Finite Group Databasebeta

← #31796220

Subgroups — #31796220

Order 17536.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#22578421C211yesmaximal, characteristic, complemented
2#22578347C211yesmaximal, characteristic, complemented
3#22578255C214yesmaximal, complemented
4#22577954C211yesmaximal, characteristic
5C2×C4⋊#10587739—21no—
6#17984155—21no—
7#17984145C2212yes—
8C4×#11764047—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11C2×#14555511—21no—
12#17984085C411yescharacteristic, complemented
13#17984077—21no—
14#17984063—21no—
15#17984063C2212yes—
16#17984050—21no—
17#17984050C412yescomplemented
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C4⋊#10587739—42no—
23#14555511—42no—
24#14555511—42no—
25C2×D548—22no—
26C2×D548—21no—
27C2×D548—21no—
28C2×D548C2×C412yes—
29C2×C4×D137—22no—
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×C4×D137C2311yescharacteristic
33#14555487—22no—
34#14555487—22no—
35C4×Dic137C2×C411yescharacteristic
36C4×Dic137D411yescharacteristic
37C4×Dic137D411yescharacteristic
38C2×Q8×C137—21no—
39C4⋊C4×C137—21no—
40C4⋊C4×C137C2×C412yes—
41C42×C137C2×C411yescharacteristic
42C2×#10587739—42no—
43C2×#10587739—42no—
44C2×#10587739—42no—
45#11764047—42no—
46C22×D137—22no—
47C22×D137C2×D411yescharacteristic
48D548—42no—
49D548—42no—
50D548—42no—
51D548—42no—
52D548—42no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C4×D137—41no—
57C4×D137—41no—
58C4×D137—44no—
59C4×D137D812yes—
60C4×D137SD1612yes—
61Q8×C137—42no—
62C2×C4×C137—22no—
63C2×C4×C137C22⋊C411yescharacteristic
64C2×C4×C137C22⋊C411yescharacteristic
65C2×C4×C137C22×C411yescharacteristic
66C22×C137C2×C22⋊C411yescharacteristic
67C2×D137—21no—
68C2×D137—21no—
69C2×D137—42no—
70C2×D137—42no—
71C2×D137—42no—
72C2×D137C2×D811yescharacteristic
73C2×D137C2×SD1611yescharacteristic
74#10587739—82no—
75C4×C137—21no—
76C4×C137—21no—
77C4×C137—42no—
78C4×C137#10212yes—
79Dic137—21no—
80Dic137—21no—
81Dic137#10212yes—
82C2×C137#41011yescharacteristic
83C2×C137C2×#10211yescharacteristic
84C2×C137#42111yescharacteristic
85D137—21no—
86D137—21no—
87D137—82no—
88C137#151711yescharacteristic, complemented, Sylow
89#1517—1371nomaximal, Sylow
90#532—1371no—
91#431—1371no—
92#382—1371no—
93#327—1374no—
94C2×#41—2741no—
95C4⋊Q8#1058773911yescharacteristic, complemented
96#127—1371no—
97#124—2741no—
98#121—2741no—
99#121—1372no—
100C4×D4—2741no—
101C4×D4—1372no—
102C2×C4⋊C4—2741no—
103C2×C42—1371no—
104C4⋊C8—2741no—
105C22⋊C8—1372no—
106C4×C8—2741no—
107#95—2741no—
108#95—1372no—
109#41—5482no—
110#41—5482no—
111C2×Q8—21no—
112C2×D4—2742no—
113C2×D4—2741no—
114C2×D4—2741no—
115C2×D4—1372no—
116C22×C4—2742no—
117C22×C4—2742no—
118C22×C4—1371no—
119C22×C4—1371no—
120C22×C4—1371no—
121C2×C8—2742no—
122C4⋊C4—21no—
123C4⋊C4—5482no—
124C4⋊C4C2×#1058773912yes—
125C22⋊C4—2742no—
126C22⋊C4—2742no—
127C42—1371no—
128C42—1371no—
129C42—1371no—
130C42C2×#1058773911yescharacteristic
131C23—2742no—
132C23—1371no—
133Q8—42no—
134D4—5482no—
135D4—5482no—
136D4—5482no—
137D4—5482no—
138D4—5482no—
139C2×C4—22no—
140C2×C4—5481no—
141C2×C4—5481no—
142C2×C4—5484no—
143C2×C4—5482no—
144C2×C4—5482no—
145C2×C4—5482no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4—1372no—
150C2×C4—1372no—
151C2×C4C22⋊#1058773911yescharacteristic
152C2×C4C22⋊#1058773911yescharacteristic
153C2×C4C22×#1058773911yescharacteristic
154C8—5482no—
155C22—2741no—
156C22—2741no—
157C22—5482no—
158C22—5482no—
159C22—5482no—
160C22—1371no—
161C22—1371no—
162C22C2×C22⋊#1058773911yescharacteristic
163C4—21no—
164C4—21no—
165C4—42no—
166C4—2741no—
167C4—2741no—
168C4—10962no—
169C4—1372no—
170C4#1798415012yes—
171C2—2741no—
172C2—2741no—
173C2—10962no—
174C2#2257838411yescharacteristic
175C2C2×#1798415011yescharacteristic
176C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577954C2
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1517
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22578421C2semidirect
2. #22578347C2semidirect
3. #22578255C2semidirect
12. #17984085C4semidirect
17. #17984050C4semidirect
19. C4⋊Q8×C137C4semidirect
88. C137#1517semidirect
95. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784212
#225783472
#225782552
#225779542
#1517137