Finite Group Databasebeta

← #31793442

Subgroups — #31793442

Order 17536.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#22578127C211yesmaximal, characteristic, complemented
2#22578110C211yesmaximal, characteristic, complemented
3#22577759C212yesmaximal, complemented
4#22577753C211yesmaximal, characteristic, complemented
5#22577708C212yesmaximal, complemented
6#22577694C211yesmaximal, characteristic, complemented
7#22577620C212yesmaximal
8#22577527C211yesmaximal, characteristic, complemented
9SD16×Dic137C212yesmaximal, complemented
10C4×#14555495C211yesmaximal, characteristic
11#461×C137C211yesmaximal, characteristic, complemented
12C2×#14555495C2211yescharacteristic
13#17984088C2211yescharacteristic
14#17984083—22no—
15#17984079C2212yes—
16#17984077C2212yes—
17#17984068—21no—
18#17984067—21no—
19#17984065—22no—
20Q8×Dic137C2212yes—
21C4×Dic274C2211yescharacteristic
22C4×D548C2211yescharacteristic
23D4×Dic137C2212yes—
24#17984046—21no—
25C2×C4⋊Dic137—21no—
26#17984021C2211yescharacteristic
27#17984020C2212yes—
28#17984017C2212yes—
29#17984012C2212yes—
30#17984012C2212yes—
31#17984010C2212yes—
32#17984009C2211yescharacteristic
33#17984008C2211yescharacteristic
34C4×#11764040C2211yescharacteristic
35C8×Dic137C2212yes—
36C2×SD16×C137C2212yes—
37C4⋊Q8×C137C2211yescharacteristic
38C4×D4×C137C2211yescharacteristic
39C4⋊C8×C137C2211yescharacteristic
40#103×C137C2212yes—
41C2×Dic274—21no—
42C2×Dic274D411yescharacteristic
43C2×D548C2311yescharacteristic
44C22×Dic137—22no—
45C2×C4×D137—21no—
46#14555495—22no—
47C2×#11764040C2311yescharacteristic
48C2×#11764040C2311yescharacteristic
49C4⋊Dic137—22no—
50C4⋊Dic137—22no—
51C4⋊Dic137—21no—
52C4⋊Dic137D411yescharacteristic
53C4⋊Dic137C2311yescharacteristic
54C4⋊Dic137C2312yes—
55#14555489—22no—
56#14555489D412yes—
57C22⋊Dic137—22no—
58#14555487—22no—
59#14555487—22no—
60#14555487—21no—
61C4×Dic137—22no—
62C4×Dic137C2312yes—
63C2×Q8×C137C2312yes—
64C2×D4×C137C2311yescharacteristic
65C22×C4×C137—21no—
66SD16×C137—24no—
67C2×C8×C137C2312yes—
68C4⋊C4×C137—21no—
69C4⋊C4×C137—21no—
70C4⋊C4×C137C2311yescharacteristic
71C22⋊C4×C137—21no—
72C42×C137C2311yescharacteristic
73C22×D137—21no—
74Dic274—22no—
75Dic274—81no—
76D548—21no—
77D548—21no—
78C2×Dic137—21no—
79C2×Dic137—22no—
80C2×Dic137—82no—
81C2×Dic137C2×D412yes—
82C2×Dic137C2×D412yes—
83C4×D137—42no—
84#11764040—21no—
85#11764040—21no—
86C23×C137—21no—
87Q8×C137—22no—
88Q8×C137—22no—
89D4×C137—21no—
90D4×C137#4112yes—
91C2×C4×C137—21no—
92C2×C4×C137—22no—
93C2×C4×C137—42no—
94C2×C4×C137C2×D411yescharacteristic
95C2×C4×C137C2×D411yescharacteristic
96C2×C4×C137C2411yescharacteristic
97C8×C137—22no—
98C22×C137—22no—
99C22×C137—41no—
100C22×C137—41no—
101C22×C137C22×D411yescharacteristic
102C2×D137—41no—
103C2×D137—41no—
104C2×D137—41no—
105C4×C137—21no—
106C4×C137—42no—
107C4×C137—41no—
108C4×C137#13512yes—
109C4×C137C2×#4111yescharacteristic
110C4×C1372−1+411yescharacteristic
111Dic137—22no—
112Dic137—22no—
113Dic137—41no—
114Dic137—42no—
115C2×C137—42no—
116C2×C137#54811yescharacteristic
117C2×C137C2×#13511yescharacteristic
