Finite Group Databasebeta

← #31793455

Subgroups — #31793455

Order 17536.

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

normal subgroups only

This group has 392 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
1Q8×Dic274C211yesmaximal, characteristic, complemented
2#22578073C211yesmaximal, characteristic
3#22577749C211yesmaximal, characteristic, complemented
4#22577748C212yesmaximal, complemented
5#22577694C211yesmaximal, characteristic, complemented
6#22577691C212yesmaximal, complemented
7#22577601C212yesmaximal, complemented
8#22577527C211yesmaximal, characteristic, complemented
9SD16×Dic137C212yesmaximal, complemented
10C4×#14555497C211yesmaximal, characteristic, complemented
11#460×C137C211yesmaximal, characteristic, complemented
12C2×#14555509—21no—
13C2×#14555497C2211yescharacteristic
14#17984088—21no—
15#17984088C2211yescharacteristic
16#17984087—22no—
17#17984087C2212yes—
18#17984084C2212yes—
19#17984060—22no—
20#17984059—21no—
21Q8×Dic137C2212yes—
22C4×Dic274—21no—
23C4×Dic274C2211yescharacteristic
24C4×D548C2211yescharacteristic
25D4×Dic137C2212yes—
26#17984020C2212yes—
27#17984019C2211yescharacteristic
28#17984017C2212yes—
29#17984012C2211yescharacteristic
30#17984010C2212yes—
31#17984009C2212yes—
32#17984009C2212yes—
33#17984008C2211yescharacteristic
34C4×#11764040C2211yescharacteristic
35C8×Dic137C2212yes—
36C2×SD16×C137C2212yes—
37#127×C137C2211yescharacteristic
38C4×Q8×C137C2211yescharacteristic
39C4⋊C8×C137C2211yescharacteristic
40#102×C137C2212yes—
41#14555509—44no—
42C2×Dic274—22no—
43C2×Dic274—21no—
44C2×Dic274D411yescharacteristic
45C2×D548C2311yescharacteristic
46C2×C22⋊D137—22no—
47C2×C22⋊D137—22no—
48C22×Dic137—22no—
49C2×C4×D137—21no—
50#14555497—22no—
51C2×#11764040C2311yescharacteristic
52C2×#11764040C2311yescharacteristic
53C4⋊Dic137—22no—
54C4⋊Dic137—21no—
55C4⋊Dic137D411yescharacteristic
56C4⋊Dic137C2312yes—
57C4⋊Dic137C2311yescharacteristic
58#14555489—22no—
59#14555489D412yes—
60C22⋊Dic137—22no—
61#14555487—21no—
62C4×Dic137—22no—
63C4×Dic137C2312yes—
64C2×Q8×C137C2311yescharacteristic
65C2×D4×C137—21no—
66C2×D4×C137—21no—
67C2×D4×C137C2312yes—
68SD16×C137—24no—
69C2×C8×C137C2312yes—
70C4⋊C4×C137—21no—
71C4⋊C4×C137C2311yescharacteristic
72C42×C137—21no—
73C42×C137C2311yescharacteristic
74C22×D137—21no—
75Dic274—82no—
76Dic274—81no—
77Dic274SD1614yescomplemented
78D548—21no—
79D548—21no—
80C22⋊D137—44no—
81C22⋊D137—82no—
82C2×Dic137—22no—
83C2×Dic137—21no—
84C2×Dic137—44no—
85C2×Dic137C2×D412yes—
86C2×Dic137C2×D412yes—
87C4×D137—42no—
88#11764040—21no—
89#11764040—21no—
90C23×C137—22no—
91Q8×C137—21no—
92Q8×C137#4112yescomplemented
93D4×C137—22no—
94D4×C137—22no—
95D4×C137—42no—
96D4×C137—81no—
97C2×C4×C137—21no—
98C2×C4×C137—21no—
99C2×C4×C137C2×D411yescharacteristic
100C2×C4×C137C2×D411yescharacteristic
101C2×C4×C137C2411yescharacteristic
102C8×C137—22no—
103C22×C137—42no—
104C22×C137—42no—
105C22×C137—42no—
106C22×C137C22×D411yescharacteristic
107C2×D137—41no—
108C2×D137—41no—
109C2×D137—41no—
110C4×C137—22no—
111C4×C137—21no—
112C4×C137—41no—
113C4×C137C2×SD1612yes—
114C4×C137C2×#4111yescharacteristic
115C4×C1372−1+411yescharacteristic
116Dic137—22no—
117Dic137—41no—
118Dic137—42no—
119Dic137C2×SD1614yes—
120C2×C137—82no—
121C2×C137#54811yescharacteristic
