Finite Group Databasebeta

← #31796148

Subgroups — #31796148

Order 17536.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578424C211yesmaximal, characteristic, complemented
2#22578419C211yesmaximal, characteristic, complemented
3#22578251C212yesmaximal, complemented
4D4⋊#11764047C212yesmaximal, complemented
5#22577948C211yesmaximal, characteristic
6C4⋊#11764047—21no—
7C4⋊#11764047C2212yes—
8C22⋊#11764047—21no—
9C22⋊#11764047—21no—
10#17984143—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
23C2×#11764047—22no—
24C2×#11764047—22no—
25#14555509—44no—
26C2×Dic274—21no—
27C2×D548C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C22×Dic137—21no—
32C22×Dic137D411yescharacteristic
33C22×Dic137D411yescharacteristic
34C2×C4×D137—21no—
35C4⋊Dic137C2×C411yescharacteristic
36#14555489—21no—
37C22⋊Dic137—21no—
38#14555487—21no—
39#14555487—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C2×D4×C137—21no—
45C2×D4×C137C2×C411yescharacteristic
46C22×C4×C137C2×C411yescharacteristic
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49#11764047—42no—
50#11764047—42no—
51C22×D137—21no—
52Dic274—42no—
53D548—22no—
54C22⋊D137—42no—
55C22⋊D137—42no—
56C22⋊D137—42no—
57C2×Dic137—22no—
58C2×Dic137—21no—
59C2×Dic137—41no—
60C2×Dic137—42no—
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C2×D411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C4×D137—42no—
66C23×C137—21no—
67C23×C137C22⋊C411yescharacteristic
68D4×C137—22no—
69D4×C137—42no—
70C2×C4×C137—22no—
71C2×C4×C137—21no—
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22×C411yescharacteristic
74C22×C137—21no—
75C22×C137—21no—
76C22×C137—21no—
77C22×C137—41no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137C2×C22⋊C411yescharacteristic
81C2×D137—41no—
82C2×D137—41no—
83C2×D137—41no—
84C4×C137—21no—
85C4×C137—41no—
86C4×C137#13112yes—
87Dic137—21no—
88Dic137—21no—
89Dic137—21no—
90Dic137—41no—
91Dic137AGL(1,ℤ/8)12yes—
92C2×C137—41no—
93C2×C137—81no—
94C2×C137#40811yescharacteristic
95C2×C137#41811yescharacteristic
96C2×C137#42111yescharacteristic
97D137—81no—
98C137#149611yescharacteristic, complemented, Sylow
99#1496—1371nomaximal, Sylow
100#532—1371no—
101#432—1371no—
102#431—1371no—
103D4⋊C8—1372no—
104D4⋊C8—1372no—
105C2×#41—2741no—
106C4⋊Q8—1371no—
107#127—1371no—
108#124—2741no—
109#121—2741no—
110#121—1371no—
111#121#1058773911yescharacteristic
112C4×D4—2741no—
113C4×D4—1371no—
114C4×D4—1371no—
115C2×C42—1371no—
116C4⋊C8—2741no—
117C4⋊C8—1372no—
118C22⋊C8—2741no—
119C22⋊C8—2741no—
120#97—2741no—
121C4×C8—1372no—
122#41—5484no—
123C2×Q8—2741no—
124C2×D4—21no—
125C2×D4—2741no—
126C2×D4—2741no—
127C2×D4—2741no—
128C2×D4—1371no—
129C2×D4C2×#1058773911yescharacteristic
130C22×C4—2741no—
131C22×C4—2741no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4C2×#1058773911yescharacteristic
135C2×C8—2742no—
136C2×C8—2742no—
137C4⋊C4—2741no—
138C4⋊C4—1371no—
139C4⋊C4C2×#1058773911yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—2741no—
142C22⋊C4—2741no—
143C22⋊C4—2741no—
144C42—1371no—
145C42—1371no—
146C42—1371no—
147C42—1371no—
148C23—21no—
149C23—2741no—
150C23C22⋊#1058773911yescharacteristic
151Q8—5482no—
152D4—22no—
153D4—42no—
154D4—2742no—
155D4—5482no—
156D4—5482no—
157D4—5482no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—2742no—
161C2×C4—2741no—
162C2×C4—5482no—
163C2×C4—5481no—
164C2×C4—5482no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4—1371no—
168C2×C4—1371no—
169C2×C4C22⋊#1058773911yescharacteristic
170C2×C4C22×#1058773911yescharacteristic
171C8—5482no—
172C8—5482no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—5481no—
180C22—5481no—
181C22—5481no—
182C22C2×C22⋊#1058773911yescharacteristic
183C4—21no—
184C4—41no—
185C4—2741no—
186C4—2741no—
187C4—2741no—
188C4—5481no—
189C4—1372no—
190C4#1798417512yes—
191C2—41no—
192C2—81no—
193C2—10961no—
194C2#2257837611yescharacteristic
195C2#2257839511yescharacteristic
196C2#2257840011yescharacteristic
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#1496
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. #22578424C2semidirect
2. #22578419C2semidirect
3. #22578251C2semidirect
4. D4⋊#11764047C2semidirect
98. C137#1496semidirect

Maximal subgroups

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

subgroupindex
#225784242
#225784192
#225782512
D4⋊#117640472
#225779482
#1496137