Finite Group Databasebeta

← D137⋊#355 #31797075

Subgroups — D137⋊#355 #31797075

Order 17536.

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

normal subgroups only

This group has 260 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, complemented
2#22578312C212yesmaximal, complemented
3#22578311C212yesmaximal, complemented
4#124×D137C211yesmaximal, characteristic
5D137⋊M16—22no—
6#17984149—22no—
7#17984149C2214yescomplemented
8C2×Q8×D137C2211yescharacteristic
9C2×D4×D137C411yescharacteristic
10#17984077C411yescharacteristic
11#17984076C2211yescharacteristic
12#17984075C2211yescharacteristic
13#17984074—22no—
14C22⋊C4×D137—22no—
15C42×D137C411yescharacteristic
16#124×C137C411yescharacteristic
17#14555520—42no—
18#14555519—24no—
19#14555518—42no—
20C23×D137—21no—
21C2×Dic274D412yes—
22C2×Dic274C2311yescharacteristic
23Q8×D137—41no—
24Q8×D137—41no—
25C2×D548C2×C411yescharacteristic
26C2×C22⋊D137—21no—
27D4×D137—41no—
28D4×D137—41no—
29C2×C4×D137—21no—
30C2×C4×D137C2×C411yescharacteristic
31C2×C4×D137D412yes—
32C22⋊Dic137—22no—
33#14555487—22no—
34#14555487—22no—
35C4×Dic137—21no—
36C4×Dic137C2×C411yescharacteristic
37C2×Q8×C137C2×C411yescharacteristic
38C2×D4×C137C2×C411yescharacteristic
39C22⋊C4×C137—22no—
40C42×C137C2×C411yescharacteristic
41#11764047—44no—
42C22×D137—21no—
43C22×D137—44no—
44C22×D137—41no—
45C22×D137—41no—
46C22×D137C22⋊C411yescharacteristic
47Dic274—42no—
48Dic274—41no—
49D548—41no—
50C22⋊D137—42no—
51C2×Dic137—21no—
52C2×Dic137C22⋊C411yescharacteristic
53C2×Dic137C2×D412yes—
54C4×D137—21no—
55C4×D137—21no—
56C4×D137—22no—
57C4×D137—22no—
58C4×D137—42no—
59C23×C137—21no—
60Q8×C137—41no—
61D4×C137—41no—
62C2×C4×C137—21no—
63C2×C4×C137C22⋊C412yes—
64C2×C4×C137C22×C411yescharacteristic
65C22×C137—42no—
66C22×C137—41no—
67C22×C137C2×C22⋊C411yescharacteristic
68C2×D137—22no—
69C2×D137—42no—
70C2×D137—41no—
71C2×D137—82no—
72C2×D137—82no—
73C2×D137#9911yescharacteristic
74C2×D137#9911yescharacteristic
75C4×C137—22no—
76C4×C137—21no—
77C4×C137—41no—
78Dic137—21no—
79Dic137—22no—
80Dic137—41no—
81C2×C137—21no—
82C2×C137—81no—
83C2×C137C2×#9911yescharacteristic
84D137—21no—
85D137—81no—
86D137#35512yescomplemented
87C137C2×#35511yescharacteristic, complemented, Sylow
88C2×#355—1371nomaximal, Sylow
89C2×#124—1371no—
90C2×#101—1372no—
91#355—1372no—
92#355—1372no—
93C22×Q8—1371no—
94C22×D4—1371no—
95C2×M16—2742no—
96#124—2742no—
97#124—1371no—
98#124—1371no—
99#124—1371no—
100#124#1058773911yescharacteristic
101C2×C22⋊C4—2742no—
102C2×C42—1371no—
103#101—2742no—
104#101—1374no—
105C24—2741no—
106C2×Q8—5481no—
107C2×Q8—5481no—
108C2×Q8—1372no—
109C2×Q8—1371no—
110C2×Q8C2×#1058773911yescharacteristic
111C2×D4—2741no—
112C2×D4—5481no—
113C2×D4—5481no—
114C2×D4—1371no—
115C2×D4C2×#1058773911yescharacteristic
116C22×C4—2741no—
117C22×C4—1372no—
118C22×C4—1371no—
119M16—2744no—
120M16—5482no—
121C2×C8—5482no—
122C22⋊C4—22no—
123C22⋊C4—2742no—
124C22⋊C4—2742no—
125C22⋊C4—2742no—
126C42—2741no—
127C42—1371no—
128C42C2×#1058773911yescharacteristic
129C23—21no—
130C23—2741no—
131C23—5484no—
132C23—5481no—
133C23—5481no—
134C23—1371no—
135Q8—41no—
136Q8—5482no—
137Q8—5481no—
138D4—41no—
139D4—5481no—
140D4—5482no—
141C2×C4—21no—
142C2×C4—2742no—
143C2×C4—2742no—
144C2×C4—2741no—
145C2×C4—2741no—
146C2×C4—2741no—
147C2×C4—5482no—
148C2×C4—1371no—
149C2×C4—1372no—
150C2×C4C22⋊#1058773912yes—
151C2×C4C22×#1058773911yescharacteristic
152C8—5484no—
153C22—42no—
154C22—41no—
155C22—2742no—
156C22—5482no—
157C22—5481no—
158C22—10962no—
159C22—10962no—
160C22—1371no—
161C22—1371no—
162C22C2×C22⋊#1058773911yescharacteristic
163C4—22no—
164C4—21no—
165C4—41no—
166C4—2742no—
167C4—2741no—
168C4—5481no—
169C2—21no—
170C2—81no—
171C2—2741no—
172C2—10961no—
173C2—1372no—
174C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137C2×#99
prime radicalC2×D4×D137C4
cosocleC2×Dic274C23
hypercenter#124#10587739
hypocenterC137C2×#355
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D137⋊#101C2semidirect
2. #22578312C2semidirect
3. #22578311C2semidirect
7. #17984149C22semidirect
86. D137#355semidirect
87. C137C2×#355semidirect

Maximal subgroups

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

subgroupindex
D137⋊#1012
#225783122
#225783112
#124×D1372
C2×#355137