Finite Group Databasebeta

← #104×#10587739 #31796551

Subgroups — #104×#10587739 #31796551

Order 17536.

360 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 92 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#41×#10587739C211yesmaximal, characteristic
2D137⋊#104C211yesmaximal, characteristic
3M16×#10587739C211yesmaximal, characteristic, complemented
4C42×#10587739C211yesmaximal, characteristic, complemented
5#22578275C211yesmaximal, characteristic, complemented
6#22578274C211yesmaximal, characteristic, complemented
7#104×D137C211yesmaximal, characteristic
8D137⋊M16C2211yescharacteristic
9Q8×#10587739C412yescomplemented
10D4×#10587739—22no—
11D4×#10587739C412yescomplemented
12#17984167—21no—
13#17984167C2211yescharacteristic
14C2×C4×#10587739—22no—
15C2×C4×#10587739—21no—
16C2×C4×#10587739C2211yescharacteristic
17C2×C4×#10587739C2211yescharacteristic
18#17984153C412yescomplemented
19#17984152C412yescomplemented
20#17984142—21no—
21C8×#10587739—21no—
22#41×D137C2211yescharacteristic
23M16×D137C2211yescharacteristic
24C42×D137C2211yescharacteristic
25#17984016C411yescharacteristic, complemented
26#17984015C411yescharacteristic, complemented
27#17984014C411yescharacteristic, complemented
28#104×C137C411yescharacteristic, complemented
29C22×#10587739—22no—
30C22×#10587739—21no—
31C22×#10587739D412yes—
32#14555520C2×C412yescomplemented
33#14555519—21no—
34#14555518—21no—
35C4⋊#10587739—22no—
36C4⋊#10587739C2×C412yes—
37C22⋊#10587739—22no—
38C22⋊#10587739—21no—
39C4×#10587739—28no—
40C4×#10587739—22no—
41C4×#10587739—21no—
42C4×#10587739—21no—
43C4×#10587739C2×C412yes—
44C4×#10587739C2×C412yescomplemented
45C4×#10587739D411yescharacteristic
46C4×#10587739D411yescharacteristic
47#14555511C2×C411yescharacteristic
48#14555510—21no—
49#14555510C2×C411yescharacteristic
50#14555509—21no—
51#14555509C2×C411yescharacteristic
52Q8×D137C2×C411yescharacteristic
53D4×D137—21no—
54D4×D137C2×C411yescharacteristic
55C2×C4×D137—21no—
56C2×C4×D137—21no—
57C2×C4×D137C2311yescharacteristic
58#14555494C2×C411yescharacteristic, complemented
59#14555493—21no—
60C8×D137—21no—
61C4×Dic137—21no—
62C4×Dic137C2×C411yescharacteristic, complemented
63#41×C137C2×C411yescharacteristic
64M16×C137C2×C411yescharacteristic, complemented
65C42×C137C2×C411yescharacteristic, complemented
66C2×#10587739—24no—
67C2×#10587739—22no—
68C2×#10587739—24no—
69C2×#10587739—22no—
70C2×#10587739—44no—
71C2×#10587739C22⋊C412yes—
72C2×#10587739C22⋊C412yes—
73#11764047—22no—
74C22×D137—21no—
75C22×D137C2×D411yescharacteristic
76Dic274—21no—
77Dic274C4211yescharacteristic, complemented
78D548—21no—
79D548C4211yescharacteristic, complemented
80C22⋊D137—21no—
81C22⋊D137—21no—
82C22⋊D137—21no—
83C2×Dic137—21no—
84C2×Dic137—21no—
85C2×Dic137#4111yescharacteristic
86C4×D137—24no—
87C4×D137—21no—
88C4×D137—21no—
89C4×D137—21no—
90C4×D137C22×C411yescharacteristic
91C4×D137C22×C411yescharacteristic
92C4×D137C2×D411yescharacteristic
93C4×D137#4111yescharacteristic
94#11764040—21no—
95Q8×C137C4211yescharacteristic, complemented
96D4×C137—21no—
97D4×C137C4211yescharacteristic, complemented
98C2×C4×C137—21no—
99C2×C4×C137—21no—
100C2×C4×C137C22×C411yescharacteristic
101C8×C137—21no—
102C22×C137—21no—
103C22×C137C4×D411yescharacteristic
104C2×D137—22no—
105C2×D137—21no—
106C2×D137—42no—
107C2×D137C2×C22⋊C411yescharacteristic
108C2×D137C4×D411yescharacteristic
109#10587739—24no—
110#10587739—22no—
111#10587739—42no—
112#10587739#10414yesdirect product factor
113C4×C137—22no—
114C4×C137—21no—
115C4×C137C2×C4211yescharacteristic
116C4×C137C4×D411yescharacteristic
117Dic137—22no—
118Dic137—21no—
119Dic137#11711yescharacteristic
120Dic137C4×D411yescharacteristic
121C2×C137—21no—
122C2×C137—41no—
123C2×C137C4×C22⋊C411yescharacteristic
