Finite Group Databasebeta

← #31797089

Subgroups — #31797089

Order 17536.

262 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 52 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#22578390C212yesmaximal, complemented
2#22578312C212yesmaximal, complemented
3#22578311C212yesmaximal, complemented
4#22577966C211yesmaximal, characteristic
5C2×#14555519—22no—
6#17984149C2214yescomplemented
7#17984148—22no—
8C2×#14555509C2211yescharacteristic
9C2×D4×D137C411yescharacteristic
10#17984084C411yescharacteristic
11#17984076C2211yescharacteristic
12#17984075C2211yescharacteristic
13#17984060—22no—
14#17984056—22no—
15#17984045C411yescharacteristic
16#127×C137C411yescharacteristic
17#14555519—24no—
18#14555519—42no—
19C2×#11764047—42no—
20C23×D137—21no—
21#14555509—42no—
22C2×Dic274C2311yescharacteristic
23C2×D548C2×C411yescharacteristic
24C2×C22⋊D137—21no—
25C2×C22⋊D137—22no—
26C2×C22⋊D137D412yes—
27D4×D137—42no—
28C22×Dic137D412yes—
29C2×C4×D137C2×C411yescharacteristic
30#14555489—21no—
31C22⋊Dic137—22no—
32#14555487—21no—
33#14555487—22no—
34C4×Dic137C2×C411yescharacteristic
35C2×D4×C137—22no—
36C2×D4×C137C2×C411yescharacteristic
37C2×D4×C137C2×C411yescharacteristic
38C42×C137C2×C411yescharacteristic
39#11764047—44no—
40C22×D137—21no—
41C22×D137—42no—
42C22×D137—82no—
43C22×D137C22⋊C411yescharacteristic
44Dic274—41no—
45D548—41no—
46C22⋊D137—42no—
47C22⋊D137—42no—
48C22⋊D137—82no—
49C2×Dic137—22no—
50C2×Dic137—22no—
51C2×Dic137—21no—
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C2×D412yes—
54C4×D137—22no—
55C23×C137—21no—
56C23×C137C22⋊C412yes—
57D4×C137—41no—
58D4×C137—41no—
59D4×C137—82no—
60C2×C4×C137—21no—
61C2×C4×C137C22×C411yescharacteristic
62C22×C137—22no—
63C22×C137—42no—
64C22×C137—42no—
65C22×C137—41no—
66C22×C137C2×C22⋊C411yescharacteristic
67C2×D137—21no—
68C2×D137—41no—
69C2×D137—42no—
70C2×D137—41no—
71C2×D137—84no—
72C4×C137—21no—
73C4×C137—41no—
74Dic137—22no—
75Dic137—41no—
76Dic137#9912yes—
77C2×C137—21no—
78C2×C137—42no—
79C2×C137—81no—
80C2×C137C2×#9911yescharacteristic
81D137—41no—
82D137—81no—
83C137#213411yescharacteristic, complemented, Sylow
84#2134—1371nomaximal, Sylow
85#535—1371no—
86#413—1372no—
87#355—1372no—
88#355—1372no—
89C2×#41—1371no—
90C22×D4—1371no—
91C2×M16—2742no—
92#127—1371no—
93#127#1058773911yescharacteristic
94#124—1371no—
95#124—1371no—
96#121—2742no—
97#120—2742no—
98#117—1371no—
99#101—1374no—
100#100—2742no—
101C24—2741no—
102#41—5482no—
103C2×Q8—1371no—
104C2×D4—22no—
105C2×D4—2742no—
106C2×D4—2741no—
107C2×D4—5482no—
108C2×D4—1372no—
109C2×D4—1371no—
110C2×D4C2×#1058773911yescharacteristic
111C2×D4C2×#1058773911yescharacteristic
112C22×C4—1372no—
113C22×C4—1371no—
114M16—2744no—
115M16—5482no—
116C2×C8—5482no—
117C4⋊C4—2741no—
118C22⋊C4—2741no—
119C22⋊C4—2742no—
120C22⋊C4—2742no—
121C42—1371no—
122C42C2×#1058773911yescharacteristic
123C23—21no—
124C23—2741no—
125C23—5482no—
126C23—10962no—
127C23—1371no—
128C23C22⋊#1058773912yes—
129Q8—5481no—
130D4—41no—
131D4—41no—
132D4—82no—
133D4—5481no—
134D4—5482no—
135D4—5482no—
136D4—10962no—
137C2×C4—21no—
138C2×C4—2742no—
139C2×C4—2742no—
140C2×C4—2742no—
141C2×C4—2741no—
142C2×C4—1372no—
143C2×C4—1371no—
144C2×C4C22×#1058773911yescharacteristic
145C8—5484no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22—41no—
150C22—2741no—
151C22—5481no—
152C22—5482no—
153C22—5481no—
154C22—10964no—
155C22C2×C22⋊#1058773911yescharacteristic
156C4—21no—
157C4—41no—
158C4—2742no—
159C4—5481no—
160C4—1372no—
161C2—21no—
162C2—42no—
163C2—81no—
164C2—5481no—
165C2—10961no—
166C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC2×C137C2×#99
prime radical#22577966C2
cosocleC2×Dic274C23
hypercenter#127#10587739
hypocenterC137#2134
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578390C2semidirect
2. #22578312C2semidirect
3. #22578311C2semidirect
6. #17984149C22semidirect
83. C137#2134semidirect

Maximal subgroups

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

subgroupindex
#225783902
#225783122
#225783112
#225779662
#2134137