Finite Group Databasebeta

← #31796966

Subgroups — #31796966

Order 17536.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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
1C4×#14555520C211yesmaximal, characteristic
2C4×C4⋊#10587739C211yesmaximal, characteristic
3#22578273C212yesmaximal
4#22578273C212yesmaximal
5#22577959C211yesmaximal, characteristic
6C2×#14555520C2212yes—
7C2×C4⋊#10587739—21no—
8C2×C4×#10587739C2212yes—
9#17984143—21no—
10C4×#11764047—21no—
11(C2×#10587739)⋊C4—21no—
12#17984087C411yescharacteristic, complemented
13#17984087C411yescharacteristic, complemented
14#17984080C411yescharacteristic, complemented
15#17984066—21no—
16#17984066—21no—
17#17984053—21no—
18#17984053—21no—
19#17984046C2211yescharacteristic
20#17984046C2211yescharacteristic
21C42×D137C2211yescharacteristic
22#125×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24#14555520—22no—
25#14555520Q814yescomplemented
26C2×#11764047—22no—
27C4⋊#10587739—24no—
28C4×#10587739—24no—
29C2×Dic274—21no—
30C2×Dic274—21no—
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137C2311yescharacteristic
34C4⋊Dic137C2×C411yescharacteristic
35C4⋊Dic137C2×C411yescharacteristic
36#14555489—21no—
37#14555489—21no—
38#14555489—21no—
39#14555489—21no—
40#14555487—21no—
41#14555487—21no—
42C4×Dic137—21no—
43C4×Dic137—21no—
44C4×Dic137C2×C411yescharacteristic
45C4×Dic137D411yescharacteristic
46C4×Dic137D411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137—21no—
49C4⋊C4×C137C2×C411yescharacteristic
50C4⋊C4×C137C2×C411yescharacteristic
51C42×C137C2×C411yescharacteristic
52C2×#10587739—24no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55#11764047—24no—
56C22×D137C2×D411yescharacteristic
57Dic274—42no—
58Dic274—42no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C2×D411yescharacteristic
64C4×D137—22no—
65C4×D137—41no—
66C4×D137C2×Q812yes—
67C4×D137#4112yes—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22⋊C411yescharacteristic
72C2×C4×C137C22×C411yescharacteristic
73C22×C137C2×C22⋊C411yescharacteristic
74C2×D137—21no—
75C2×D137—21no—
76C2×D137#12411yescharacteristic
77C2×D137C4⋊Q811yescharacteristic
78#10587739—44no—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—41no—
82C4×C137—41no—
83C4×C137C4×Q812yes—
84Dic137—21no—
85Dic137—41no—
86Dic137—41no—
87Dic137#10412yes—
88Dic137C22⋊Q812yes—
89C2×C137#39111yescharacteristic
90C2×C137C2×#10411yescharacteristic
91C2×C137#42111yescharacteristic
92D137—21no—
93D137—21no—
94C137#198611yescharacteristic, complemented, Sylow
95#1986—1371nomaximal, Sylow
96#533—1371no—
97C4×M16—1371no—
98C4×C4⋊C4—1371no—
99#339—1372no—
100#339—1372no—
101C2×M16—1372no—
102C4⋊Q8—1371no—
103C4⋊Q8—1371no—
104#125—1371no—
105#125#1058773911yescharacteristic, complemented
106C22⋊Q8—2741no—
107C22⋊Q8—2741no—
108C4×Q8—2741no—
109C4×Q8—2741no—
110#117—1371no—
111#117—1371no—
112C2×C4⋊C4—2741no—
113C2×C42—1372no—
114C2×C42—1371no—
115#97—2741no—
116C4×C8—2741no—
117#95—2741no—
118C2×Q8—2741no—
119C2×Q8—2741no—
120C22×C4—2742no—
121C22×C4—1371no—
122C22×C4—1371no—
123C22×C4—1371no—
124M16—2742no—
125M16—1374no—
126C2×C8—2742no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C4⋊C4—2744no—
132C4⋊C4—2741no—
133C4⋊C4—2741no—
134C4⋊C4—1371no—
135C4⋊C4—1371no—
136C4⋊C4C2×#1058773911yescharacteristic
137C4⋊C4C2×#1058773911yescharacteristic
138C22⋊C4—2741no—
139C22⋊C4—2741no—
140C42—2741no—
141C42—2741no—
142C42—2744no—
143C42—1371no—
144C42—1371no—
145C42—1371no—
146C42C2×#1058773911yescharacteristic
147C23—1371no—
148Q8—5482no—
149Q8—5482no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—2742no—
153C2×C4—2744no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—5481no—
157C2×C4—5482no—
158C2×C4—5482no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1372no—
163C2×C4—1372no—
164C2×C4C22⋊#1058773911yescharacteristic
165C2×C4C22⋊#1058773911yescharacteristic
166C2×C4C22×#1058773911yescharacteristic
167C8—2744no—
168C22—2741no—
169C22—2741no—
170C22—1371no—
171C22—1371no—
172C22C2×C22⋊#1058773911yescharacteristic
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—2741no—
178C4—5484no—
179C4—5481no—
180C4—5481no—
181C4—1372no—
182C4—1372no—
183C4Q8×#1058773912yes—
184C2—2741no—
185C2—2741no—
186C2(C2×#10587739)⋊Q811yescharacteristic
187C2D137⋊#10411yescharacteristic
188C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#125×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenter#125#10587739
hypocenterC137#1986
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
12. #17984087C4semidirect
13. #17984087C4semidirect
14. #17984080C4semidirect
22. #125×C137C4semidirect
25. #14555520Q8semidirect
94. C137#1986semidirect
105. #125#10587739semidirect

Maximal subgroups

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

subgroupindex
C4×#145555202
C4×C4⋊#105877392
#225782732
#225779592
#1986137