Finite Group Databasebeta

← #31791113

Subgroups — #31791113

Order 17536.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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, complemented
2#22577954C211yesmaximal, characteristic
3#22577759C214yesmaximal, complemented
4#22577753C211yesmaximal, characteristic, complemented
5#22577751C211yesmaximal, characteristic, complemented
6#22577750C211yesmaximal, characteristic, complemented
7#22577398C214yesmaximal, complemented
8C4×C8×D137C211yesmaximal, characteristic, complemented
9#487×C137C211yesmaximal, characteristic, complemented
10C2×#14555511C2212yes—
11C2×C8×D137C2212yes—
12#17984088C2211yescharacteristic
13#17984085C2211yescharacteristic
14#17984079C2212yes—
15#17984077C2212yes—
16#17984068—21no—
17#17984064—21no—
18#17984063—21no—
19#17984063—21no—
20Q8×Dic137—21no—
21#17984053—21no—
22#17984050—21no—
23#17984050—21no—
24#17984046C2212yes—
25C42×D137C2211yescharacteristic
26#17984012C2214yes—
27#17984010C2214yes—
28#17984008C2214yes—
29C4×#11764040C2211yescharacteristic
30C8×Dic137C2212yes—
31C4⋊Q8×C137C2211yescharacteristic
32C4⋊Q8×C137C2211yescharacteristic
33#103×C137C2214yes—
34C4×C8×C137C2211yescharacteristic
35#14555511—28no—
36C2×Dic274—21no—
37C2×D548—22no—
38C2×D548—21no—
39C2×D548—21no—
40C2×D548C2312yes—
41C2×C4×D137—22no—
42C2×C4×D137D411yescharacteristic
43C2×C4×D137D411yescharacteristic
44C2×C4×D137C2311yescharacteristic
45C2×#11764040C2312yes—
46C8×D137—24no—
47C4⋊Dic137—22no—
48C4⋊Dic137—21no—
49C4⋊Dic137C2312yes—
50#14555489—22no—
51#14555487—22no—
52#14555487—22no—
53#14555487—22no—
54C4×Dic137—22no—
55C4×Dic137D411yescharacteristic
56C4×Dic137D411yescharacteristic
57C4×Dic137C2311yescharacteristic
58C2×Q8×C137—21no—
59C2×Q8×C137C2312yes—
60C2×C8×C137C2312yes—
61C4⋊C4×C137—21no—
62C4⋊C4×C137—21no—
63C4⋊C4×C137—21no—
64C4⋊C4×C137C2312yes—
65C42×C137C2311yescharacteristic
66C22×D137—22no—
67C22×D137C2×D411yescharacteristic
68Dic274—42no—
69D548—24no—
70D548—44no—
71D548—42no—
72D548—42no—
73C2×Dic137—22no—
74C2×Dic137C2×D411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C2×Dic137C2×D411yescharacteristic
77C4×D137—22no—
78C4×D137—22no—
79C4×D137—44no—
80C4×D137#4114yes—
81#11764040—22no—
82Q8×C137—24no—
83Q8×C137—42no—
84C2×C4×C137—22no—
85C2×C4×C137—22no—
86C2×C4×C137C2×D411yescharacteristic
87C2×C4×C137C2×D411yescharacteristic
88C2×C4×C137C2411yescharacteristic
89C8×C137—22no—
90C22×C137C22×D411yescharacteristic
91C2×D137—21no—
92C2×D137—21no—
93C2×D137—42no—
94C2×D137—42no—
95C2×D137—42no—
96C2×D137#12412yes—
97C4×C137—42no—
98C4×C137—42no—
99C4×C137#13512yes—
100C4×C137#13512yes—
101C4×C137C2×#4112yes—
102Dic137—21no—
103Dic137—21no—
104Dic137—42no—
105Dic137#12412yes—
106C2×C137C2×#12411yescharacteristic
