Finite Group Databasebeta

← #31790301

Subgroups — #31790301

Order 17536.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#17984114C211yesmaximal, characteristic
2C22×#14555498C211yesmaximal, characteristic, complemented
3#22577979C211yesmaximal, characteristic, complemented
4#22577618C214yesmaximal, complemented
5#22577608C214yesmaximal, complemented
6#22577430C211yesmaximal, characteristic, complemented
7#22577411C211yesmaximal, characteristic, complemented
8#22577397C211yesmaximal, characteristic, complemented
9#518×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11C22×Dic274C2211yescharacteristic
12#17984114—24no—
13C2×#14555498—24no—
14C2×#14555498C2214yes—
15C2×#14555498C2212yes—
16C2×#14555495C2212yes—
17C2×#14555494C2211yescharacteristic
18C22×#11764040C2211yescharacteristic
19#17984088C2212yescomplemented
20#17984076C2212yescomplemented
21#17984072—22no—
22#17984068—22no—
23C4⋊#11764040C2214yescomplemented
24#17984011C2214yescomplemented
25#17984011C2212yescomplemented
26#17984008C2212yescomplemented
27C22×D4×C137C2211yescharacteristic
28C4×D4×C137—22no—
29C4×D4×C137C2214yes—
30#117×C137C2211yescharacteristic
31#117×C137C2211yescharacteristic
32C2×C22⋊C4×C137—22no—
33#14555510—22no—
34#14555510—22no—
35C2×Dic274—21no—
36C2×Dic274—22no—
37C2×Dic274—22no—
38C2×Dic274C2312yes—
39C2×Dic274C2311yescharacteristic
40C2×D548C2311yescharacteristic
41C2×C22⋊D137—21no—
42C22×Dic137—21no—
43C2×C4×D137—21no—
44#14555498—28no—
45#14555498—24no—
46#14555495—24no—
47#14555494—22no—
48C2×#11764040—22no—
49C2×#11764040C2312yes—
50C2×#11764040C2312yes—
51C4⋊Dic137—24no—
52#14555487—24no—
53C24×C137—21no—
54C2×D4×C137—21no—
55C2×D4×C137—22no—
56C2×D4×C137D414yes—
57C2×D4×C137C2312yes—
58C22×C4×C137—24no—
59C22×C4×C137C2311yescharacteristic
60C4⋊C4×C137D412yes—
61C4⋊C4×C137C2312yes—
62C22⋊C4×C137—21no—
63C22⋊C4×C137—24no—
64C22⋊C4×C137D412yes—
65C42×C137—21no—
66C42×C137C2312yes—
67C22×D137—21no—
68Dic274—22no—
69Dic274—21no—
70Dic274—22no—
71Dic274—21no—
72Dic274—42no—
73D548—21no—
74D548—21no—
75C22⋊D137—42no—
76C2×Dic137—22no—
77C2×Dic137—21no—
78C2×Dic137—42no—
79C4×D137—42no—
80#11764040—22no—
81#11764040—22no—
82C23×C137—22no—
83C23×C137—24no—
84C23×C137—82no—
85C23×C137C2×D411yescharacteristic
86D4×C137—24no—
87D4×C137—22no—
88D4×C137C2×D414yes—
89C2×C4×C137—22no—
90C2×C4×C137—84no—
91C2×C4×C137C2×D414yes—
92C2×C4×C137C2×D411yescharacteristic
93C2×C4×C137C2×D412yes—
94C2×C4×C137#4112yes—
95C2×C4×C137C2411yescharacteristic
96C22×C137—22no—
97C22×C137—24no—
98C22×C137—44no—
99C22×C137—84no—
100C22×C137#12112yes—
101C22×C137C22×D411yescharacteristic
102C2×D137—41no—
103C2×D137—42no—
104C4×C137—24no—
105C4×C137—42no—
106C4×C137#12112yes—
107C4×C137C22×D411yescharacteristic
108C4×C137C2×#4111yescharacteristic
109Dic137—41no—
110Dic137—42no—
111C2×C137—22no—
112C2×C137—44no—
113C2×C137C2×#12111yescharacteristic
114C2×C137#57712yes—
115D137—81no—
116C137#304211yescharacteristic, complemented, Sylow