122C2×C137C22×SD1611yescharacteristic
123C2×C137#57611yescharacteristic
124D137—81no—
125C137#336211yescharacteristic, complemented, Sylow
126#3362—1371nomaximal, Sylow
127Q82—1371no—
128#550—1371no—
129#486—1371no—
130#486—1372no—
131#476—1371no—
132#476—1372no—
133#460—1372no—
134#460D13711yescharacteristic, complemented
135C8×Q8—1371no—
136C4×SD16—1372no—
137C4×SD16—1371no—
138C2×#41—2741no—
139C2×SD16—1371no—
140C2×SD16C2×D13712yes—
141C4⋊Q8—2741no—
142C4⋊Q8—2742no—
143C4⋊Q8—1372no—
144C4⋊Q8—1371no—
145#127—1372no—
146#127C2×D13711yescharacteristic
147#121—2741no—
148#121—2742no—
149C4×Q8—2741no—
150C4×Q8—1372no—
151C4×Q8—1371no—
152C4×Q8C2×D13711yescharacteristic
153C4×D4—1372no—
154C4×D4—1371no—
155#106—1372no—
156#106—1371no—
157C4⋊C8—1372no—
158C4⋊C8C2×D13711yescharacteristic
159#103—1371no—
160#102—1372no—
161#102—1372no—
162#102—1372no—
163#102—1371no—
164#102C2×D13712yes—
165C4×C8—1372no—
166C4×C8—1371no—
167#41—5484no—
168C2×Q8—2742no—
169C2×Q8—2741no—
170C2×Q8—1371no—
171C2×Q8C22×D13711yescharacteristic
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—2742no—
175C2×D4—2742no—
176C2×D4—1371no—
177C2×D4C22×D13712yes—
178C22×C4—2742no—
179C22×C4—2741no—
180SD16—24no—
181SD16—2742no—
182C2×C8—1371no—
183C2×C8—1371no—
184C2×C8C22×D13712yes—
185C4⋊C4—21no—
186C4⋊C4—2742no—
187C4⋊C4—2742no—
188C4⋊C4—2741no—
189C4⋊C4—1372no—
190C4⋊C4—1371no—
191C4⋊C4—1372no—
192C4⋊C4—1371no—
193C4⋊C4C22×D13711yescharacteristic
194C22⋊C4—2742no—
195C22⋊C4—2741no—
196C42—21no—
197C42—2742no—
198C42—1372no—
199C42C22×D13711yescharacteristic
200C23—22no—
201C23—2741no—
202Q8—21no—
203Q8—10961no—
204Q8—10962no—
205Q8—1374no—
206Q8#1455550912yescomplemented
207D4—22no—
208D4—22no—
209D4—42no—
210D4—81no—
211D4—2741no—
212D4—2741no—
213D4—5484no—
214D4—10962no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—2742no—
218C2×C4—2741no—
219C2×C4—5484no—
220C2×C4—5482no—
221C2×C4—1372no—
222C2×C4—1372no—
223C2×C4D4×D13711yescharacteristic
224C2×C4D4×D13711yescharacteristic
225C2×C4C23×D13711yescharacteristic
226C8—22no—
227C8—2741no—
228C8—2741no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—5481no—
233C22—5481no—
234C22—5481no—
235C22C2×D4×D13711yescharacteristic
236C4—21no—
237C4—22no—
238C4—41no—
239C4—2742no—
240C4—5482no—
241C4—5481no—
242C4—1374no—
243C4SD16×D13712yes—
244C4C2×#1455550911yescharacteristic
245C4#1798413511yescharacteristic
246C2—82no—
247C2—10961no—
248C2#2257805911yescharacteristic
249C2C2×SD16×D13711yescharacteristic
250C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#460×C137C2
socleC22×C137C22×D4
prime radical#22578073C2
hypercenter#460D137
hypocenterC137#3362
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×Dic274C2semidirect
3. #22577749C2semidirect
4. #22577748C2semidirect
5. #22577694C2semidirect
6. #22577691C2semidirect
7. #22577601C2semidirect
8. #22577527C2semidirect
9. SD16×Dic137C2semidirect
10. C4×#14555497C2semidirect
11. #460×C137C2semidirect
77. Dic274SD16semidirect
92. Q8×C137#41semidirect
125. C137#3362semidirect
134. #460D137semidirect
206. Q8#14555509semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2742
#225780732
#225777492
#225777482
#225776942
#225776912
#225776012
#225775272
SD16×Dic1372
C4×#145554972
#460×C1372
#3362137