Finite Group Databasebeta

← #31796108

Subgroups — #31796108

Order 17536.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#22578419C211yesmaximal, characteristic, complemented
2C4×#14555519C211yesmaximal, characteristic, complemented
3#22578251C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5#22577948C211yesmaximal, characteristic
6C2×#14555519—21no—
7C4⋊#11764047C2212yes—
8C22⋊#11764047—21no—
9#17984143—21no—
10C4×#11764047—21no—
11C4×#11764047C2212yes—
12C2×#14555509—21no—
13#17984087C411yescharacteristic
14#17984084C411yescharacteristic
15#17984074—21no—
16#17984062C411yescharacteristic
17#17984060—21no—
18D4×Dic137C2211yescharacteristic
19#17984050C2211yescharacteristic
20#17984048—21no—
21C2×C4×Dic137C2211yescharacteristic
22#121×C137C411yescharacteristic
23#14555519—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26#14555509—44no—
27C2×Dic274—21no—
28C2×D548C2×C411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137—21no—
33C22×Dic137D411yescharacteristic
34C22×Dic137D411yescharacteristic
35C2×C4×D137—21no—
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C2×D4×C137—21no—
46C2×D4×C137C2×C411yescharacteristic
47C22×C4×C137C2×C411yescharacteristic
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50#11764047—24no—
51#11764047—42no—
52C22×D137—21no—
53Dic274—42no—
54D548—22no—
55C22⋊D137—42no—
56C22⋊D137—42no—
57C22⋊D137—42no—
58C2×Dic137—22no—
59C2×Dic137—22no—
60C2×Dic137—21no—
61C2×Dic137—42no—
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C2×Dic137C2×D411yescharacteristic
66C4×D137—42no—
67C23×C137—21no—
68C23×C137C22⋊C411yescharacteristic
69D4×C137—22no—
70D4×C137—42no—
71C2×C4×C137—22no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22×C411yescharacteristic
75C22×C137—21no—
76C22×C137—21no—
77C22×C137—21no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137—41no—
81C22×C137C2×C22⋊C411yescharacteristic
82C2×D137—41no—
83C2×D137—41no—
84C2×D137—41no—
85C4×C137—21no—
86C4×C137—41no—
87C4×C137#13112yes—
88Dic137—21no—
89Dic137—21no—
90Dic137—41no—
91Dic137#10412yes—
92Dic137AGL(1,ℤ/8)12yes—
93C2×C137—41no—
94C2×C137—81no—
95C2×C137#40811yescharacteristic
96C2×C137#41811yescharacteristic
97C2×C137C2×#10411yescharacteristic
98D137—81no—
99C137#148411yescharacteristic, complemented, Sylow
100#1484—1371nomaximal, Sylow
101#532—1371no—
102#431—1371no—
103C4×M16—1371no—
104D4⋊C8—1372no—
105D4⋊C8—1372no—
106C2×#41—2741no—
107C2×M16—2741no—
108C4⋊Q8—1371no—
109#127—1371no—
110#124—2741no—
111#121—2741no—
112#121—1371no—
113#121#1058773911yescharacteristic
114C4×D4—2741no—
115C4×D4—1371no—
116C4×D4—1371no—
117C2×C42—1371no—
118C4⋊C8—1372no—
119C22⋊C8—2741no—
120#97—2741no—
121C4×C8—2741no—
122C4×C8—1372no—
123#41—5484no—
124C2×Q8—2741no—
125C2×D4—21no—
126C2×D4—2741no—
127C2×D4—2741no—
128C2×D4—2741no—
129C2×D4—1371no—
130C2×D4C2×#1058773911yescharacteristic
131C22×C4—2741no—
132C22×C4—2741no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4C2×#1058773911yescharacteristic
136M16—2744no—
137C2×C8—2742no—
138C2×C8—2742no—
139C4⋊C4—2741no—
140C4⋊C4—1371no—
141C4⋊C4C2×#1058773911yescharacteristic
142C22⋊C4—21no—
143C22⋊C4—2741no—
144C22⋊C4—2741no—
145C22⋊C4—2741no—
146C42—1371no—
147C42—1371no—
148C42—1371no—
149C42—1371no—
150C23—21no—
151C23—2741no—
152C23C22⋊#1058773911yescharacteristic
153Q8—5482no—
154D4—22no—
155D4—42no—
156D4—2742no—
157D4—5482no—
158D4—5482no—
159D4—5482no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—2742no—
163C2×C4—2742no—
164C2×C4—2741no—
165C2×C4—5482no—
166C2×C4—5482no—
167C2×C4—1371no—
168C2×C4—1371no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4C22⋊#1058773911yescharacteristic
172C2×C4C22×#1058773911yescharacteristic
173C8—2744no—
174C8—5482no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—5481no—
182C22—5481no—
183C22—5481no—
184C22C2×C22⋊#1058773911yescharacteristic
185C4—21no—
186C4—41no—
187C4—2741no—
188C4—2741no—
189C4—5481no—
190C4—1372no—
191C4—1372no—
192C4#1798417512yes—
193C2—41no—
194C2—81no—
195C2—10961no—
196C2#2257837611yescharacteristic
197C2#2257839511yescharacteristic
198C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#121×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577948C2
cosocleC4×Dic137C23
hypercenter#121#10587739
hypocenterC137#1484
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22578419C2semidirect
2. C4×#14555519C2semidirect
3. #22578251C2semidirect
4. D4⋊#11764047C2semidirect
99. C137#1484semidirect

Maximal subgroups

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

subgroupindex
#225784192
C4×#145555192
#225782512
D4⋊#117640472
#225779482
#1484137