no—
76C22⋊D157—22no—
77C22⋊D157—41no—
78C22⋊D157—41no—
79C2×Dic157—21no—
80C2×Dic157—21no—
81C4×D157—22no—
82C4×D157—22no—
83C4×D157—41no—
84C4×D157—41no—
85#11929434—22no—
86#11929434—41no—
87C23×C157#4111yescharacteristic
88C2×C4×C157—41no—
89C2×C4×C157C22×C411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157#4111yescharacteristic
95C8×C157—22no—
96C8×C157—22no—
97C22×C157—41no—
98C22×C157C4×D411yescharacteristic
99C22×C157#12111yescharacteristic
100C22×C157#12311yescharacteristic
101C2×D157—22no—
102C2×D157—42no—
103C2×D157—41no—
104C2×D157—81no—
105C4×C157—81no—
106C4×C157C2×C22⋊C411yescharacteristic
107C4×C157C4×D411yescharacteristic
108C4×C157#12011yescharacteristic
109C4×C157#12111yescharacteristic
110Dic157—22no—
111Dic157—41no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157#38611yescharacteristic
116D157—42no—
117D157—81no—
118C157#188111yescharacteristic, complemented, Sylow
119#1881—1571nomaximal, Sylow
120#575—1571no—
121#568—1571no—
122#421—1571no—
123#416—1571no—
124#408—1571no—
125#344D15711yescharacteristic, complemented
126#341—1571no—
127C2×#41—1571no—
128C2×#41—1571no—
129#137—3141no—
130#137—1572no—
131AGL(1,ℤ/8)—3141no—
132AGL(1,ℤ/8)—1572no—
133#135—3142no—
134#131—3142no—
135#131—3142no—
136C2×M16—3142no—
137C2×M16—1571no—
138C2×M16C2×D15711yescharacteristic
139C2×M16C2×D15711yescharacteristic
140C22×C8—1571no—
141#117C2×D15711yescharacteristic
142#104—3142no—
143#103—3141no—
144#102—3141no—
145C22⋊C8—3142no—
146#41—3141no—
147#41—3141no—
148#41—3141no—
149#41—3141no—
150#41—3141no—
151#41—3141no—
152#41—1572no—
153#41—1572no—
154C2×Q8—1571no—
155C2×Q8—1571no—
156C2×D4—3141no—
157C2×D4—3141no—
158C2×D4—1571no—
159C2×D4—1571no—
160C22×C4—3141no—
161C22×C4—3141no—
162C22×C4C22×D15711yescharacteristic
163Q16—3142no—
164SD16—3142no—
165SD16—3142no—
166D8—3142no—
167M16—21no—
168M16—21no—
169M16—21no—
170M16—3144no—
171M16—3141no—
172M16—1572no—
173M16C4×D15712yes—
174C2×C8—21no—
175C2×C8—3144no—
176C2×C8—3141no—
177C2×C8—3141no—
178C2×C8—3141no—
179C2×C8—3141no—
180C2×C8D62812yes—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—3141no—
185C23—3141no—
186C23#1457895411yescharacteristic
187Q8—3141no—
188Q8—3141no—
189Q8—3141no—
190Q8—1572no—
191D4—3142no—
192D4—3142no—
193D4—3141no—
194D4—3141no—
195D4—3141no—
196D4—6281no—
197D4—6281no—
198D4—1572no—
199C2×C4—41no—
200C2×C4—3141no—
201C2×C4—3142no—
202C2×C4—3142no—
203C2×C4—3141no—
204C2×C4—6281no—
205C2×C4—6281no—
206C2×C4C2×C4×D15711yescharacteristic
207C2×C4D4×D15711yescharacteristic
208C2×C4D4×D15711yescharacteristic
209C2×C4D4×D15711yescharacteristic
210C2×C4C2×D62811yescharacteristic
211C2×C4#1457895511yescharacteristic
212C8—22no—
213C8—22no—
214C8—3142no—
215C8—6281no—
216C22—41no—
217C22—3142no—
218C22—6282no—
219C22—6281no—
220C22—12561no—
221C22C4×D62811yescharacteristic
222C22#1909774411yescharacteristic
223C22#1909775611yescharacteristic
224C4—81no—
225C4—3142no—
226C4—6281no—
227C4C22⋊C4×D15711yescharacteristic
228C4#1909773311yescharacteristic
229C4#1909774111yescharacteristic
230C4#1909774611yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—6282no—
235C2—12561no—
236C2#2375715111yescharacteristic
subgroupG/H
centerC4#19097741
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#344×C157C2
socleC2×C157#386
prime radical#23758073C2
cosocleC22×C4×C157C23
hypercenter#344D157
hypocenterC157#1881
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. #23758039C2semidirect
3. #23757291C2semidirect
4. #23757267C2semidirect
5. #23757241C2semidirect
6. #23757037C2semidirect
7. #344×C157C2semidirect
11. #19097804C4semidirect
13. #19097799C4semidirect
118. C157#1881semidirect
125. #344D157semidirect

Maximal subgroups

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

subgroupindex
#237580732
#237580392
#237572912
#237572672
#237572412
#237570372
#344×C1572
#1881157