Finite Group Databasebeta

← #31791184

Subgroups — #31791184

Order 17536.

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

normal subgroups only

This group has 424 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#22577954C211yesmaximal, characteristic
2C4⋊Q8×D137C211yesmaximal, characteristic, complemented
3#22577764C214yesmaximal, complemented
4#22577753C211yesmaximal, characteristic, complemented
5#22577751C211yesmaximal, characteristic, complemented
6#22577750C211yesmaximal, characteristic, complemented
7#22577403C214yesmaximal, complemented
8C4×#14555493C211yesmaximal, characteristic, complemented
9#487×C137C211yesmaximal, characteristic, complemented
10C2×#14555511C2212yes—
11C2×Q8×D137—21no—
12C2×#14555493C2212yes—
13#17984088C2211yescharacteristic
14#17984087—21no—
15#17984087C2212yes—
16#17984086—21no—
17#17984085C2211yescharacteristic
18#17984077C2212yes—
19#17984063—21no—
20#17984063—21no—
21#17984050—21no—
22#17984050—21no—
23C4⋊C4×D137—21no—
24C4⋊C4×D137C2212yes—
25C42×D137C2211yescharacteristic
26#17984012C2214yes—
27#17984010C2214yes—
28#17984008C2214yes—
29#17983998C2212yes—
30C4×#11764040C2211yescharacteristic
31C4⋊Q8×C137C2211yescharacteristic
32C4⋊Q8×C137C2211yescharacteristic
33#103×C137C2214yes—
34C4×C8×C137C2211yescharacteristic
35#14555511—28no—
36C2×Dic274—22no—
37C2×Dic274—21no—
38Q8×D137—44no—
39C2×D548—22no—
40C2×D548—21no—
41C2×D548—21no—
42C2×D548C2312yes—
43C2×C4×D137—22no—
44C2×C4×D137—22no—
45C2×C4×D137D411yescharacteristic
46C2×C4×D137D411yescharacteristic
47C2×C4×D137C2311yescharacteristic
48#14555493—24no—
49C2×#11764040C2312yes—
50C4⋊Dic137—21no—
51C4⋊Dic137C2312yes—
52#14555489—22no—
53#14555489—22no—
54#14555487—22no—
55#14555487—22no—
56C4×Dic137D411yescharacteristic
57C4×Dic137D411yescharacteristic
58C4×Dic137C2311yescharacteristic
59C2×Q8×C137—21no—
60C2×Q8×C137C2312yes—
61C2×C8×C137C2312yes—
62C4⋊C4×C137—21no—
63C4⋊C4×C137—21no—
64C4⋊C4×C137—21no—
65C4⋊C4×C137C2312yes—
66C42×C137C2311yescharacteristic
67C22×D137—22no—
68C22×D137C2×D411yescharacteristic
69Dic274—44no—
70Dic274—42no—
71D548—24no—
72D548—44no—
73D548—42no—
74D548—42no—
75C2×Dic137—22no—
76C2×Dic137C2×D411yescharacteristic
77C2×Dic137C2×D411yescharacteristic
78C2×Dic137C2×D411yescharacteristic
79C4×D137—22no—
80C4×D137—22no—
81C4×D137—44no—
82C4×D137—44no—
83C4×D137#4114yes—
84#11764040—22no—
85Q8×C137—24no—
86Q8×C137—42no—
87C2×C4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137C2×D411yescharacteristic
90C2×C4×C137C2×D411yescharacteristic
91C2×C4×C137C2411yescharacteristic
92C8×C137—22no—
93C22×C137C22×D411yescharacteristic
94C2×D137—21no—
95C2×D137—21no—
96C2×D137—42no—
97C2×D137—42no—
98C2×D137—42no—
99C2×D137#12412yes—
100C4×C137—42no—
101C4×C137—42no—
102C4×C137AGL(1,ℤ/8)12yes—
103C4×C137#13712yes—
104C4×C137C2×#4112yes—
105Dic137—21no—
106Dic137—21no—
107Dic137—42no—
108Dic137#12412yes—
