Finite Group Databasebeta

← #31792255

Subgroups — #31792255

Order 17536.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#22578026C212yesmaximal, complemented
2#22577692C212yesmaximal, complemented
3#22577686C212yesmaximal, complemented
4#22577656C211yesmaximal, characteristic
5#22577634C212yesmaximal, complemented
6#22577627C212yesmaximal, complemented
7#22577621C212yesmaximal, complemented
8#22577431C211yesmaximal, characteristic, complemented
9#470×C137C211yesmaximal, characteristic, complemented
10C2×#14555497C2212yes—
11C2×D1096C2211yescharacteristic
12C22×#11764040C2211yescharacteristic
13#17984087C2212yes—
14#17984083—22no—
15#17984077C2212yes—
16#17984074—22no—
17#17984070—22no—
18#17984068—22no—
19#17984062C2212yes—
20Q8×Dic137C2212yes—
21#17984050C2212yes—
22#17984047C2211yescharacteristic
23#17984020C2211yescharacteristic
24#17984019C2212yes—
25#17984017C2212yes—
26#17984017C2212yes—
27#17984012C2212yes—
28#17984010C2212yes—
29#17984008C2212yes—
30C2×Q16×C137C2211yescharacteristic
31C2×M16×C137C2211yescharacteristic
32C22⋊Q8×C137C2212yes—
33#106×C137C2211yescharacteristic
34#103×C137C2212yes—
35C2×Dic274—22no—
36C2×D548C2312yes—
37C2×C22⋊D137—22no—
38C2×C4×D137—22no—
39#14555497—42no—
40D1096—41no—
41C2×#11764040—41no—
42C2×#11764040C2312yes—
43C4⋊Dic137—22no—
44C4⋊Dic137D412yes—
45C4⋊Dic137C2312yes—
46#14555489—22no—
47C22⋊Dic137D411yescharacteristic
48C22⋊Dic137D411yescharacteristic
49#14555487—22no—
50#14555487—22no—
51#14555487—22no—
52#14555487—22no—
53C4×Dic137—22no—
54C4×Dic137C2312yes—
55C2×Q8×C137C2312yes—
56C22×C4×C137C2311yescharacteristic
57Q16×C137—41no—
58M16×C137—41no—
59C2×C8×C137C2311yescharacteristic
60C2×C8×C137C2311yescharacteristic
61C4⋊C4×C137—22no—
62C4⋊C4×C137C2312yes—
63C22⋊C4×C137—22no—
64C22×D137—22no—
65Dic274—82no—
66D548—22no—
67D548—22no—
68C22⋊D137—82no—
69C2×Dic137—22no—
70C2×Dic137C2×D412yes—
71C2×Dic137C2×D412yes—
72C4×D137—82no—
73#11764040—22no—
74C23×C137C2×D411yescharacteristic
75Q8×C137—22no—
76Q8×C137—22no—
77C2×C4×C137—21no—
78C2×C4×C137—21no—
79C2×C4×C137—22no—
80C2×C4×C137—22no—
81C2×C4×C137C2×D411yescharacteristic
82C2×C4×C137C2411yescharacteristic
83C8×C137—21no—
84C8×C137—21no—
85C22×C137—21no—
86C22×C137—22no—
87C22×C137C22×D411yescharacteristic
88C2×D137—42no—
89C2×D137—42no—
90C2×D137—42no—
91C4×C137—21no—
92C4×C137—42no—
93C4×C137—42no—
94C4×C137H(5,𝔽2)11yescharacteristic
95C4×C1372−1+411yescharacteristic
96Dic137—22no—
97Dic137—22no—
98Dic137—42no—
99C2×C137—41no—
100C2×C137#53711yescharacteristic
101C2×C137#57712yes—
102D137—82no—
103C137#326311yescharacteristic, complemented, Sylow
104#3263—1371nomaximal, Sylow
105#542—1372no—
106#476—1372no—
107#475—1372no—
108#470D13711yescharacteristic, complemented
109#469—1371no—
110#465—1372no—
111#464—1372no—
112#463—1372no—
113#424—1371no—
114C2×Q16C2×D13711yescharacteristic
115C2×SD16—1372no—
116C2×D8—1371no—
117C2×M16C2×D13711yescharacteristic
118C22×C8—1371no—
119C4⋊Q8—1372no—
120#126—2742no—
121#124—2742no—
122#124—1372no—
123#123—2742no—
124C22⋊Q8—2742no—
125C22⋊Q8C2×D13712yes—
126#121—1372no—
127C4×Q8—1372no—
128C4×D4—1372no—
129#117—1371no—
130#106—1371no—
131#106—1372no—
132#106C2×D13711yescharacteristic
133C4⋊C8—1372no—
134C4⋊C8—1372no—
135#103—1372no—
136#103C2×D13712yes—
137#102—1372no—
138#102—1372no—
139C2×Q8—2742no—
140C2×Q8C22×D13712yes—
141C2×D4—2742no—
142C2×D4—1372no—
143C22×C4—2742no—
144C22×C4C22×D13711yescharacteristic
145Q16—41no—
146SD16—5482no—
147D8—5481no—
148M16—41no—
149C2×C8—5481no—
150C2×C8—1372no—
151C2×C8C22×D13711yescharacteristic
152C2×C8C22×D13711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—2742no—
155C4⋊C4—2742no—
156C4⋊C4—1372no—
157C4⋊C4—1372no—
158C4⋊C4C22×D13712yes—
159C22⋊C4—22no—
160C22⋊C4—2742no—
161C22⋊C4—2742no—
162C22⋊C4—2742no—
163C22⋊C4—2742no—
164C22⋊C4—1371no—
165C22⋊C4—1371no—
166C42—2742no—
167C42—1372no—
168C23—2742no—
169C23D4×D13711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—10962no—
173D4—2742no—
174D4—2742no—
175D4—10962no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—2742no—
181C2×C4—10962no—
182C2×C4—1372no—
183C2×C4—1372no—
184C2×C4D4×D13711yescharacteristic
185C2×C4C23×D13711yescharacteristic
186C8—21no—
187C8—21no—
188C8—2742no—
189C22—22no—
190C22—21no—
191C22—5482no—
192C22—5482no—
193C22—5482no—
194C22C2×D4×D13711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—2742no—
199C4—2742no—
200C4—5482no—
201C4#1798413311yescharacteristic
202C4#1798413511yescharacteristic
203C2—41no—
204C2—10962no—
205C2#2257798011yescharacteristic
206C2#2257821212yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#470×C137C2
socleC22×C137C22×D4
prime radical#22577656C2
hypercenter#470D137
hypocenterC137#3263
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #22578026C2semidirect
2. #22577692C2semidirect
3. #22577686C2semidirect
5. #22577634C2semidirect
6. #22577627C2semidirect
7. #22577621C2semidirect
8. #22577431C2semidirect
9. #470×C137C2semidirect
103. C137#3263semidirect
108. #470D137semidirect

Maximal subgroups

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

subgroupindex
#225780262
#225776922
#225776862
#225776562
#225776342
#225776272
#225776212
#225774312
#470×C1372
#3263137