Finite Group Databasebeta

← #31796748

Subgroups — #31796748

Order 17536.

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

normal subgroups only

This group has 302 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#22578472C211yesmaximal, characteristic
2#22578414C211yesmaximal, characteristic, complemented
3#22578394C211yesmaximal, characteristic
4#22578349C211yesmaximal, characteristic, complemented
5#22578282C211yesmaximal, characteristic, complemented
6#22578274C211yesmaximal, characteristic, complemented
7#104×D137C211yesmaximal, characteristic
8D137⋊M16C2211yescharacteristic
9Q8×#10587739—21no—
10D4×#10587739—22no—
11D4×#10587739—21no—
12#17984167—21no—
13#17984167C2211yescharacteristic
14C2×C4⋊#10587739—21no—
15C2×C4⋊#10587739—22no—
16C2×C4⋊#10587739C2211yescharacteristic
17C2×C4⋊#10587739C2211yescharacteristic
18#17984157—21no—
19#17984156—21no—
20#17984151—21no—
21#17984150—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#14555520—21no—
33#14555519—21no—
34#14555518—21no—
35C4⋊#10587739—22no—
36C4⋊#10587739—21no—
37C4⋊#10587739—21no—
38C4⋊#10587739—21no—
39C4⋊#10587739—21no—
40C4⋊#10587739—44no—
41C4⋊#10587739D411yescharacteristic
42C4⋊#10587739D411yescharacteristic
43C22⋊#10587739—22no—
44C22⋊#10587739—21no—
45C4×#10587739—22no—
46C4×#10587739—21no—
47#14555511Q811yescharacteristic
48#14555510—21no—
49#14555510C2×C411yescharacteristic
50#14555509—21no—
51#14555509Q811yescharacteristic
52Q8×D137D411yescharacteristic
53D4×D137—21no—
54D4×D137D411yescharacteristic
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—21no—
67C2×#10587739—22no—
68C2×#10587739—21no—
69C2×#10587739—42no—
70C2×#10587739—42no—
71C2×#10587739—41no—
72C2×#10587739—82no—
73#11764047—41no—
74C22×D137—21no—
75C22×D137C2×D411yescharacteristic
76Dic274—21no—
77Dic274C4⋊C411yescharacteristic, complemented
78D548—21no—
79D548C4⋊C411yescharacteristic, complemented
80C22⋊D137—21no—
81C22⋊D137—21no—
82C22⋊D137—21no—
83C2×Dic137—21no—
84C2×Dic137—21no—
85C2×Dic137#4111yescharacteristic
86C4×D137—22no—
87C4×D137—22no—
88C4×D137—21no—
89C4×D137—21no—
90C4×D137—21no—
91C4×D137C2×D411yescharacteristic
92C4×D137C2×D411yescharacteristic
93C4×D137C2×Q811yescharacteristic
94C4×D137#4111yescharacteristic
95#11764040—21no—
96Q8×C137C4⋊C411yescharacteristic, complemented
97D4×C137—21no—
98D4×C137C4⋊C411yescharacteristic, complemented
99C2×C4×C137—21no—
100C2×C4×C137—21no—
101C2×C4×C137C22×C411yescharacteristic
102C8×C137—21no—
103C22×C137—21no—
104C22×C137C4×D411yescharacteristic
105C2×D137—21no—
106C2×D137—21no—
107C2×D137—21no—
108C2×D137—41no—
109C2×D137—41no—
110C2×D137#12011yescharacteristic
111C2×D137C22⋊Q811yescharacteristic
112#10587739—42no—
113#10587739—41no—
114#10587739—41no—
115#10587739—81no—
116C4×C137—22no—
117C4×C137—21no—
118C4×C137C2×C4⋊C411yescharacteristic
119C4×C137C4×D411yescharacteristic
120Dic137—22no—
121Dic137—21no—
122Dic137C22⋊Q811yescharacteristic
123Dic137#12311yescharacteristic
124C2×C137—21no—
125C2×C137—41no—
126C2×C137#38511yescharacteristic
127D137—21no—
128D137—41no—
129D137#45311yescharacteristic
130D137#45611yescharacteristic
131C137#185911yescharacteristic, complemented, Sylow
132#1859—1371nomaximal, Sylow
133#520—1371no—
134#428—1371no—
135C2×#104—1371no—
136#417—1371no—
137#384—1371no—