109C2×C137C2×#12411yescharacteristic
110C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C137C2×#13711yescharacteristic
112D137—22no—
113D137—82no—
114C137#314111yescharacteristic, complemented, Sylow
115#3141—1371nomaximal, Sylow
116#532—1371no—
117C2×C4⋊Q8—1371no—
118#489—1374no—
119#487—1371no—
120#487D13711yescharacteristic, complemented
121#486—1371no—
122#486—1371no—
123#417—1374no—
124C4×M16—1371no—
125C2×#41—1372no—
126C22×Q8—2741no—
127C2×M16—1372no—
128C4⋊Q8—2741no—
129C4⋊Q8—2741no—
130C4⋊Q8—1372no—
131C4⋊Q8—1371no—
132C4⋊Q8C2×D13711yescharacteristic
133C4⋊Q8C2×D13711yescharacteristic
134#127—1371no—
135#124—1372no—
136#121—2741no—
137#121—2741no—
138C4×D4—2741no—
139C4×D4—2741no—
140C2×C4⋊C4—2741no—
141C2×C4⋊C4—1372no—
142C2×C42—1371no—
143#103—1374no—
144#103C2×D13714yes—
145#102—1374no—
146#102—1374no—
147#97—1372no—
148C4×C8—1371no—
149C4×C8C2×D13711yescharacteristic
150#41—2748no—
151C2×Q8—21no—
152C2×Q8—2742no—
153C2×Q8—2741no—
154C2×Q8—5484no—
155C2×Q8C22×D13712yes—
156C2×D4—2742no—
157C2×D4—2741no—
158C2×D4—2741no—
159C2×D4—1372no—
160C22×C4—2742no—
161C22×C4—2742no—
162C22×C4—1371no—
163C22×C4—1371no—
164C22×C4—1371no—
165M16—2744no—
166C2×C8—1372no—
167C2×C8C22×D13712yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—2742no—
172C4⋊C4—2742no—
173C4⋊C4—2741no—
174C4⋊C4—1372no—
175C4⋊C4C22×D13712yes—
176C22⋊C4—2742no—
177C22⋊C4—2742no—
178C42—1371no—
179C42—1371no—
180C42—1371no—
181C42C22×D13711yescharacteristic
182C23—2742no—
183C23—1371no—
184Q8—24no—
185Q8—42no—
186Q8—5484no—
187Q8—5482no—
188D4—2744no—
189D4—5484no—
190D4—5482no—
191D4—5482no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—2742no—
195C2×C4—2742no—
196C2×C4—2742no—
197C2×C4—5484no—
198C2×C4—5484no—
199C2×C4—1371no—
200C2×C4—1371no—
201C2×C4—1371no—
202C2×C4—1374no—
203C2×C4D4×D13711yescharacteristic
204C2×C4D4×D13711yescharacteristic
205C2×C4C23×D13711yescharacteristic
206C8—22no—
207C8—2742no—
208C22—2741no—
209C22—2741no—
210C22—5482no—
211C22—5482no—
212C22—5482no—
213C22—1372no—
214C22C2×D4×D13711yescharacteristic
215C4—42no—
216C4—42no—
217C4—2741no—
218C4—2741no—
219C4—5482no—
220C4—1372no—
221C4#1798411512yes—
222C4#1798412012yes—
223C4#41×D13712yes—
224C2—2742no—
225C2—10962no—
226C2#124×D13711yescharacteristic
227C2C2×#1798411511yescharacteristic
228C2C2×#1798412011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
prime radical#22577954C2
hypercenter#487D137
hypocenterC137#3141
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
2. C4⋊Q8×D137C2semidirect
3. #22577764C2semidirect
4. #22577753C2semidirect
5. #22577751C2semidirect
6. #22577750C2semidirect
7. #22577403C2semidirect
8. C4×#14555493C2semidirect
9. #487×C137C2semidirect
114. C137#3141semidirect
120. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225779542
C4⋊Q8×D1372
#225777642
#225777532
#225777512
#225777502
#225774032
C4×#145554932
#487×C1372
#3141137