Finite Group Databasebeta

← #31797081

Subgroups — #31797081

Order 17536.

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

normal subgroups only

This group has 250 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#22578314C212yesmaximal, complemented
3#22578311C212yesmaximal, complemented
4#22577951C211yesmaximal, characteristic
5C2×#14555519—22no—
6#17984149C2214yes—
7#17984148—22no—
8C2×#14555509C411yescharacteristic
9C2×#14555509C2211yescharacteristic
10#17984088C2211yescharacteristic
11#17984075C411yescharacteristic
12#17984075C2211yescharacteristic
13#17984059—22no—
14D4×Dic137—22no—
15C42×D137C411yescharacteristic
16#127×C137C411yescharacteristic
17#14555519—24no—
18#14555519—42no—
19C2×#11764047—42no—
20#14555509—41no—
21#14555509—41no—
22#14555509—41no—
23#14555509—41no—
24C2×Dic274C2×C411yescharacteristic
25C2×Dic274C2311yescharacteristic
26C2×C22⋊D137—21no—
27C2×C22⋊D137D412yes—
28C22×Dic137—21no—
29C22×Dic137D412yes—
30C2×C4×D137—21no—
31C2×C4×D137C2×C411yescharacteristic
32C4⋊Dic137—22no—
33C22⋊Dic137—22no—
34C22⋊Dic137—22no—
35C4×Dic137—21no—
36C4×Dic137C2×C411yescharacteristic
37C2×D4×C137—22no—
38C2×D4×C137C2×C411yescharacteristic
39C2×D4×C137C2×C411yescharacteristic
40C42×C137C2×C411yescharacteristic
41#11764047—44no—
42C22×D137C22⋊C411yescharacteristic
43Dic274—41no—
44Dic274—41no—
45C22⋊D137—42no—
46C22⋊D137—42no—
47C2×Dic137—22no—
48C2×Dic137—22no—
49C2×Dic137—21no—
50C2×Dic137—21no—
51C2×Dic137—41no—
52C2×Dic137—41no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C2×D412yes—
55C4×D137—21no—
56C4×D137—21no—
57C4×D137—42no—
58C23×C137—21no—
59C23×C137C22⋊C412yes—
60D4×C137—41no—
61D4×C137—41no—
62D4×C137—44no—
63C2×C4×C137—21no—
64C2×C4×C137C22×C411yescharacteristic
65C22×C137—22no—
66C22×C137—42no—
67C22×C137—42no—
68C22×C137—41no—
69C22×C137C2×C22⋊C411yescharacteristic
70C2×D137—21no—
71C2×D137—41no—
72C4×C137—22no—
73C4×C137—21no—
74Dic137—22no—
75Dic137—41no—
76Dic137—41no—
77Dic137#9911yescharacteristic
78Dic137#9911yescharacteristic
79C2×C137—21no—
80C2×C137—42no—
81C2×C137—81no—
82C2×C137C2×#9911yescharacteristic
83D137—41no—
84C137#213211yescharacteristic, complemented, Sylow
85#2132—1371nomaximal, Sylow
86#532—1371no—
87#413—1372no—
88#356—1372no—
89#355—1372no—
90C2×#41—1371no—
91C2×#41—1371no—
92C2×M16—2742no—
93C4⋊Q8—1371no—
94#127#1058773911yescharacteristic
95#124—1371no—
96#124—1371no—
97#121—2742no—
98C4×D4—2742no—
99C2×C42—1371no—
100#101—1374no—
101#100—2742no—
102#41—5481no—
103#41—5481no—
104#41—5481no—
105#41—5481no—
106C2×Q8—1371no—
107C2×Q8—1371no—
108C2×D4—22no—
109C2×D4—2741no—
110C2×D4—1372no—
111C2×D4C2×#1058773911yescharacteristic
112C2×D4C2×#1058773911yescharacteristic
113C22×C4—2741no—
114C22×C4—2741no—
115C22×C4—1372no—
116C22×C4—1371no—
117M16—2744no—
118M16—5482no—
119C2×C8—5482no—
120C4⋊C4—2742no—
121C22⋊C4—2742no—
122C22⋊C4—2742no—
123C42—2741no—
124C42—1371no—
125C42C2×#1058773911yescharacteristic
126C23—21no—
127C23—1371no—
128C23C22⋊#1058773912yes—
129Q8—5481no—
130Q8—5481no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—5482no—
135D4—5482no—
136C2×C4—21no—
137C2×C4—2741no—
138C2×C4—2741no—
139C2×C4—2742no—
140C2×C4—2742no—
141C2×C4—2741no—
142C2×C4—2741no—
143C2×C4—5482no—
144C2×C4—5481no—
145C2×C4—5481no—
146C2×C4—1372no—
147C2×C4—1371no—
148C2×C4C22×#1058773911yescharacteristic
149C8—5484no—
150C22—22no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22—2741no—
155C22—5481no—
156C22C2×C22⋊#1058773911yescharacteristic
157C4—22no—
158C4—21no—
159C4—2742no—
160C4—5481no—
161C4—5481no—
162C4—1371no—
163C4—1371no—
164C2—21no—
165C2—42no—
166C2—81no—
167C2—5481no—
168C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC2×C137C2×#99
prime radical#22577951C2
cosocleC2×Dic274C23
hypercenter#127#10587739
hypocenterC137#2132
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. #22578314C2semidirect
3. #22578311C2semidirect
84. C137#2132semidirect

Maximal subgroups

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

subgroupindex
#225783902
#225783142
#225783112
#225779512
#2132137