124D137—21no—
125D137—41no—
126D137C2×#10412yes—
127C137C4×#10411yescharacteristic, complemented, Sylow
128C4×#104—1371nomaximal, Sylow
129C4×#41—1371no—
130C2×#104—1371no—
131C2×#104—1371no—
132C4×M16—1371no—
133C43—1371no—
134#339—1371no—
135#339—1371no—
136C2×#41—1371no—
137C2×M16—1371no—
138C2×M16—1371no—
139C4×Q8—1372no—
140C4×D4—2742no—
141C4×D4—1372no—
142#117—2741no—
143#117—1371no—
144C2×C42—2742no—
145C2×C42—2741no—
146C2×C42—1371no—
147C2×C42—1371no—
148C2×C42—1371no—
149#104—1372no—
150#104—1372no—
151#104—1371no—
152#104—1371no—
153#104—1371no—
154#104#1058773911yescharacteristic, direct product factor
155#97—2741no—
156C4×C8—2741no—
157#41—2741no—
158#41—2741no—
159#41—1371no—
160#41—1371no—
161#41—1371no—
162#41C2×#1058773911yescharacteristic
163C2×Q8—1371no—
164C2×D4—2741no—
165C2×D4—1371no—
166C22×C4—2741no—
167C22×C4—2741no—
168C22×C4—2741no—
169C22×C4—2742no—
170C22×C4—1372no—
171C22×C4—1371no—
172M16—2741no—
173M16—2741no—
174M16—1372no—
175M16—1371no—
176M16C2×#1058773911yescharacteristic, complemented
177C2×C8—2741no—
178C2×C8—2741no—
179C4⋊C4—2742no—
180C4⋊C4—1372no—
181C22⋊C4—2741no—
182C22⋊C4—2742no—
183C42—2741no—
184C42—2748no—
185C42—2741no—
186C42—2742no—
187C42—2741no—
188C42—1372no—
189C42—1371no—
190C42—1371no—
191C42—1371no—
192C42—1372no—
193C42C2×#1058773911yescharacteristic, complemented
194C23—2741no—
195C23—1371no—
196Q8—2741no—
197Q8—1371no—
198Q8C4×#1058773911yescharacteristic, complemented
199D4—21no—
200D4—2741no—
201D4—2741no—
202D4—2741no—
203D4—2741no—
204D4—1371no—
205D4C4×#1058773911yescharacteristic, complemented
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—2744no—
209C2×C4—2741no—
210C2×C4—2741no—
211C2×C4—2744no—
212C2×C4—2742no—
213C2×C4—2741no—
214C2×C4—2741no—
215C2×C4—2744no—
216C2×C4—2741no—
217C2×C4—2742no—
218C2×C4—5484no—
219C2×C4—1372no—
220C2×C4—1372no—
221C2×C4—1371no—
222C2×C4—1371no—
223C2×C4—1371no—
224C2×C4—1371no—
225C2×C4—1371no—
226C2×C4C22×#1058773911yescharacteristic
227C8—21no—
228C8—2742no—
229C8—2741no—
230C22—21no—
231C22—2742no—
232C22—2741no—
233C22—5482no—
234C22—1371no—
235C22—1371no—
236C22D4×#1058773911yescharacteristic
237C4—22no—
238C4—21no—
239C4—2742no—
240C4—2744no—
241C4—2742no—
242C4—2741no—
243C4—5482no—
244C4—1374no—
245C4—1371no—
246C4—1371no—
247C4C2×C4×#1058773911yescharacteristic
248C4D4×#1058773911yescharacteristic
249C2—21no—
250C2—41no—
251C2—2741no—
252C2—5481no—
253C2—1372no—
254C2C22⋊C4×#1058773911yescharacteristic
subgroupG/H
centerC4D4×#10587739
commutatorC4×C137C2×C42
FrattiniC2×C4C22×#10587739
Fitting#104×C137C4
socleC2×C137C4×C22⋊C4
prime radical#41×D137C22
cosocleC2×C4×D137C23
hypercenter#104#10587739
hypocenterC137C4×#104
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
3. M16×#10587739C2semidirect
4. C42×#10587739C2semidirect
5. #22578275C2semidirect
6. #22578274C2semidirect
9. Q8×#10587739C4semidirect
11. D4×#10587739C4semidirect
18. #17984153C4semidirect
19. #17984152C4semidirect
25. #17984016C4semidirect
26. #17984015C4semidirect
27. #17984014C4semidirect
28. #104×C137C4semidirect
32. #14555520C2×C4semidirect
44. C4×#10587739C2×C4semidirect
58. #14555494C2×C4semidirect
62. C4×Dic137C2×C4semidirect
64. M16×C137C2×C4semidirect
65. C42×C137C2×C4semidirect
77. Dic274C42semidirect
79. D548C42semidirect
95. Q8×C137C42semidirect
97. D4×C137C42semidirect
112. #10587739#104direct (N × H)
127. C137C4×#104semidirect
154. #104#10587739direct (N × H)
176. M16C2×#10587739semidirect
193. C42C2×#10587739semidirect
198. Q8C4×#10587739semidirect
205. D4C4×#10587739semidirect

Maximal subgroups

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

subgroupindex
#41×#105877392
D137⋊#1042
M16×#105877392
C42×#105877392
#225782752
#225782742
#104×D1372
C4×#104137