Finite Group Databasebeta

← #31792705

Subgroups — #31792705

Order 17536.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#22578127C211yesmaximal, characteristic, complemented
2#22578056C211yesmaximal, characteristic
3#22577742C212yesmaximal, complemented
4#22577680C211yesmaximal, characteristic, complemented
5#22577643C212yesmaximal, complemented
6#22577610C211yesmaximal, characteristic, complemented
7#22577568C212yesmaximal, complemented
8C4×#14555499C211yesmaximal, characteristic, complemented
9C8×D548C211yesmaximal, characteristic, complemented
10#22577398C212yesmaximal, complemented
11#487×C137C211yesmaximal, characteristic, complemented
12C2×#14555511C2212yes—
13C2×#14555500C2212yes—
14C2×#14555499C2211yescharacteristic
15C2×C8×D137C2212yes—
16#17984083—22no—
17#17984077—21no—
18#17984073—22no—
19#17984068C2212yes—
20#17984064C2212yes—
21#17984063—21no—
22Q8×Dic137—21no—
23C4×Dic274C2211yescharacteristic
24C4×D548C2211yescharacteristic
25C4×D548C2211yescharacteristic
26#17984046C2212yes—
27C4⋊C4×D137—21no—
28#17984021C2212yes—
29#17984020C2211yescharacteristic
30C4⋊#11764040C2211yescharacteristic
31#17984013C2211yescharacteristic
32#17984012C2212yes—
33#17984010C2211yescharacteristic
34#17984008C2212yes—
35#17984000C2212yes—
36C4⋊Q8×C137C2211yescharacteristic
37C4⋊Q8×C137C2211yescharacteristic
38#103×C137C2212yes—
39#103×C137C2212yes—
40C4×C8×C137C2211yescharacteristic
41#14555511—24no—
42#14555511—42no—
43C2×Dic274C2311yescharacteristic
44C2×D548—22no—
45C2×D548D411yescharacteristic
46C2×D548C2311yescharacteristic
47C2×C4×D137—22no—
48C2×C4×D137—21no—
49C2×C4×D137C2312yes—
50#14555500—42no—
51#14555499—22no—
52C2×#11764040C2312yes—
53C8×D137—42no—
54C4⋊Dic137—22no—
55C4⋊Dic137D411yescharacteristic
56C4⋊Dic137C2311yescharacteristic
57C4⋊Dic137C2311yescharacteristic
58#14555489—22no—
59#14555489—21no—
60#14555487—22no—
61#14555487—22no—
62#14555487—22no—
63#14555487—21no—
64#14555487D412yes—
65C4×Dic137—22no—
66C4×Dic137—21no—
67C2×Q8×C137—21no—
68C2×Q8×C137C2312yes—
69C2×C8×C137C2311yescharacteristic
70C2×C8×C137C2311yescharacteristic
71C4⋊C4×C137—21no—
72C4⋊C4×C137—21no—
73C4⋊C4×C137—21no—
74C4⋊C4×C137C2312yes—
75C42×C137C2311yescharacteristic
76C22×D137—21no—
77C22×D137C2×D412yes—
78Dic274—21no—
79Dic274—21no—
80D548—22no—
81D548—21no—
82D548—44no—
83D548#4112yes—
84C2×Dic137—21no—
85C2×Dic137—21no—
86C2×Dic137—21no—
87C2×Dic137C2×D412yes—
88C4×D137—22no—
89C4×D137—22no—
90C4×D137—44no—
91C4×D137—42no—
92#11764040—22no—
93Q8×C137—22no—
94Q8×C137—22no—
95Q8×C137—42no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137C2×D411yescharacteristic
99C2×C4×C137C2×D411yescharacteristic
100C2×C4×C137C2411yescharacteristic
101C8×C137—21no—
102C8×C137—21no—
103C22×C137C22×D411yescharacteristic
104C2×D137—22no—
105C2×D137—22no—
106C2×D137—22no—
107C2×D137—22no—
108C2×D137—41no—
109C2×D137—41no—
110C4×C137—21no—
111C4×C137—42no—
112C4×C137—42no—
113C4×C137#13512yes—
114C4×C137C2×#4111yescharacteristic
115C4×C137H(5,𝔽2)11yescharacteristic
116Dic137—22no—
117Dic137—41no—
118Dic137—41no—
119Dic137—41no—
120C2×C137#54611yescharacteristic
