Finite Group Databasebeta

← #31796856

Subgroups — #31796856

Order 17536.

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

normal subgroups only

This group has 186 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#22578410C211yesmaximal, characteristic
2#22578368C211yesmaximal, characteristic
3#22578279C211yesmaximal, characteristic
4#22578273C211yesmaximal, characteristic
5#22578273C211yesmaximal, characteristic
6#22578268C211yesmaximal, characteristic
7#22577438C211yesmaximal, characteristic
8C2×#14555520C2211yescharacteristic
9C2×#14555520C2211yescharacteristic
10C2×#14555518C2211yescharacteristic
11C2×C4×#10587739C2211yescharacteristic
12#17984155—21no—
13C4⋊#11764047—21no—
14#17984143—21no—
15C4×#11764047—21no—
16C2×C8×D137C2211yescharacteristic
17#17984046C2211yescharacteristic
18#17984046C2211yescharacteristic
19#17984021C411yescharacteristic, complemented
20#17984021C411yescharacteristic, complemented
21#17984020C411yescharacteristic, complemented
22#106×C137C411yescharacteristic, complemented
23C22×#10587739—21no—
24#14555520—21no—
25#14555520—21no—
26#14555520—21no—
27#14555520Q812yes—
28#14555518—21no—
29#14555518—21no—
30C2×#11764047—21no—
31C2×#11764047—21no—
32C2×#11764047D411yescharacteristic
33C2×#11764047D411yescharacteristic
34C4×#10587739—21no—
35C4×#10587739—21no—
36C2×C4×D137C2311yescharacteristic
37C2×#11764040C2×C411yescharacteristic
38C8×D137—21no—
39C8×D137—21no—
40C4⋊Dic137C2×C411yescharacteristic
41C4⋊Dic137C2×C411yescharacteristic
42#14555487—21no—
43#14555487—21no—
44C4×Dic137—21no—
45C4×Dic137—21no—
46C2×C8×C137C2×C411yescharacteristic
47C4⋊C4×C137C2×C411yescharacteristic
48C4⋊C4×C137C2×C411yescharacteristic
49C2×#10587739—22no—
50C2×#10587739—41no—
51C2×#10587739—41no—
52#11764047—22no—
53#11764047—21no—
54#11764047—21no—
55#11764047—41no—
56C22×D137#4111yescharacteristic
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C2×D411yescharacteristic
60C4×D137C2×Q811yescharacteristic
61C4×D137#4111yescharacteristic
62C4×D137#4111yescharacteristic
63C4×D137#4111yescharacteristic
64#11764040—21no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137C22×C411yescharacteristic
68C8×C137—21no—
69C22×C137C4×D411yescharacteristic
70C2×D137—21no—
71C2×D137—21no—
72C2×D137#12511yescharacteristic
73C2×D137#12611yescharacteristic
74#10587739—42no—
75C4×C137—41no—
76C4×C137—41no—
77C4×C137#11711yescharacteristic
78C4×C137C4×Q811yescharacteristic
79Dic137—41no—
80Dic137—41no—
81Dic137C22⋊Q811yescharacteristic
82Dic137#12311yescharacteristic
83C2×C137#38711yescharacteristic
84C2×C137#44311yescharacteristic
85C2×C137#44411yescharacteristic
86D137—21no—
87D137—21no—
88C137#192211yescharacteristic, complemented, Sylow
89#1922—1371nomaximal, Sylow
90#427—1371no—
91#424—1371no—
92#405—1371no—
93#341—1371no—
94#339—1371no—
95#339—1371no—
96#336—1371no—
97C2×M16—1371no—
98C2×M16—1371no—
99C22×C8—1371no—
100C22×C8—1371no—
101#117—1371no—
102#117—1371no—
103C2×C42—1371no—
104#107—1371no—
105#107—1371no—
106#106—1371no—
107#106#1058773911yescharacteristic, complemented
108C4⋊C8—2741no—
109C4⋊C8—2741no—
110#97—2741no—
111C4×C8—2741no—
112C22×C4—2741no—
113C22×C4—1371no—
114M16—2741no—
115M16—2741no—
116M16—2741no—
117M16—1372no—
118C2×C8—2741no—
119C2×C8—2741no—
120C2×C8—2741no—
121C2×C8—2741no—
122C2×C8—2741no—
123C2×C8—2741no—
124C2×C8—1371no—
125C2×C8—1371no—
126C2×C8—1371no—
127C2×C8C2×#1058773911yescharacteristic
128C4⋊C4—1371no—
129C4⋊C4—1371no—
130C4⋊C4C2×#1058773911yescharacteristic
131C4⋊C4C2×#1058773911yescharacteristic
132C22⋊C4—2741no—
133C22⋊C4—2741no—
134C42—2741no—
135C42—2741no—
136C42—2741no—
137C42—2741no—
138C23—1371no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—2742no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—5481no—
145C2×C4—5481no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1371no—
151C2×C4C22×#1058773911yescharacteristic
152C8—21no—
153C8—2741no—
154C8—2741no—
155C8—2742no—
156C8—2741no—
157C8—5481no—
158C22—2741no—
159C22—2741no—
160C22—1371no—
161C22—1371no—
162C22D4×#1058773911yescharacteristic
163C4—41no—
164C4—41no—
165C4—5482no—
166C4—5481no—
167C4—5481no—
168C4—1371no—
169C4—1371no—
170C4#1798416711yescharacteristic
171C4Q8×#1058773911yescharacteristic
172C2—2741no—
173C2—2741no—
174C2#2257835211yescharacteristic
175C2#2257844611yescharacteristic
176C2#2257844811yescharacteristic
subgroupG/H
centerC22D4×#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#106×C137C4
socleC22×C137C4×D4
prime radicalC22×D137#41
cosocleC2×C4×D137C23
hypercenter#106#10587739
hypocenterC137#1922
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
19. #17984021C4semidirect
20. #17984021C4semidirect
21. #17984020C4semidirect
22. #106×C137C4semidirect
88. C137#1922semidirect
107. #106#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784102
#225783682
#225782792
#225782732
#225782682
#225774382
#1922137