117#3042—1371nomaximal, Sylow
118C2×AGL(1,ℤ/8)—1371no—
119C22×SD16—1371no—
120#536—1371no—
121#518D13711yescharacteristic, complemented
122#463—1374no—
123#461—1374no—
124#424—1371no—
125#419—1371no—
126#416—1371no—
127C2×#41—1371no—
128C22×Q8—1371no—
129C22×D4C2×D13711yescharacteristic
130AGL(1,ℤ/8)—2744no—
131C2×SD16—2744no—
132C2×SD16—1374no—
133C2×SD16—1372no—
134C2×D8—1372no—
135C2×M16—1371no—
136C22×C8—1371no—
137C4⋊Q8—1372no—
138#124—1372no—
139#123—2742no—
140C22⋊Q8—2742no—
141C4×D4—22no—
142C4×D4C2×D13714yes—
143#117C2×D13711yescharacteristic
144#117C2×D13711yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—1374no—
147#103—1372no—
148#103—1374no—
149#102—1372no—
150C24—21no—
151#41—2742no—
152#41—2742no—
153C2×Q8—2741no—
154C2×Q8—2742no—
155C2×Q8—2742no—
156C2×Q8—1372no—
157C2×Q8—1371no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—2741no—
161C2×D4—1371no—
162C2×D4D54814yes—
163C2×D4C22×D13712yes—
164C22×C4—24no—
165C22×C4—2741no—
166C22×C4—2741no—
167C22×C4C22×D13711yescharacteristic
168SD16—2748no—
169SD16—2744no—
170D8—2744no—
171M16—2742no—
172C2×C8—2742no—
173C2×C8—1372no—
174C2×C8—1372no—
175C4⋊C4—2744no—
176C4⋊C4C22⋊D13712yes—
177C4⋊C4C22×D13712yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—2744no—
181C22⋊C4C22⋊D13712yes—
182C42—21no—
183C42C22×D13712yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2741no—
188C23C2×C22⋊D13711yescharacteristic
189Q8—2742no—
190Q8—2741no—
191Q8—2742no—
192Q8—2741no—
193Q8—5482no—
194D4—24no—
195D4—22no—
196D4—2741no—
197D4—2741no—
198D4—5482no—
199D4C2×D54814yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—2742no—
203C2×C4—2741no—
204C2×C4—5482no—
205C2×C4—5482no—
206C2×C4C2×C22⋊D13714yes—
207C2×C4C2×C22⋊D13711yescharacteristic
208C2×C4C2×D54812yes—
209C2×C4#1455551012yes—
210C2×C4C23×D13711yescharacteristic
211C8—2742no—
212C8—2742no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—5482no—
218C22—5481no—
219C22#1798406212yes—
220C22C22×C22⋊D13711yescharacteristic
221C4—24no—
222C4—42no—
223C4—5481no—
224C4—5482no—
225C4#1798406212yes—
226C4C22×D54811yescharacteristic
227C4C2×#1455551011yescharacteristic
228C2—22no—
229C2—44no—
230C2—10961no—
231C2C2×#1798406211yescharacteristic
232C2#2257821412yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#518×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984114C2
hypercenter#518D137
hypocenterC137#3042
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#14555498C2semidirect
3. #22577979C2semidirect
4. #22577618C2semidirect
5. #22577608C2semidirect
6. #22577430C2semidirect
7. #22577411C2semidirect
8. #22577397C2semidirect
9. #518×C137C2semidirect
19. #17984088C22semidirect
20. #17984076C22semidirect
23. C4⋊#11764040C22semidirect
24. #17984011C22semidirect
25. #17984011C22semidirect
26. #17984008C22semidirect
116. C137#3042semidirect
121. #518D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C22×#145554982
#225779792
#225776182
#225776082
#225774302
#225774112
#225773972
#518×C1372
#3042137