118C2×C137#57711yescharacteristic
119D137—81no—
120C137#336011yescharacteristic, complemented, Sylow
121#3360—1371nomaximal, Sylow
122#556—1371no—
123#554—1371no—
124#488—1372no—
125#487—1371no—
126#479—1372no—
127#476—1371no—
128#463—1372no—
129#461D13711yescharacteristic, complemented
130C8×Q8—1371no—
131C4×SD16—1372no—
132C4×D8—1371no—
133C2×SD16C2×D13712yes—
134C2×D8—1371no—
135C4⋊Q8—1371no—
136C4⋊Q8C2×D13711yescharacteristic
137#126—2742no—
138#125—1372no—
139#124—1372no—
140C22⋊Q8—2742no—
141C22⋊Q8—2741no—
142C22⋊Q8—2741no—
143C4×Q8—1372no—
144C4×Q8—1371no—
145C4×D4—1372no—
146C4×D4—1371no—
147C4×D4C2×D13711yescharacteristic
148#117—2741no—
149C2×C4⋊C4—2741no—
150#107—1371no—
151#106—1372no—
152C4⋊C8—1372no—
153C4⋊C8C2×D13711yescharacteristic
154#103—1372no—
155#103—1372no—
156#103C2×D13712yes—
157#102—1371no—
158#102—1372no—
159#102—1371no—
160C4×C8—1372no—
161C4×C8—1371no—
162C2×Q8—2741no—
163C2×Q8—1371no—
164C2×Q8C22×D13712yes—
165C2×D4—1371no—
166C2×D4C22×D13711yescharacteristic
167C22×C4—21no—
168C22×C4—2741no—
169C22×C4—2742no—
170SD16—24no—
171D8—2742no—
172C2×C8—1371no—
173C2×C8—1371no—
174C2×C8C22×D13712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—2742no—
178C4⋊C4—2742no—
179C4⋊C4—2742no—
180C4⋊C4—2741no—
181C4⋊C4—1372no—
182C4⋊C4—1371no—
183C4⋊C4—1372no—
184C4⋊C4—1371no—
185C4⋊C4C22×D13711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—2742no—
188C22⋊C4—2742no—
189C22⋊C4—2742no—
190C22⋊C4—2741no—
191C42—2742no—
192C42—1372no—
193C42C22×D13711yescharacteristic
194C23—21no—
195C23—2741no—
196Q8—22no—
197Q8—22no—
198Q8—2742no—
199Q8—10961no—
200D4—21no—
201D4—2741no—
202D4—2741no—
203D4#1455550912yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—2741no—
208C2×C4—2742no—
209C2×C4—5482no—
210C2×C4—10962no—
211C2×C4—1372no—
212C2×C4—1372no—
213C2×C4D4×D13711yescharacteristic
214C2×C4D4×D13711yescharacteristic
215C2×C4C23×D13711yescharacteristic
216C8—22no—
217C8—2741no—
218C8—2741no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—5481no—
223C22—5481no—
224C22—5481no—
225C22C2×D4×D13711yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—2742no—
230C4—2742no—
231C4—5481no—
232C4—5482no—
233C4#1798410912yes—
234C4C2×#1455550911yescharacteristic
235C4#1798413511yescharacteristic
236C2—42no—
237C2—10961no—
238C2#2257805911yescharacteristic
239C2C2×#1798410911yescharacteristic
240C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#461×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555495C22
hypercenter#461D137
hypocenterC137#3360
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22578127C2semidirect
2. #22578110C2semidirect
3. #22577759C2semidirect
4. #22577753C2semidirect
5. #22577708C2semidirect
6. #22577694C2semidirect
8. #22577527C2semidirect
9. SD16×Dic137C2semidirect
11. #461×C137C2semidirect
120. C137#3360semidirect
129. #461D137semidirect

Maximal subgroups

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

subgroupindex
#225781272
#225781102
#225777592
#225777532
#225777082
#225776942
#225776202
#225775272
SD16×Dic1372
C4×#145554952
#461×C1372
#3360137