Finite Group Databasebeta

← #31791340

Subgroups — #31791340

Order 17536.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#22577958C211yesmaximal, characteristic
2#125×D137C211yesmaximal, characteristic
3#22577813C211yesmaximal, characteristic, complemented
4#22577812C211yesmaximal, characteristic, complemented
5#22577809C211yesmaximal, characteristic, complemented
6#22577804C212yesmaximal, complemented
7#22577803C212yesmaximal, complemented
8#22577444C212yesmaximal
9#22577442C212yesmaximal
10#22577355C211yesmaximal, characteristic
11#501×C137C211yesmaximal, characteristic, complemented
12C2×#14555493C2212yes—
13#17984088C2211yescharacteristic
14#17984080C2211yescharacteristic
15#17984079C2211yescharacteristic
16#17984079C2211yescharacteristic
17#17984079C2212yes—
18#17984078—21no—
19#17984078—21no—
20#17984068—21no—
21#17984064—21no—
22Q8×Dic137—21no—
23#17984053—21no—
24#17984046C2212yes—
25C4⋊C4×D137—21no—
26C4⋊C4×D137—21no—
27C4⋊C4×D137C2211yescharacteristic
28C4⋊C4×D137C2211yescharacteristic
29C42×D137C2211yescharacteristic
30#17984022C2212yes—
31#17984021C2212yes—
32#17984021C2212yes—
33#17984020C2212yes—
34#17984019C2212yes—
35#17984019C2212yes—
36#17983999C2211yescharacteristic
37C8×Dic137C2212yes—
38C4⋊Q8×C137C2211yescharacteristic
39#125×C137C2211yescharacteristic
40#107×C137C2212yes—
41#106×C137C2212yes—
42#97×C137C2211yescharacteristic
43C2×Dic274—21no—
44C2×C4×D137—21no—
45C2×C4×D137—21no—
46C2×C4×D137D411yescharacteristic
47C2×C4×D137D411yescharacteristic
48C2×C4×D137C2311yescharacteristic
49#14555493Q818yescomplemented
50C2×#11764040C2312yes—
51C4⋊Dic137—21no—
52C4⋊Dic137—21no—
53C4⋊Dic137—22no—
54C4⋊Dic137—21no—
55C4⋊Dic137C2311yescharacteristic
56C4⋊Dic137C2312yes—
57C4⋊Dic137C2311yescharacteristic
58#14555489—21no—
59#14555489—21no—
60#14555489—21no—
61#14555489—21no—
62#14555489—22no—
63#14555489—21no—
64#14555489—21no—
65#14555487—22no—
66C4×Dic137—22no—
67C4×Dic137D411yescharacteristic
68C4×Dic137D411yescharacteristic
69C4×Dic137C2311yescharacteristic
70C2×Q8×C137—21no—
71C2×C8×C137C2312yes—
72C4⋊C4×C137—21no—
73C4⋊C4×C137—21no—
74C4⋊C4×C137—21no—
75C4⋊C4×C137C2311yescharacteristic
76C4⋊C4×C137C2312yes—
77C4⋊C4×C137C2311yescharacteristic
78C42×C137C2311yescharacteristic
79C22×D137C2×D411yescharacteristic
80Dic274—42no—
81C2×Dic137—22no—
82C2×Dic137—21no—
83C2×Dic137—21no—
84C2×Dic137C2×D411yescharacteristic
85C2×Dic137C2×D411yescharacteristic
86C2×Dic137C2×D411yescharacteristic
87C4×D137—41no—
88C4×D137—41no—
89C4×D137—42no—
90C4×D137—42no—
91C4×D137C2×Q814yes—
92#11764040C2×Q814yes—
93Q8×C137—42no—
94C2×C4×C137—21no—
95C2×C4×C137—21no—
96C2×C4×C137—22no—
97C2×C4×C137C2×D411yescharacteristic
98C2×C4×C137C2×D411yescharacteristic
99C2×C4×C137C2411yescharacteristic
100C8×C137C2×Q814yes—
101C22×C137C22×D411yescharacteristic
102C2×D137—21no—
103C2×D137—21no—
104C2×D137C4⋊Q812yes—
105C4×C137—21no—
106C4×C137—21no—
107C4×C137—42no—
108C4×C137—41no—
109C4×C137—41no—
110C4×C137C22×Q812yes—
111Dic137—21no—
112Dic137—21no—
113Dic137—41no—
