Finite Group Databasebeta

← #31790797

Subgroups — #31790797

Order 17536.

448 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 145 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#22577960C211yesmaximal, characteristic, complemented
2#22577733C212yesmaximal, complemented
3#22577693C212yesmaximal, complemented
4#22577608C212yesmaximal, complemented
5#22577554C212yesmaximal
6C4×#14555498C214yesmaximal, complemented
7#22577456C211yesmaximal, characteristic, complemented
8C2×C4×D4×C137C211yesmaximal, characteristic
9C2×#14555498C2214yes—
10#17984088C2211yescharacteristic, complemented
11#17984088C2211yescharacteristic, complemented
12#17984080—21no—
13#17984067—21no—
14#17984067C2212yes—
15C4×Dic274—21no—
16C4×Dic274C2212yescomplemented
17#17984047—21no—
18#17984019C2214yescomplemented
19C4⋊#11764040C2212yescomplemented
20#17984011C2214yescomplemented
21#17984009C2214yes—
22C22⋊#11764040C2212yes—
23C4×#11764040C2212yescomplemented
24C22×D4×C137C2211yescharacteristic
25C23×C4×C137—21no—
26C4×D4×C137—21no—
27C4×D4×C137—21no—
28C4×D4×C137C2212yes—
29C4×D4×C137C2212yes—
30C2×C4⋊C4×C137C2211yescharacteristic
31C2×C22⋊C4×C137—21no—
32C2×C42×C137C2211yescharacteristic
33C2×Dic274—21no—
34C2×Dic274C2312yes—
35#14555498—28no—
36C2×#11764040C2314yes—
37C4⋊Dic137—21no—
38C4⋊Dic137—21no—
39C4⋊Dic137—21no—
40C4⋊Dic137C2312yes—
41#14555489—22no—
42#14555489—22no—
43C22⋊Dic137—22no—
44C4×Dic137—22no—
45C24×C137—21no—
46C2×D4×C137—24no—
47C2×D4×C137—21no—
48C2×D4×C137C2312yes—
49C22×C4×C137—21no—
50C22×C4×C137—22no—
51C22×C4×C137—22no—
52C22×C4×C137—44no—
53C22×C4×C137D411yescharacteristic
54C22×C4×C137D411yescharacteristic
55C22×C4×C137C2311yescharacteristic
56C4⋊C4×C137—21no—
57C4⋊C4×C137C2312yes—
58C22⋊C4×C137—22no—
59C22⋊C4×C137—22no—
60C42×C137D411yescharacteristic
61C42×C137D411yescharacteristic
62C42×C137C2311yescharacteristic
63C42×C137C2311yescharacteristic
64Dic274—24no—
65Dic274—42no—
66C2×Dic137—22no—
67C2×Dic137—22no—
68#11764040—24no—
69C23×C137—24no—
70C23×C137—22no—
71C23×C137—42no—
72C23×C137—42no—
73C23×C137C2×D411yescharacteristic
74D4×C137—24no—
75D4×C137—22no—
76D4×C137#4114yes—
77C2×C4×C137—22no—
78C2×C4×C137—22no—
79C2×C4×C137—41no—
80C2×C4×C137—42no—
81C2×C4×C137—44no—
82C2×C4×C137—44no—
83C2×C4×C137SD1614yescomplemented
84C2×C4×C137C2×D411yescharacteristic
85C2×C4×C137C2×D411yescharacteristic
86C2×C4×C137C2×D411yescharacteristic
87C2×C4×C137C2×D411yescharacteristic
88C2×C4×C137C2×D411yescharacteristic
89C2×C4×C137C2411yescharacteristic
90C22×C137—21no—
91C22×C137—21no—
92C22×C137—24no—
93C22×C137—48no—
94C22×C137—42no—
95C22×C137—42no—
96C22×C137C2×SD1612yes—
97C22×C137C22×D411yescharacteristic
98C4×C137—21no—
99C4×C137—21no—
100C4×C137—21no—
101C4×C137—42no—
102C4×C137C2×SD1612yes—
103C4×C137C2×SD1612yes—
104C4×C137C2×#4112yes—
105Dic137—42no—
106Dic137—42no—
107C2×C137—22no—
108C2×C137—44no—
109C2×C137#52511yescharacteristic
110C2×C137C22×SD1611yescharacteristic
111C2×C137#57511yescharacteristic
112C137#310111yescharacteristic, complemented, Sylow
113#3101—1371nomaximal, Sylow
