Finite Group Databasebeta

← D137⋊#356 #31797078

Subgroups — D137⋊#356 #31797078

Order 17536.

240 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D137⋊#101C212yesmaximal
2#22578314C212yesmaximal, complemented
3#22578313C212yesmaximal, complemented
4C4⋊Q8×D137C211yesmaximal, characteristic
5D137⋊M16—22no—
6#17984149—22no—
7#17984149C2214yes—
8C2×Q8×D137C411yescharacteristic
9C2×Q8×D137C2211yescharacteristic
10#17984088C2211yescharacteristic
11#17984087—22no—
12#17984086C411yescharacteristic
13#17984086C2211yescharacteristic
14C4⋊C4×D137—22no—
15C42×D137C411yescharacteristic
16C4⋊Q8×C137C411yescharacteristic
17#14555520—42no—
18#14555519—24no—
19#14555518—42no—
20C2×Dic274—21no—
21C2×Dic274C2×C411yescharacteristic
22C2×Dic274D412yes—
23C2×Dic274C2311yescharacteristic
24Q8×D137—41no—
25Q8×D137—41no—
26Q8×D137—41no—
27Q8×D137—41no—
28C2×C4×D137—21no—
29C2×C4×D137—21no—
30C2×C4×D137C2×C411yescharacteristic
31C2×C4×D137D412yes—
32C4⋊Dic137—22no—
33#14555489—22no—
34#14555489—22no—
35C4×Dic137—21no—
36C4×Dic137C2×C411yescharacteristic
37C2×Q8×C137C2×C411yescharacteristic
38C2×Q8×C137C2×C411yescharacteristic
39C4⋊C4×C137—22no—
40C42×C137C2×C411yescharacteristic
41#11764047—44no—
42C22×D137C22⋊C411yescharacteristic
43Dic274—42no—
44Dic274—41no—
45Dic274—42no—
46Dic274—41no—
47C2×Dic137—21no—
48C2×Dic137—21no—
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C2×D412yes—
51C4×D137—24no—
52C4×D137—21no—
53C4×D137—21no—
54C4×D137—22no—
55C4×D137—22no—
56C4×D137—41no—
57C4×D137—41no—
58Q8×C137—41no—
59Q8×C137—41no—
60C2×C4×C137—21no—
61C2×C4×C137—21no—
62C2×C4×C137C22⋊C412yes—
63C2×C4×C137C22×C411yescharacteristic
64C22×C137C2×C22⋊C411yescharacteristic
65C2×D137—22no—
66C2×D137#9911yescharacteristic
67C2×D137#9911yescharacteristic
68C4×C137—22no—
69C4×C137—22no—
70C4×C137—21no—
71C4×C137—41no—
72Dic137—22no—
73Dic137—21no—
74Dic137—22no—
75Dic137—41no—
76C2×C137—21no—
77C2×C137C2×#9911yescharacteristic
78D137—21no—
79D137#35612yescomplemented
80C137C2×#35611yescharacteristic, complemented, Sylow
81C2×#356—1371nomaximal, Sylow
82C2×C4⋊Q8—1371no—
83C2×#101—1372no—
84#356—1372no—
85#356—1372no—
86C22×Q8—1371no—
87C22×Q8—1371no—
88C2×M16—2742no—
89C4⋊Q8—2742no—
90C4⋊Q8—1371no—
91C4⋊Q8—1371no—
92C4⋊Q8—1371no—
93C4⋊Q8#1058773911yescharacteristic
94C2×C4⋊C4—2742no—
95C2×C42—1371no—
96#101—2742no—
97#101—1374no—
98C2×Q8—2741no—
99C2×Q8—5481no—
100C2×Q8—5481no—
101C2×Q8—5481no—
102C2×Q8—5481no—
103C2×Q8—1371no—
104C2×Q8—1372no—
105C2×Q8—1371no—
106C2×Q8C2×#1058773911yescharacteristic
107C2×Q8C2×#1058773911yescharacteristic
108C22×C4—2741no—
109C22×C4—2741no—
110C22×C4—1372no—
111C22×C4—1371no—
112M16—2744no—
113M16—5482no—
114C2×C8—5482no—
115C4⋊C4—22no—
116C4⋊C4—2742no—
117C4⋊C4—2742no—
118C4⋊C4—2742no—
119C42—2741no—
120C42—1371no—
121C42C2×#1058773911yescharacteristic
122C23—1371no—
123Q8—41no—
124Q8—41no—
125Q8—5482no—
126Q8—5481no—
127Q8—5481no—
128Q8—5482no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—2744no—
132C2×C4—2742no—
133C2×C4—2742no—
134C2×C4—2741no—
135C2×C4—2741no—
136C2×C4—2741no—
137C2×C4—2741no—
138C2×C4—5481no—
139C2×C4—5481no—
140C2×C4—1371no—
141C2×C4—1372no—
142C2×C4C22⋊#1058773912yes—
143C2×C4C22×#1058773911yescharacteristic
144C8—5484no—
145C22—2742no—
146C22—1371no—
147C22—1371no—
148C22C2×C22⋊#1058773911yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—41no—
153C4—2742no—
154C4—2742no—
155C4—2741no—
156C4—5481no—
157C2—21no—
158C2—2741no—
159C2—1372no—
160C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC2×C137C2×#99
prime radicalC22×D137C22⋊C4
cosocleC2×Dic274C23
hypercenterC4⋊Q8#10587739
hypocenterC137C2×#356
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22578314C2semidirect
3. #22578313C2semidirect
79. D137#356semidirect
80. C137C2×#356semidirect

Maximal subgroups

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

subgroupindex
D137⋊#1012
#225783142
#225783132
C4⋊Q8×D1372
C2×#356137