114Dic137—41no—
115Dic137—42no—
116Dic137C4⋊Q812yes—
117C2×C137C2×C4⋊Q811yescharacteristic
118C2×C137#57511yescharacteristic
119C2×C137#57511yescharacteristic
120D137—22no—
121C137#316611yescharacteristic, complemented, Sylow
122#3166—1371nomaximal, Sylow
123#533—1371no—
124C2×#125—1371no—
125#501—1371no—
126#501—1371no—
127#501—1371no—
128#501D13711yescharacteristic, complemented
129#499—1372no—
130#499—1372no—
131#428—1372no—
132#428—1372no—
133C4×M16—1371no—
134C2×M16—1372no—
135C4⋊Q8—1371no—
136C4⋊Q8C2×D13711yescharacteristic
137#125—2741no—
138#125—2741no—
139#125—1371no—
140#125—1371no—
141#125—1372no—
142#125—1371no—
143#125C2×D13711yescharacteristic
144C22⋊Q8—2741no—
145C22⋊Q8—2741no—
146C4×Q8—2741no—
147C4×Q8—2741no—
148#117—1372no—
149C2×C4⋊C4—2741no—
150C2×C4⋊C4—2741no—
151C2×C4⋊C4—1371no—
152C2×C4⋊C4—1371no—
153C2×C42—1371no—
154#107—1372no—
155#107—1372no—
156#107—1372no—
157#107C2×D13712yes—
158#106—1372no—
159#106—1372no—
160#106—1372no—
161#106C2×D13712yes—
162#97—1371no—
163#97C2×D13711yescharacteristic
164C4×C8—1372no—
165C2×Q8—21no—
166C2×Q8—2741no—
167C22×C4—2741no—
168C22×C4—2741no—
169C22×C4—1371no—
170C22×C4—1371no—
171C22×C4—1371no—
172M16—1378no—
173C2×C8—1372no—
174C2×C8C22×D13712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—2741no—
179C4⋊C4—2741no—
180C4⋊C4—2741no—
181C4⋊C4—2741no—
182C4⋊C4—2742no—
183C4⋊C4—2741no—
184C4⋊C4—2741no—
185C4⋊C4—2742no—
186C4⋊C4—2741no—
187C4⋊C4—2741no—
188C4⋊C4—2741no—
189C4⋊C4—1371no—
190C4⋊C4—1371no—
191C4⋊C4—1372no—
192C4⋊C4C22×D13711yescharacteristic
193C4⋊C4C22×D13712yes—
194C4⋊C4C22×D13711yescharacteristic
195C22⋊C4—2742no—
196C42—2742no—
197C42—1371no—
198C42—1371no—
199C42—1371no—
200C42C22×D13711yescharacteristic
201C23—1371no—
202Q8—42no—
203Q8—5482no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—2742no—
208C2×C4—2741no—
209C2×C4—2741no—
210C2×C4—5481no—
211C2×C4—5481no—
212C2×C4—5482no—
213C2×C4—5482no—
214C2×C4—1371no—
215C2×C4—1371no—
216C2×C4—1371no—
217C2×C4—1374no—
218C2×C4D4×D13711yescharacteristic
219C2×C4D4×D13711yescharacteristic
220C2×C4C23×D13711yescharacteristic
221C8—1374no—
222C8Q8×D13714yes—
223C22—2741no—
224C22—2741no—
225C22—1372no—
226C22C2×D4×D13711yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—2741no—
233C4—2741no—
234C4—5481no—
235C4—5481no—
236C4—5482no—
237C4—1372no—
238C4C2×Q8×D13712yes—
239C2—2742no—
240C2C4⋊Q8×D13711yescharacteristic
241C2#2257819911yescharacteristic
242C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#501×C137C2
socleC22×C137C22×D4
prime radicalC22×D137C2×D4
hypercenter#501D137
hypocenterC137#3166
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #22577813C2semidirect
4. #22577812C2semidirect
5. #22577809C2semidirect
6. #22577804C2semidirect
7. #22577803C2semidirect
11. #501×C137C2semidirect
49. #14555493Q8semidirect
121. C137#3166semidirect
128. #501D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
#125×D1372
#225778132
#225778122
#225778092
#225778042
#225778032
#225774442
#225774422
#225773552
#501×C1372
#3166137