Finite Group Databasebeta

← #34116136

Subgroups — #34116136

Order 20096.

370 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 68 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#23758342C211yesmaximal, characteristic, complemented
2D157⋊#104C211yesmaximal, characteristic, complemented
3#23758265C211yesmaximal, characteristic, complemented
4#23758225C211yesmaximal, characteristic, complemented
5#23758152C211yesmaximal, characteristic, complemented
6#23758145C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×D157C211yesmaximal, characteristic
8D157⋊M16C2211yescharacteristic
9Q8×#10597845—21no—
10D4×#10597845—22no—
11D4×#10597845—21no—
12#19097850—21no—
13#19097850C2211yescharacteristic
14C2×C4⋊#10597845—21no—
15C2×C4⋊#10597845C2211yescharacteristic
16C2×C22⋊#10597845—22no—
17C2×C4×#10597845C2211yescharacteristic
18#19097836—21no—
19#19097835—21no—
20#19097833—21no—
21#19097833—21no—
22#41×D157C2211yescharacteristic
23C2×D4×D157C2211yescharacteristic
24#19097799C411yescharacteristic, complemented
25#19097798—21no—
26#19097797C411yescharacteristic, complemented
27#19097796C411yescharacteristic, complemented
28#19097795—21no—
29SD16×D157—21no—
30D8×D157—21no—
31M16×D157C2211yescharacteristic
32AGL(1,ℤ/8)×C157C411yescharacteristic, complemented
33C22×#10597845—22no—
34C22×#10597845—21no—
35C22×#10597845D412yes—
36#14578964—21no—
37#14578963—21no—
38D157⋊C8—21no—
39C4⋊#10597845—22no—
40C4⋊#10597845—21no—
41C4⋊#10597845—21no—
42C4⋊#10597845D411yescharacteristic
43C4⋊#10597845D411yescharacteristic
44C22⋊#10597845—22no—
45C22⋊#10597845—21no—
46C22⋊#10597845—44no—
47C4×#10597845—22no—
48C4×#10597845—21no—
49C4×#10597845—21no—
50C4×#10597845—21no—
51C23×D157—21no—
52#14578955D411yescharacteristic
53#14578954—21no—
54#14578954C2×C411yescharacteristic
55#14578953—21no—
56#14578953D411yescharacteristic
57Q8×D157D411yescharacteristic
58C2×D628C2×C411yescharacteristic, complemented
59C2×C22⋊D157—21no—
60D4×D157—21no—
61D4×D157—21no—
62D4×D157—21no—
63D4×D157—21no—
64D4×D157—21no—
65D4×D157D411yescharacteristic
66C2×C4×D157—21no—
67C2×C4×D157C2311yescharacteristic
68#14578943—21no—
69#14578942—21no—
70#14578941—21no—
71D1256—21no—
72#14578939—21no—
73#14578939—21no—
74#14578938C2×C411yescharacteristic, complemented
75#14578937—21no—
76C8×D157—21no—
77#41×C157C2×C411yescharacteristic
78C2×D4×C157C2×C411yescharacteristic, complemented
79SD16×C157—21no—
80D8×C157—21no—
81M16×C157C2×C411yescharacteristic, complemented
82C2×#10597845—22no—
83C2×#10597845—21no—
84C2×#10597845—22no—
85C2×#10597845—21no—
86C2×#10597845—41no—
87C2×#10597845—41no—
88C2×#10597845—42no—
89C2×#10597845—82no—
90#11929441—41no—
91C22×D157—21no—
92C22×D157—21no—
93C22×D157—42no—
94C22×D157—42no—
95C22×D157—41no—
96C22×D157—41no—
97C22×D157C2×D411yescharacteristic
98Dic314—21no—