107C2×C137C2×#13511yescharacteristic
108C2×C137C2×#13511yescharacteristic
109D137—22no—
110D137—82no—
111C137#313111yescharacteristic, complemented, Sylow
112#3131—1371nomaximal, Sylow
113#533—1371no—
114#532—1371no—
115#488—1374no—
116#487—1371no—
117#487D13711yescharacteristic, complemented
118#486—1371no—
119#486—1371no—
120#416—1374no—
121C2×C4×C8—1371no—
122C2×#41—1372no—
123C22×C8—1372no—
124C4⋊Q8—1371no—
125C4⋊Q8C2×D13711yescharacteristic
126C4⋊Q8C2×D13711yescharacteristic
127#127—1371no—
128#125—1372no—
129#124—1372no—
130C22⋊Q8—2741no—
131C22⋊Q8—2741no—
132#121—2741no—
133#121—2741no—
134C4×Q8—2741no—
135C4×Q8—2741no—
136C4×D4—2741no—
137C4×D4—2741no—
138#117—1372no—
139C2×C42—1371no—
140#103—1374no—
141#103C2×D13714yes—
142#102—1374no—
143#102—1374no—
144C4×C8—1372no—
145C4×C8—1371no—
146C4×C8C2×D13711yescharacteristic
147#41—2748no—
148C2×Q8—21no—
149C2×Q8—2741no—
150C2×Q8C22×D13712yes—
151C2×D4—2742no—
152C2×D4—2741no—
153C2×D4—2741no—
154C2×D4—1372no—
155C22×C4—2742no—
156C22×C4—1371no—
157C22×C4—1371no—
158C22×C4—1371no—
159C2×C8—2744no—
160C2×C8—1372no—
161C2×C8C22×D13712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—2742no—
166C4⋊C4—2742no—
167C4⋊C4—2741no—
168C4⋊C4—1372no—
169C4⋊C4C22×D13712yes—
170C22⋊C4—2742no—
171C22⋊C4—2742no—
172C22⋊C4—2742no—
173C42—2742no—
174C42—1371no—
175C42—1371no—
176C42—1371no—
177C42C22×D13711yescharacteristic
178C23—2742no—
179C23—1371no—
180Q8—24no—
181Q8—42no—
182Q8—5482no—
183D4—2744no—
184D4—5484no—
185D4—5482no—
186D4—5482no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—2742no—
190C2×C4—2742no—
191C2×C4—2742no—
192C2×C4—5484no—
193C2×C4—1371no—
194C2×C4—1371no—
195C2×C4—1371no—
196C2×C4—1374no—
197C2×C4D4×D13711yescharacteristic
198C2×C4D4×D13711yescharacteristic
199C2×C4C23×D13711yescharacteristic
200C8—22no—
201C8—2742no—
202C22—2741no—
203C22—2741no—
204C22—5482no—
205C22—5482no—
206C22—5482no—
207C22—1372no—
208C22C2×D4×D13711yescharacteristic
209C4—42no—
210C4—42no—
211C4—2741no—
212C4—2741no—
213C4—5482no—
214C4—1372no—
215C4#1798410912yes—
216C4#1798411112yes—
217C4#41×D13712yes—
218C2—2742no—
219C2—10962no—
220C2#124×D13711yescharacteristic
221C2C2×#1798410911yescharacteristic
222C2C2×#1798411111yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
prime radical#22577954C2
hypercenter#487D137
hypocenterC137#3131
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. #22577958C2semidirect
3. #22577759C2semidirect
4. #22577753C2semidirect
5. #22577751C2semidirect
6. #22577750C2semidirect
7. #22577398C2semidirect
8. C4×C8×D137C2semidirect
9. #487×C137C2semidirect
111. C137#3131semidirect
117. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225779582
#225779542
#225777592
#225777532
#225777512
#225777502
#225773982
C4×C8×D1372
#487×C1372
#3131137