Finite Group Databasebeta

← #34111939

Subgroups — #34111939

Order 20096.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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#23758090C211yesmaximal, characteristic, complemented
2#23758089C212yesmaximal, complemented
3#23758086C211yesmaximal, characteristic, complemented
4#23757651C211yesmaximal, characteristic, complemented
5#23757549C211yesmaximal, characteristic, complemented
6#23757547C212yesmaximal, complemented
7#23757458C212yesmaximal, complemented
8#23757454C212yesmaximal, complemented
9#23757389C211yesmaximal, characteristic, complemented
10#23757388C211yesmaximal, characteristic, complemented
11#473×C157C211yesmaximal, characteristic, complemented
12#19097819C2212yes—
13#19097818C2212yes—
14#19097804—22no—
15#19097804C2212yes—
16#19097804C2212yes—
17#19097803—22no—
18#19097803—22no—
19#19097802C2212yes—
20#19097800—22no—
21#19097794—21no—
22#19097793—21no—
23#19097793C2211yescharacteristic
24#19097792—22no—
25#19097791C2211yescharacteristic
26C2×Dic628—21no—
27C2×Dic628C2211yescharacteristic
28C2×#14578944C2212yes—
29Q16×D157—21no—
30Q16×D157—22no—
31#19097779C2212yes—
32#19097777C2211yescharacteristic
33#19097769C2212yes—
34#19097706C2211yescharacteristic
35#19097698C2211yescharacteristic
36#19097697C2211yescharacteristic
37#19097697C2212yes—
38#19097689C2212yes—
39C8×Dic157C2211yescharacteristic
40#137×C157C2212yes—
41C2×Q16×C157C2211yescharacteristic
42#131×C157C2211yescharacteristic
43#108×C157C2211yescharacteristic
44#101×C157C2212yes—
45#14578955—22no—
46#14578955—22no—
47#14578954—22no—
48#14578954—42no—
49#14578954C2311yescharacteristic
50#14578954C2312yes—
51#14578953—22no—
52#14578953—42no—
53C2×Dic314—22no—
54C2×Dic314C2312yes—
55Q8×D157—22no—
56Q8×D157—22no—
57Q8×D157—22no—
58Dic628—22no—
59Dic628—22no—
60Dic628—21no—
61Dic628—21no—
62Dic628D411yescharacteristic
63#14578944—22no—
64#14578944—22no—
65#14578944—22no—
66#14578944D411yescharacteristic
67#14578943—22no—
68#14578943—22no—
69#14578943D412yes—
70#14578942D411yescharacteristic
71#14578941—22no—
72#14578941D411yescharacteristic
73D1256D411yescharacteristic
74#14578939D411yescharacteristic
75#14578938C2311yescharacteristic
76#14578937—22no—
77#14578937—21no—
78C2×#11929434C2311yescharacteristic
79C8×D157—22no—
80C8×D157—21no—
81#14578933—42no—
82C4×Dic157C2311yescharacteristic
83#41×C157C2311yescharacteristic
84C2×Q8×C157C2312yes—
85Q16×C157—22no—
86Q16×C157—21no—
87Q16×C157—21no—
88SD16×C157—22no—
89M16×C157—21no—
90M16×C157C2311yescharacteristic
91M16×C157C2312yes—
92C2×C8×C157—21no—
93C2×C8×C157C2311yescharacteristic
94Dic314—22no—
95Dic314—22no—
96Dic314—42no—
97Dic314—42no—
98Dic314C2×D412yes—
99Dic314C2×D411yescharacteristic
100D628—42no—
101D628C2×D412yes—
102D628C2×D411yescharacteristic
103C22⋊D157—22no—
104C22⋊D157—21no—
105C22⋊D157—42no—
106C2×Dic157—21no—
107C2×Dic157—22no—
108C2×Dic157—42no—
109C4×D157—22no—
110C4×D157—21no—
111C4×D157—42no—
112C4×D157—42no—
113C4×D157—42no—
114#11929434—21no—
115#11929434C2×D411yescharacteristic
116#11929434C2×D411yescharacteristic
117Q8×C157—22no—
118Q8×C157—22no—
119Q8×C157C2×D411yescharacteristic
120D4×C157—21no—
121D4×C157C2×D411yescharacteristic
122C2×C4×C157—21no—
123C2×C4×C157—22no—
124C2×C4×C157C2411yescharacteristic
125C8×C157—21no—
126C8×C157—22no—
127C8×C157C2×D411yescharacteristic
128C8×C157C2×D411yescharacteristic
129C22×C157—21no—
130C22×C157H(5,𝔽2)11yescharacteristic
131C2×D157—22no—
132C2×D157—21no—
133C4×C157—21no—
134C4×C157—42no—
135C4×C157C22×D411yescharacteristic
136C4×C157C22×D411yescharacteristic