99Dic314C22⋊C411yescharacteristic, complemented
100D628—21no—
101D628—21no—
102D628—21no—
103D628C22⋊C411yescharacteristic, complemented
104C22⋊D157—21no—
105C22⋊D157—21no—
106C22⋊D157—21no—
107C22⋊D157—41no—
108C22⋊D157—41no—
109C2×Dic157—21no—
110C2×Dic157#4111yescharacteristic
111C4×D157—21no—
112C4×D157—21no—
113C4×D157—21no—
114C4×D157C2×D411yescharacteristic
115C4×D157C2×D411yescharacteristic
116C4×D157C2×D411yescharacteristic
117C4×D157#4111yescharacteristic
118#11929434—21no—
119C23×C157—21no—
120Q8×C157C22⋊C411yescharacteristic, complemented
121D4×C157—21no—
122D4×C157—21no—
123D4×C157—21no—
124D4×C157C22⋊C411yescharacteristic, complemented
125C2×C4×C157—21no—
126C2×C4×C157C22×C411yescharacteristic
127C8×C157—21no—
128C22×C157—21no—
129C22×C157—42no—
130C22×C157—41no—
131C22×C157C4×D411yescharacteristic
132C2×D157—21no—
133C2×D157—21no—
134C2×D157—21no—
135C2×D157—41no—
136C2×D157—41no—
137C2×D157—42no—
138C2×D157—41no—
139C2×D157—82no—
140C2×D157—81no—
141C2×D157—81no—
142C2×D157#12011yescharacteristic
143C2×D157#12111yescharacteristic
144#10597845—42no—
145#10597845—42no—
146#10597845—41no—
147#10597845—41no—
148C4×C157—21no—
149C4×C157C2×C22⋊C411yescharacteristic
150C4×C157C4×D411yescharacteristic
151Dic157—21no—
152Dic157#12111yescharacteristic
153Dic157#12311yescharacteristic
154C2×C157—21no—
155C2×C157—41no—
156C2×C157—81no—
157C2×C157#38611yescharacteristic
158D157—21no—
159D157—41no—
160D157—81no—
161D157#45311yescharacteristic
162D157#45411yescharacteristic
163C157#188311yescharacteristic, complemented, Sylow
164#1883—1571nomaximal, Sylow
165C2×AGL(1,ℤ/8)—1571no—
166#520—1571no—
167C2×#104—1571no—
168#418—1571no—
169#390—1571no—
170#343—1571no—
171#339—1571no—
172C2×#41—1571no—
173C22×D4—1571no—
174AGL(1,ℤ/8)—3141no—
175AGL(1,ℤ/8)—3141no—
176AGL(1,ℤ/8)—1571no—
177AGL(1,ℤ/8)—1571no—
178AGL(1,ℤ/8)—1571no—
179AGL(1,ℤ/8)#1059784511yescharacteristic, complemented
180C2×SD16—3141no—
181C2×D8—3141no—
182C2×M16—1571no—
183C2×M16—1571no—
184C4×Q8—3141no—
185C4×D4—3141no—
186C4×D4—3142no—
187#117—3141no—
188#117—1571no—
189C2×C4⋊C4—3141no—
190C2×C4⋊C4—1571no—
191C2×C22⋊C4—3142no—
192C2×C42—1571no—
193#104—3141no—
194#104—3141no—
195#102—3141no—
196#102—3141no—
197C24—3141no—
198#41—3141no—
199#41—3141no—
200#41—1571no—
201#41—1571no—
202#41—1571no—
203#41C2×#1059784511yescharacteristic
204C2×Q8—1571no—
205C2×D4—3141no—
206C2×D4—3141no—
207C2×D4—3141no—
208C2×D4—3141no—
209C2×D4—3141no—
210C2×D4—3141no—
211C2×D4—1571no—
212C2×D4—1571no—
213C2×D4C2×#1059784511yescharacteristic, complemented
214C22×C4—3141no—
215C22×C4—3141no—
216C22×C4—3142no—
217C22×C4—1572no—
218C22×C4—1571no—
219SD16—21no—
220SD16—3141no—
221SD16—3141no—
222SD16—3141no—
223D8—21no—
224D8—3141no—
225D8—3141no—
226D8—3141no—
227M16—3141no—
228M16—3141no—
229M16—3141no—
230M16—1571no—
