Finite Group Databasebeta

← #31796308

Subgroups — #31796308

Order 17536.

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

normal subgroups only

This group has 252 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#22578423C211yesmaximal, characteristic, complemented
2C2×#17984155C211yesmaximal, characteristic, complemented
3#22578263C212yesmaximal, complemented
4#22578261C212yesmaximal, complemented
5#22577949C211yesmaximal, characteristic
6C22×#11764047—21no—
7#17984155—21no—
8#17984155C2212yes—
9#17984155C2212yes—
10C22⋊#11764047—21no—
11#17984143—21no—
12C2×#14555510C411yescharacteristic
13#17984086C2211yescharacteristic
14#17984084C2211yescharacteristic
15#17984077C411yescharacteristic
16#17984075C411yescharacteristic
17#17984060—21no—
18#17984060—21no—
19C4×C22⋊D137—21no—
20C4×C22⋊D137—21no—
21C2×C4×Dic137C2211yescharacteristic
22C2×#41×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26#14555510—42no—
27C2×Dic274C2×C411yescharacteristic
28C2×D548C2×C411yescharacteristic
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137D411yescharacteristic
33C22×Dic137D411yescharacteristic
34C2×C4×D137—21no—
35#14555489—21no—
36#14555489—21no—
37C22⋊Dic137—21no—
38C22⋊Dic137—21no—
39#14555487—21no—
40#14555487—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45#41×C137—42no—
46C2×Q8×C137C2×C411yescharacteristic
47C2×D4×C137—21no—
48C2×D4×C137C2×C411yescharacteristic
49C22×C4×C137—21no—
50C22×C4×C137C2×C411yescharacteristic
51#11764047—42no—
52#11764047—42no—
53C22×D137—21no—
54Dic274—41no—
55D548—41no—
56C22⋊D137—42no—
57C22⋊D137—42no—
58C22⋊D137—41no—
59C22⋊D137—41no—
60C2×Dic137—22no—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—21no—
64C2×Dic137C22⋊C411yescharacteristic
65C2×Dic137C22⋊C411yescharacteristic
66C2×Dic137C2×D411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C4×D137—42no—
69C23×C137—21no—
70C23×C137C22⋊C411yescharacteristic
71Q8×C137—41no—
72D4×C137—41no—
73D4×C137—42no—
74C2×C4×C137—22no—
75C2×C4×C137—21no—
76C2×C4×C137—42no—
77C2×C4×C137C22⋊C411yescharacteristic
78C2×C4×C137C22×C411yescharacteristic
79C22×C137—21no—
80C22×C137—21no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137#10012yes—
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—41no—
87C2×D137—41no—
88C2×D137—41no—
89C4×C137—21no—
90C4×C137—21no—
91C4×C137—41no—
92Dic137—21no—
93Dic137—21no—
94Dic137—41no—
95Dic137#13512yes—
96Dic137AGL(1,ℤ/8)11yescharacteristic
97Dic137#13711yescharacteristic
98C2×C137—22no—
99C2×C137—81no—
100C2×C137C2×#10011yescharacteristic
101C2×C137#41611yescharacteristic
102C2×C137#41711yescharacteristic
103D137—81no—
104C137#155111yescharacteristic, complemented, Sylow
105#1551—1371nomaximal, Sylow
106#532—1371no—
107#432—1371no—
108C2×C4⋊C8—1371no—
109#333—1372no—
110#331—1372no—
111C2×#41—1371no—
112C2×#41#1058773911yescharacteristic
113C22×C8—2741no—
114C4⋊Q8—1371no—
115#127—1371no—
116#124—1371no—
117#124—1371no—
118#121—2741no—
119#121—2741no—
120C4×D4—2741no—
121C4×D4—2741no—
122C2×C42—1371no—
123C4⋊C8—2741no—
124C4⋊C8—1372no—
125C4⋊C8—1372no—
126C22⋊C8—2741no—
127#97—2741no—
128#41—42no—
129#41—5482no—
130C2×Q8—1371no—
131C2×Q8C2×#1058773911yescharacteristic
132C2×D4—21no—
133C2×D4—2741no—
134C2×D4—2741no—
135C2×D4—2741no—
136C2×D4—1371no—
137C2×D4C2×#1058773911yescharacteristic
138C22×C4—21no—
139C22×C4—2741no—
140C22×C4—1371no—
141C22×C4—1371no—
142C22×C4C2×#1058773911yescharacteristic
143C2×C8—2742no—
144C2×C8—2742no—
145C2×C8—5482no—
146C4⋊C4—2741no—
147C4⋊C4—2741no—
148C22⋊C4—2741no—
149C22⋊C4—2741no—
150C22⋊C4—2741no—
151C22⋊C4—2741no—
152C42—1371no—
153C42—1371no—
154C42—1371no—
155C42—1371no—
156C23—21no—
157C23—2741no—
158C23C22⋊#1058773911yescharacteristic
159Q8—41no—
160Q8—5481no—
161D4—41no—
162D4—42no—
163D4—5481no—
164D4—5482no—
165D4—5482no—
166D4—5481no—
167D4—5481no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—2741no—
172C2×C4—2742no—
173C2×C4—2741no—
174C2×C4—2741no—
175C2×C4—5482no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4—1371no—
179C2×C4—1371no—
180C2×C4C22⋊#1058773911yescharacteristic
181C2×C4C22×#1058773911yescharacteristic
182C8—5482no—
183C8—5482no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—5481no—
190C22—5481no—
191C22—5481no—
192C22#1798414812yes—
193C22C2×C22⋊#1058773911yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—2741no—
198C4—2741no—
199C4—5481no—
200C4—1372no—
201C4—1371no—
202C4—1371no—
203C2—22no—
204C2—81no—
205C2—10961no—
206C2C2×#1798414811yescharacteristic
207C2#2257839311yescharacteristic
208C2#2257839411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×#41×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577949C2
cosocleC4×Dic137C23
hypercenterC2×#41#10587739
hypocenterC137#1551
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #22578423C2semidirect
2. C2×#17984155C2semidirect
3. #22578263C2semidirect
4. #22578261C2semidirect
104. C137#1551semidirect

Maximal subgroups

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

subgroupindex
#225784232
C2×#179841552
#225782632
#225782612
#225779492
#1551137