Finite Group Databasebeta

← #31792259

Subgroups — #31792259

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#22578020C212yesmaximal, complemented
2#22577691C212yesmaximal, complemented
3#22577687C212yesmaximal, complemented
4#22577661C211yesmaximal, characteristic, complemented
5#22577630C212yesmaximal, complemented
6#22577624C212yesmaximal, complemented
7#22577616C212yesmaximal, complemented
8#22577431C211yesmaximal, characteristic, complemented
9#469×C137C211yesmaximal, characteristic
10C2×Dic548C2211yescharacteristic
11C2×#14555498C2212yes—
12C22×#11764040C2211yescharacteristic
13#17984087C2212yes—
14#17984081—22no—
15#17984075—22no—
16#17984075C2212yes—
17#17984069—22no—
18#17984067C2212yes—
19#17984065—22no—
20#17984053C2212yes—
21D4×Dic137C2212yes—
22#17984047C2211yescharacteristic
23#17984020C2211yescharacteristic
24#17984019C2212yes—
25#17984017C2212yes—
26#17984017C2212yes—
27#17984013C2212yes—
28#17984011C2212yes—
29#17984009C2212yes—
30C2×D8×C137C2211yescharacteristic
31C2×M16×C137C2211yescharacteristic
32#121×C137C2212yes—
33#106×C137C2211yescharacteristic
34#102×C137C2212yes—
35C2×Dic274—22no—
36C2×Dic274C2312yes—
37C22×Dic137—22no—
38Dic548—41no—
39#14555498—42no—
40C2×#11764040—41no—
41C2×#11764040C2312yes—
42C4⋊Dic137D412yes—
43C4⋊Dic137C2312yes—
44#14555489—22no—
45#14555489—22no—
46#14555489—22no—
47C22⋊Dic137—22no—
48C22⋊Dic137—22no—
49C22⋊Dic137—22no—
50C22⋊Dic137—22no—
51C22⋊Dic137D411yescharacteristic
52C22⋊Dic137D411yescharacteristic
53C4×Dic137—22no—
54C4×Dic137C2312yes—
55C2×D4×C137—22no—
56C2×D4×C137C2312yes—
57C22×C4×C137C2311yescharacteristic
58D8×C137—41no—
59M16×C137—41no—
60C2×C8×C137C2311yescharacteristic
61C2×C8×C137C2311yescharacteristic
62C4⋊C4×C137C2312yes—
63C22⋊C4×C137—22no—
64Dic274—22no—
65Dic274—22no—
66Dic274—82no—
67C2×Dic137—22no—
68C2×Dic137—22no—
69C2×Dic137—82no—
70C2×Dic137C2×D412yes—
71C2×Dic137C2×D412yes—
72#11764040—22no—
73C23×C137—22no—
74C23×C137C2×D411yescharacteristic
75D4×C137—22no—
76D4×C137—22no—
77D4×C137—82no—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—22no—
81C2×C4×C137C2×D411yescharacteristic
82C2×C4×C137C2411yescharacteristic
83C8×C137—21no—
84C8×C137—21no—
85C22×C137—21no—
86C22×C137—22no—
87C22×C137—42no—
88C22×C137—42no—
89C22×C137—42no—
90C22×C137C22×D411yescharacteristic
91C4×C137—21no—
92C4×C137—42no—
93C4×C137H(5,𝔽2)11yescharacteristic
94C4×C1372−1+411yescharacteristic
95Dic137—22no—
96Dic137—22no—
97Dic137—42no—
98Dic137—42no—
99C2×C137—41no—
100C2×C137—82no—
101C2×C137#53711yescharacteristic
102C2×C137#57712yes—
103C137#326311yescharacteristic, complemented, Sylow
104#3263—1371nomaximal, Sylow
105#542—1372no—
106#476—1372no—
107#475—1372no—
108#470—1371no—
109#469D13711yescharacteristic
110#465—1372no—
111#464—1372no—
112#463—1372no—
113#424—1371no—
114C2×Q16—1371no—
115C2×SD16—1372no—
116C2×D8C2×D13711yescharacteristic
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⋊Q8—1372no—
126#121C2×D13712yes—
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#103—1372no—
137#102—1372no—
138#102C2×D13712yes—
139C2×Q8—2742no—
140C2×Q8—1372no—
141C2×D4—22no—
142C2×D4C22×D13712yes—
143C22×C4—2742no—
144C22×C4C22×D13711yescharacteristic
145Q16—5481no—
146SD16—5482no—
147D8—41no—
148M16—41no—
149C2×C8—5481no—
150C2×C8—1372no—
151C2×C8C22×D13711yescharacteristic
152C2×C8C22×D13711yescharacteristic
153C4⋊C4—2742no—
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—22no—
169C23D4×D13711yescharacteristic
170Q8—2742no—
171Q8—2742no—
172Q8—10962no—
173D4—22no—
174D4—22no—
175D4—82no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—2742no—
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—42no—
192C22—42no—
193C22—42no—
194C22C2×D4×D13711yescharacteristic
195C4—21no—
196C4—42no—
197C4—2742no—
198C4—2742no—
199C4—5482no—
200C4—5482no—
201C4#1798413311yescharacteristic
202C4#1798413511yescharacteristic
203C2—41no—
204C2—82no—
205C2#2257798011yescharacteristic
206C2#2257821312yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting, prime radical#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3263
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22578020C2semidirect
2. #22577691C2semidirect
3. #22577687C2semidirect
4. #22577661C2semidirect
5. #22577630C2semidirect
6. #22577624C2semidirect
7. #22577616C2semidirect
8. #22577431C2semidirect
103. C137#3263semidirect

Maximal subgroups

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

subgroupindex
#225780202
#225776912
#225776872
#225776612
#225776302
#225776242
#225776162
#225774312
#469×C1372
#3263137