231M16C2×#1059784511yescharacteristic, complemented
232C2×C8—3141no—
233C2×C8—3141no—
234C4⋊C4—3142no—
235C4⋊C4—3141no—
236C4⋊C4—3141no—
237C4⋊C4—1571no—
238C4⋊C4—1571no—
239C22⋊C4—3142no—
240C22⋊C4—3141no—
241C22⋊C4—6284no—
242C42—3142no—
243C42—3141no—
244C42—3141no—
245C42—3141no—
246C23—21no—
247C23—3141no—
248C23—3141no—
249C23—6282no—
250C23—6282no—
251C23—6281no—
252C23—6281no—
253C23—1571no—
254Q8—3141no—
255Q8—1571no—
256Q8C22⋊#1059784511yescharacteristic, complemented
257D4—21no—
258D4—21no—
259D4—21no—
260D4—3141no—
261D4—3141no—
262D4—3141no—
263D4—3141no—
264D4—3141no—
265D4—3141no—
266D4—6281no—
267D4—6281no—
268D4—1571no—
269D4C22⋊#1059784511yescharacteristic, complemented
270C2×C4—21no—
271C2×C4—3141no—
272C2×C4—3141no—
273C2×C4—3141no—
274C2×C4—3141no—
275C2×C4—3142no—
276C2×C4—3141no—
277C2×C4—3142no—
278C2×C4—3141no—
279C2×C4—6281no—
280C2×C4—6281no—
281C2×C4—6282no—
282C2×C4—1571no—
283C2×C4—1571no—
284C2×C4—1571no—
285C2×C4—1571no—
286C2×C4—1571no—
287C2×C4—12562no—
288C2×C4C22×#1059784511yescharacteristic
289C8—21no—
290C8—3141no—
291C8—6281no—
292C22—21no—
293C22—42no—
294C22—41no—
295C22—3141no—
296C22—3141no—
297C22—3141no—
298C22—6282no—
299C22—6281no—
300C22—6281no—
301C22—6281no—
302C22—1571no—
303C22—1571no—
304C22—12562no—
305C22—12561no—
306C22—12561no—
307C22D4×#1059784511yescharacteristic
308C4—21no—
309C4—3141no—
310C4—6282no—
311C4—6282no—
312C4—6281no—
313C4—6281no—
314C4—1571no—
315C4—1571no—
316C4C2×C22⋊#1059784511yescharacteristic
317C4D4×#1059784511yescharacteristic
318C2—21no—
319C2—41no—
320C2—81no—
321C2—3141no—
322C2—6281no—
323C2—1571no—
324C2—1571no—
325C2—12561no—
326C2#2375822111yescharacteristic
subgroupG/H
centerC2#23758221
commutatorC2×C4×C157C22×C4
FrattiniC2×C4C22×#10597845
FittingAGL(1,ℤ/8)×C157C4
socleC2×C157#386
prime radicalAGL(1,ℤ/8)×D157C2
cosocleC2×C4×D157C23
hypercenterAGL(1,ℤ/8)#10597845
hypocenterC157#1883
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23758342C2semidirect
2. D157⋊#104C2semidirect
3. #23758265C2semidirect
4. #23758225C2semidirect
5. #23758152C2semidirect
6. #23758145C2semidirect
24. #19097799C4semidirect
26. #19097797C4semidirect
27. #19097796C4semidirect
32. AGL(1,ℤ/8)×C157C4semidirect
58. C2×D628C2×C4semidirect
74. #14578938C2×C4semidirect
78. C2×D4×C157C2×C4semidirect
81. M16×C157C2×C4semidirect
99. Dic314C22⋊C4semidirect
103. D628C22⋊C4semidirect
120. Q8×C157C22⋊C4semidirect
124. D4×C157C22⋊C4semidirect
163. C157#1883semidirect
179. AGL(1,ℤ/8)#10597845semidirect
213. C2×D4C2×#10597845semidirect
231. M16C2×#10597845semidirect
256. Q8C22⋊#10597845semidirect
269. D4C22⋊#10597845semidirect

Maximal subgroups

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

subgroupindex
#237583422
D157⋊#1042
#237582652
#237582252
#237581522
#237581452
AGL(1,ℤ/8)×D1572
#1883157