138#343—1371no—
139#339—1371no—
140C2×#41—1371no—
141C2×M16—1371no—
142C2×M16—1371no—
143C4×Q8—2741no—
144C4×D4—2741no—
145C4×D4—2742no—
146#117—2741no—
147#117—1371no—
148C2×C4⋊C4—2741no—
149C2×C4⋊C4—2742no—
150C2×C4⋊C4—1371no—
151C2×C4⋊C4—1371no—
152C2×C42—1371no—
153#107—2741no—
154#106—2741no—
155#104—1371no—
156#104—1371no—
157#104—1371no—
158#104#1058773911yescharacteristic, complemented
159#103—2741no—
160#102—2741no—
161#41—2741no—
162#41—2741no—
163#41—1371no—
164#41—1371no—
165#41—1371no—
166#41C2×#1058773911yescharacteristic
167C2×Q8—1371no—
168C2×D4—2741no—
169C2×D4—1371no—
170C22×C4—2741no—
171C22×C4—2741no—
172C22×C4—2742no—
173C22×C4—2741no—
174C22×C4—1372no—
175C22×C4—1371no—
176M16—2741no—
177M16—2741no—
178M16—2741no—
179M16—1371no—
180M16C2×#1058773911yescharacteristic, complemented
181C2×C8—2741no—
182C2×C8—2741no—
183C4⋊C4—2742no—
184C4⋊C4—2741no—
185C4⋊C4—2741no—
186C4⋊C4—2741no—
187C4⋊C4—2741no—
188C4⋊C4—5484no—
189C4⋊C4—1371no—
190C4⋊C4—1371no—
191C22⋊C4—2742no—
192C22⋊C4—2741no—
193C42—2741no—
194C42—2742no—
195C42—2741no—
196C42—1371no—
197C42C2×#1058773911yescharacteristic, complemented
198C23—2741no—
199C23—1371no—
200Q8—2741no—
201Q8—1371no—
202Q8C4⋊#1058773911yescharacteristic, complemented
203D4—21no—
204D4—2741no—
205D4—2741no—
206D4—2741no—
207D4—2741no—
208D4—1371no—
209D4C4⋊#1058773911yescharacteristic, complemented
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—2742no—
213C2×C4—2742no—
214C2×C4—2741no—
215C2×C4—2741no—
216C2×C4—2741no—
217C2×C4—2741no—
218C2×C4—2741no—
219C2×C4—2741no—
220C2×C4—2742no—
221C2×C4—2741no—
222C2×C4—5482no—
223C2×C4—5482no—
224C2×C4—5481no—
225C2×C4—10962no—
226C2×C4—1371no—
227C2×C4—1371no—
228C2×C4—1371no—
229C2×C4—1371no—
230C2×C4—1371no—
231C2×C4C22×#1058773911yescharacteristic
232C8—21no—
233C8—2741no—
234C8—5481no—
235C22—21no—
236C22—2741no—
237C22—2741no—
238C22—2741no—
239C22—5481no—
240C22—5481no—
241C22—1371no—
242C22—1371no—
243C22D4×#1058773911yescharacteristic
244C4—22no—
245C4—21no—
246C4—2742no—
247C4—2741no—
248C4—5482no—
249C4—5481no—
250C4—5481no—
251C4—10961no—
252C4—1371no—
253C4—1371no—
254C4C2×C4⋊#1058773911yescharacteristic
255C4D4×#1058773911yescharacteristic
256C2—21no—
257C2—41no—
258C2—2741no—
259C2—5481no—
260C2—1371no—
261C2—1371no—
262C2#2257835011yescharacteristic
subgroupG/H
centerC2#22578350
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#104×C137C4
socleC2×C137#385
prime radical#41×D137C22
cosocleC2×C4×D137C23
hypercenter#104#10587739
hypocenterC137#1859
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #22578414C2semidirect
4. #22578349C2semidirect
5. #22578282C2semidirect
6. #22578274C2semidirect
25. #17984016C4semidirect
26. #17984015C4semidirect
27. #17984014C4semidirect
28. #104×C137C4semidirect
58. #14555494C2×C4semidirect
62. C4×Dic137C2×C4semidirect
64. M16×C137C2×C4semidirect
65. C42×C137C2×C4semidirect
77. Dic274C4⋊C4semidirect
79. D548C4⋊C4semidirect
96. Q8×C137C4⋊C4semidirect
98. D4×C137C4⋊C4semidirect
131. C137#1859semidirect
158. #104#10587739semidirect
180. M16C2×#10587739semidirect
197. C42C2×#10587739semidirect
202. Q8C4⋊#10587739semidirect
209. D4C4⋊#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784722
#225784142
#225783942
#225783492
#225782822
#225782742
#104×D1372
#1859137