114#533—1371no—
115C2×C4×D4D13711yescharacteristic
116#483—1372no—
117#476—1372no—
118#461—1372no—
119#450—1372no—
120C4×SD16—1374no—
121#431—1371no—
122C22×D4C2×D13711yescharacteristic
123C23×C4—21no—
124C2×SD16—1374no—
125C4⋊Q8—1371no—
126C4⋊Q8—1371no—
127#125—2741no—
128C22⋊Q8—2741no—
129C22⋊Q8—1372no—
130C4×Q8—2741no—
131C4×Q8—1372no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4C2×D13712yes—
135C4×D4C2×D13712yes—
136#117—2741no—
137C2×C4⋊C4C2×D13711yescharacteristic
138C2×C22⋊C4—21no—
139C2×C42C2×D13711yescharacteristic
140#106—1374no—
141C4⋊C8—1372no—
142#103—1374no—
143#102—1374no—
144C22⋊C8—1372no—
145C4×C8—1372no—
146C24—21no—
147C2×Q8—2741no—
148C2×Q8—1372no—
149C2×D4—24no—
150C2×D4—21no—
151C2×D4C22×D13712yes—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—44no—
156C22×C4C22⋊D13711yescharacteristic
157C22×C4C22⋊D13711yescharacteristic
158C22×C4C22×D13711yescharacteristic
159SD16—2748no—
160C2×C8—1374no—
161C4⋊C4—21no—
162C4⋊C4—2742no—
163C4⋊C4—2742no—
164C4⋊C4—2741no—
165C4⋊C4—2741no—
166C4⋊C4—2741no—
167C4⋊C4—1372no—
168C4⋊C4C22×D13712yes—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—2742no—
172C42—2742no—
173C42C22⋊D13711yescharacteristic
174C42C22⋊D13711yescharacteristic
175C42C22×D13711yescharacteristic
176C42C22×D13711yescharacteristic
177C23—24no—
178C23—22no—
179C23—42no—
180C23—42no—
181C23C2×C22⋊D13711yescharacteristic
182Q8—2744no—
183Q8—5482no—
184D4—24no—
185D4—22no—
186D4#1455551014yes—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—2742no—
194C2×C4—2742no—
195C2×C4#1455549814yescomplemented
196C2×C4C2×C22⋊D13711yescharacteristic
197C2×C4C2×C22⋊D13711yescharacteristic
198C2×C4C2×C22⋊D13711yescharacteristic
199C2×C4C2×C22⋊D13711yescharacteristic
200C2×C4C2×C22⋊D13711yescharacteristic
201C2×C4C23×D13711yescharacteristic
202C8—2744no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—48no—
207C22—42no—
208C22—42no—
209C22C2×#1455549812yes—
210C22C22×C22⋊D13711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—5482no—
216C4—5482no—
217C4C2×#1455549812yes—
218C4C2×#1455549812yes—
219C4C2×#1455551012yes—
220C2—22no—
221C2—44no—
222C2#2257791411yescharacteristic
223C2C22×#1455549811yescharacteristic
224C2#2257820111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×C4×D4×C137C2
socleC22×C137C22×D4
prime radicalC22×D4×C137C22
hypercenterC2×C4×D4D137
hypocenterC137#3101
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22577960C2semidirect
2. #22577733C2semidirect
3. #22577693C2semidirect
4. #22577608C2semidirect
6. C4×#14555498C2semidirect
7. #22577456C2semidirect
10. #17984088C22semidirect
11. #17984088C22semidirect
16. C4×Dic274C22semidirect
18. #17984019C22semidirect
19. C4⋊#11764040C22semidirect
20. #17984011C22semidirect
23. C4×#11764040C22semidirect
83. C2×C4×C137SD16semidirect
112. C137#3101semidirect
195. C2×C4#14555498semidirect

Maximal subgroups

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

subgroupindex
#225779602
#225777332
#225776932
#225776082
#225775542
C4×#145554982
#225774562
C2×C4×D4×C1372
#3101137