121C2×C137C2×#13511yescharacteristic
122C2×C137#57811yescharacteristic
123D137—42no—
124D137—42no—
125C137#330211yescharacteristic, complemented, Sylow
126#3302—1371nomaximal, Sylow
127#556—1371no—
128#547—1371no—
129#487D13711yescharacteristic, complemented
130#485—1372no—
131#474—1371no—
132#467—1372no—
133#462—1371no—
134#452—1372no—
135C4×SD16—1371no—
136C8×D4—1371no—
137#416—1372no—
138C2×#41—1372no—
139C2×Q16—1372no—
140C2×SD16—1371no—
141C22×C8—1372no—
142C4⋊Q8C2×D13711yescharacteristic
143C4⋊Q8C2×D13711yescharacteristic
144#126—2742no—
145#124—2741no—
146#123—2742no—
147C22⋊Q8—1372no—
148C22⋊Q8—1372no—
149#121—2741no—
150C4×Q8—2741no—
151C4×Q8—1371no—
152C4×D4—1371no—
153C4×D4—1371no—
154#117—1372no—
155C2×C4⋊C4—2741no—
156#107—1372no—
157#106—1371no—
158C4⋊C8—1371no—
159#103—1372no—
160#103—1371no—
161#103C2×D13712yes—
162#103C2×D13712yes—
163#102—1371no—
164#102—1372no—
165C22⋊C8—1372no—
166C4×C8C2×D13711yescharacteristic
167#41—2744no—
168#41—5482no—
169C2×Q8—21no—
170C2×Q8—1371no—
171C2×Q8C22×D13712yes—
172C2×D4—2742no—
173C2×D4—1371no—
174C2×D4—1371no—
175C22×C4—2742no—
176C22×C4—2741no—
177C22×C4—1372no—
178Q16—5482no—
179SD16—2742no—
180C2×C8—5482no—
181C2×C8—1372no—
182C2×C8C22×D13711yescharacteristic
183C2×C8C22×D13711yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—2742no—
188C4⋊C4—2742no—
189C4⋊C4—2741no—
190C4⋊C4—1371no—
191C4⋊C4—1371no—
192C4⋊C4—1371no—
193C4⋊C4C22×D13712yes—
194C22⋊C4—2742no—
195C22⋊C4—2742no—
196C22⋊C4—2742no—
197C22⋊C4—2741no—
198C22⋊C4—1372no—
199C42—2742no—
200C42—2741no—
201C42C22×D13711yescharacteristic
202C23—2741no—
203C23—1372no—
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—2741no—
208Q8—2741no—
209D4—2742no—
210D4—2741no—
211D4—5484no—
212D4—1372no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—2742no—
216C2×C4—2742no—
217C2×C4—2741no—
218C2×C4—2741no—
219C2×C4—2741no—
220C2×C4—5484no—
221C2×C4—5482no—
222C2×C4—1372no—
223C2×C4D4×D13711yescharacteristic
224C2×C4D4×D13711yescharacteristic
225C2×C4C23×D13711yescharacteristic
226C8—21no—
227C8—21no—
228C8—2742no—
229C22—2742no—
230C22—2742no—
231C22—2742no—
232C22—2742no—
233C22—5481no—
234C22—5481no—
235C22C2×D4×D13711yescharacteristic
236C4—21no—
237C4—42no—
238C4—42no—
239C4—2742no—
240C4—5481no—
241C4—5481no—
242C4—5481no—
243C4#1798410912yes—
244C4#41×D13711yescharacteristic
245C4#1798413411yescharacteristic
246C2—5482no—
247C2—5482no—
248C2#2257804611yescharacteristic
249C2C2×#1798410911yescharacteristic
250C2#2257821611yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
prime radical#22578056C2
hypercenter#487D137
hypocenterC137#3302
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #22578127C2semidirect
3. #22577742C2semidirect
4. #22577680C2semidirect
5. #22577643C2semidirect
6. #22577610C2semidirect
7. #22577568C2semidirect
8. C4×#14555499C2semidirect
9. C8×D548C2semidirect
10. #22577398C2semidirect
11. #487×C137C2semidirect
125. C137#3302semidirect
129. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225781272
#225780562
#225777422
#225776802
#225776432
#225776102
#225775682
C4×#145554992
C8×D5482
#225773982
#487×C1372
#3302137