137Dic157—22no—
138Dic157—21no—
139Dic157—22no—
140Dic157—42no—
141C2×C157—21no—
142C2×C157—41no—
143C2×C157D4211yescharacteristic
144D157—42no—
145D157—41no—
146C157#329311yescharacteristic, complemented, Sylow
147#3293—1571nomaximal, Sylow
148#578—1572no—
149#578—1571no—
150#578—1571no—
151#494—1571no—
152#473—1571no—
153#473—1572no—
154#473D15711yescharacteristic, complemented
155#456—1572no—
156#456—1572no—
157#443—1571no—
158#443—1571no—
1592−1+4—1572no—
1602−1+4—1572no—
161#137—3142no—
162#137—3142no—
163#137—3142no—
164#137—3142no—
165#137—1572no—
166#137—1572no—
167#137—1572no—
168#137C2×D15712yes—
169#135—3141no—
170#135—3142no—
171#135—3141no—
172#135—1571no—
173#135—1571no—
174C2×Q16—3141no—
175C2×Q16—3142no—
176C2×Q16—3141no—
177C2×Q16—1572no—
178C2×Q16—1571no—
179C2×Q16C2×D15711yescharacteristic
180#131—1572no—
181#131—1571no—
182#131C2×D15711yescharacteristic
183C4⋊Q8—1572no—
184#108—1571no—
185#108C2×D15711yescharacteristic
186#104—1571no—
187#104—1572no—
188#104—1571no—
189#101—1572no—
190#101C2×D15712yes—
191C4×C8—1571no—
192#41—3142no—
193#41—3142no—
194#41—3142no—
195#41—3142no—
196#41—6282no—
197#41—6282no—
198#41—1571no—
199#41—1572no—
200#41C22×D15711yescharacteristic
201C2×Q8—3142no—
202C2×Q8—3142no—
203C2×Q8—3142no—
204C2×Q8—3142no—
205C2×Q8—1572no—
206C2×Q8C22×D15712yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—3142no—
211Q16—3142no—
212Q16—3142no—
213Q16—3142no—
214Q16—3142no—
215Q16—3141no—
216Q16—3141no—
217Q16—1571no—
218Q16—1571no—
219SD16—22no—
220SD16—3142no—
221SD16—3142no—
222SD16—3142no—
223SD16—1571no—
224SD16—1571no—
225SD16—1572no—
226D8—1571no—
227D8—1571no—
228M16—21no—
229M16—3142no—
230M16—3141no—
231M16—1571no—
232M16C22×D15712yes—
233M16C22×D15711yescharacteristic
234C2×C8—21no—
235C2×C8—3142no—
236C2×C8—3141no—
237C2×C8—1571no—
238C2×C8C22×D15711yescharacteristic
239C4⋊C4—6282no—
240C42—1571no—
241Q8—22no—
242Q8—22no—
243Q8—3142no—
244Q8—3142no—
245Q8—6282no—
246Q8—6282no—
247Q8—1571no—
248Q8—1572no—
249Q8D4×D15711yescharacteristic
250D4—21no—
251D4—3141no—
252D4—3142no—
253D4—6282no—
254D4—6282no—
255D4—1571no—
256D4—1572no—
257D4D4×D15711yescharacteristic
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—3141no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—3142no—
264C2×C4—6282no—
265C2×C4—6282no—
266C2×C4—6282no—
267C2×C4—6282no—
268C2×C4C23×D15711yescharacteristic
269C8—21no—
270C8—22no—
271C8—3141no—
272C8—1571no—
273C8—1571no—
274C8D4×D15711yescharacteristic
275C8D4×D15711yescharacteristic
276C22—21no—
277C22—3142no—
278C22—3141no—
279C22#1909781711yescharacteristic
280C4—21no—
281C4—42no—
282C4—3142no—
283C4—3141no—
284C4—3142no—
285C4—6282no—
286C4C2×D4×D15711yescharacteristic
287C4C2×D4×D15711yescharacteristic
288C2—21no—
289C2—41no—
290C2—6282no—
291C2—6281no—
292C2#2375790711yescharacteristic
subgroupG/H
centerC2#23757907
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#473×C157C2
socleC2×C157D42
hypercenter#473D157
hypocenterC157#3293
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #23758090C2semidirect
2. #23758089C2semidirect
3. #23758086C2semidirect
4. #23757651C2semidirect
5. #23757549C2semidirect
6. #23757547C2semidirect
7. #23757458C2semidirect
8. #23757454C2semidirect
9. #23757389C2semidirect
10. #23757388C2semidirect
11. #473×C157C2semidirect
146. C157#3293semidirect
154. #473D157semidirect

Maximal subgroups

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

subgroupindex
#237580902
#237580892
#237580862
#237576512
#237575492
#237575472
#237574582
#237574542
#237573892
#237573882
#473×C1572
#3293157