Finite Group Databasebeta

← #31796320

Subgroups — #31796320

Order 17536.

246 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578426C211yesmaximal, characteristic, complemented
2#22578408C211yesmaximal, characteristic, complemented
3#22578262C212yesmaximal, complemented
4#22578261C212yesmaximal, complemented
5#22577950C211yesmaximal, characteristic
6C2×#14555520—21no—
7#17984155—21no—
8#17984155C2212yes—
9#17984155C2212yes—
10#17984145—21no—
11#17984143—21no—
12C2×#14555510—21no—
13#17984087C2211yescharacteristic
14#17984084C2211yescharacteristic
15#17984074—21no—
16#17984063C411yescharacteristic
17#17984061—21no—
18#17984059C411yescharacteristic
19C4×D548C411yescharacteristic
20C4×C22⋊D137—21no—
21C42×D137C2211yescharacteristic
22C4×D4×C137C411yescharacteristic
23#14555520—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26#14555510—44no—
27C2×Dic274—21no—
28C2×D548—21no—
29C2×D548C2×C411yescharacteristic
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C2×C22⋊D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137C2×C411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137D411yescharacteristic
37C4⋊Dic137C2×C411yescharacteristic
38#14555489—21no—
39C22⋊Dic137—21no—
40#14555487—21no—
41#14555487—21no—
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C2×D4×C137C2×C411yescharacteristic
46C22×C4×C137—21no—
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49C42×C137C2×C411yescharacteristic
50#11764047—42no—
51#11764047—42no—
52C22×D137—21no—
53C22×D137C22⋊C411yescharacteristic
54Dic274—42no—
55D548—41no—
56D548—42no—
57C22⋊D137—42no—
58C22⋊D137—41no—
59C22⋊D137—42no—
60C22⋊D137—41no—
61C2×Dic137—21no—
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C2×D411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C4×D137—22no—
66C4×D137—21no—
67C4×D137—21no—
68C4×D137—41no—
69C4×D137—42no—
70C23×C137—21no—
71D4×C137—41no—
72C2×C4×C137—21no—
73C2×C4×C137—42no—
74C2×C4×C137C22⋊C411yescharacteristic
75C2×C4×C137C22⋊C411yescharacteristic
76C2×C4×C137C22×C411yescharacteristic
77C22×C137—41no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137C2×C22⋊C411yescharacteristic
81C2×D137—21no—
82C2×D137—21no—
83C2×D137—21no—
84C2×D137—41no—
85C2×D137—41no—
86C2×D137—41no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—21no—
90C4×C137—41no—
91Dic137—21no—
92Dic137—41no—
93Dic137AGL(1,ℤ/8)11yescharacteristic
94Dic137AGL(1,ℤ/8)12yes—
95Dic137#13711yescharacteristic
96C2×C137—81no—
97C2×C137#41311yescharacteristic
98C2×C137#41711yescharacteristic
99C2×C137#41811yescharacteristic
100D137—41no—
101D137—81no—
102C137#155711yescharacteristic, complemented, Sylow
103#1557—1371nomaximal, Sylow
104#532—1371no—
105#432—1371no—
106#423—1371no—
107#332—1372no—
108#331—1372no—
109C2×#41—2741no—
110C2×M16—2741no—
111C4⋊Q8—1371no—
112#127—1371no—
113#124—2741no—
114#121—2741no—
115#121—1371no—
116#121—1371no—
117C4×D4—2741no—
118C4×D4—1371no—
119C4×D4#1058773911yescharacteristic
120C2×C42—1371no—
121C4⋊C8—2741no—
122C4⋊C8—1372no—
123C4⋊C8—1372no—
124C22⋊C8—2741no—
125#97—2741no—
126#41—5484no—
127C2×Q8—2741no—
128C2×D4—2741no—
129C2×D4—2741no—
130C2×D4—2741no—
131C2×D4—2741no—
132C2×D4—1371no—
133C2×D4C2×#1058773911yescharacteristic
134C22×C4—21no—
135C22×C4—2741no—
136C22×C4—1371no—
137C22×C4—1371no—
138C22×C4—1371no—
139M16—5482no—
140C2×C8—2742no—
141C2×C8—2742no—
142C4⋊C4—2741no—
143C4⋊C4—1371no—
144C4⋊C4C2×#1058773911yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2741no—
147C22⋊C4—2741no—
148C22⋊C4—2741no—
149C42—1371no—
150C42—1371no—
151C42—1371no—
152C42C2×#1058773911yescharacteristic
153C23—21no—
154C23—2741no—
155C23—1371no—
156Q8—5482no—
157D4—41no—
158D4—5481no—
159D4—5482no—
160D4—5482no—
161D4—5482no—
162D4—5481no—
163D4—5481no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—2741no—
167C2×C4—2742no—
168C2×C4—2741no—
169C2×C4—2741no—
170C2×C4—5481no—
171C2×C4—5482no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4C22⋊#1058773911yescharacteristic
176C2×C4C22⋊#1058773911yescharacteristic
177C2×C4C22×#1058773911yescharacteristic
178C8—5482no—
179C8—5482no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—2741no—
184C22—2741no—
185C22—2741no—
186C22—5481no—
187C22—5481no—
188C22—5481no—
189C22C2×C22⋊#1058773911yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—41no—
194C4—2741no—
195C4—5481no—
196C4—1371no—
197C4—1371no—
198C4—1372no—
199C2—81no—
200C2—5481no—
201C2—10961no—
202C2#2257838911yescharacteristic
203C2#2257839411yescharacteristic
204C2#2257839511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577950C2
cosocleC4×Dic137C23
hypercenterC4×D4#10587739
hypocenterC137#1557
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22578426C2semidirect
2. #22578408C2semidirect
3. #22578262C2semidirect
4. #22578261C2semidirect
102. C137#1557semidirect

Maximal subgroups

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

subgroupindex
#225784262
#225784082
#225782622
#225782612
#225779502
#1557137