Finite Group Databasebeta

← D137⋊#353 #31797055

Subgroups — D137⋊#353 #31797055

Order 17536.

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

normal subgroups only

This group has 368 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⋊#99C212yesmaximal, complemented
2#22578308C212yesmaximal, complemented
3#22578307C212yesmaximal, complemented
4#127×D137C211yesmaximal, characteristic
5C2×C22⋊#10587739—22no—
6#17984147C2214yescomplemented
7#17984146—22no—
8C2×D4×D137—22no—
9C2×D4×D137C411yescharacteristic, complemented
10C2×D4×D137C2211yescharacteristic
11#17984085C2211yescharacteristic
12#17984084—22no—
13#17984084C411yescharacteristic, complemented
14#17984084C2211yescharacteristic
15C42×D137C411yescharacteristic, complemented
16#127×C137C411yescharacteristic, complemented
17C22×#10587739—42no—
18C22⋊#10587739—24no—
19C22⋊#10587739—42no—
20C23×D137—21no—
21C23×D137D412yes—
22C2×D548—22no—
23C2×D548C2×C411yescharacteristic, complemented
24C2×D548C2311yescharacteristic
25C2×C22⋊D137—22no—
26C2×C22⋊D137—21no—
27C2×C22⋊D137—22no—
28C2×C22⋊D137D412yes—
29D4×D137—41no—
30D4×D137—41no—
31D4×D137—41no—
32D4×D137—41no—
33D4×D137—48no—
34C2×C4×D137—21no—
35C2×C4×D137C2×C411yescharacteristic
36C4×Dic137—21no—
37C4×Dic137C2×C411yescharacteristic, complemented
38C2×D4×C137—22no—
39C2×D4×C137C2×C411yescharacteristic
40C2×D4×C137C2×C411yescharacteristic, complemented
41C42×C137C2×C411yescharacteristic, complemented
42C2×#10587739—44no—
43C2×#10587739—84no—
44C22×D137—22no—
45C22×D137—22no—
46C22×D137—21no—
47C22×D137—44no—
48C22×D137—44no—
49C22×D137—41no—
50C22×D137—41no—
51C22×D137C22⋊C411yescharacteristic
52C22×D137C2×D412yes—
53D548—41no—
54D548—41no—
55D548—44no—
56C22⋊D137—42no—
57C22⋊D137—44no—
58C22⋊D137—44no—
59C22⋊D137—42no—
60C2×Dic137—21no—
61C2×Dic137C22⋊C411yescharacteristic
62C4×D137—24no—
63C4×D137—21no—
64C4×D137—21no—
65C23×C137—21no—
66C23×C137C22⋊C412yes—
67D4×C137—41no—
68D4×C137—41no—
69D4×C137—44no—
70C2×C4×C137—21no—
71C2×C4×C137C22×C411yescharacteristic
72C22×C137—22no—
73C22×C137—42no—
74C22×C137—42no—
75C22×C137—41no—
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—22no—
78C2×D137—22no—
79C2×D137—44no—
80C2×D137—42no—
81C2×D137—42no—
82C2×D137—41no—
83C2×D137—82no—
84C2×D137—82no—
85C2×D137—82no—
86C2×D137#9911yescharacteristic
87C2×D137#9911yescharacteristic
88#10587739—84no—
89C4×C137—22no—
90C4×C137—21no—
91Dic137—22no—
92Dic137—21no—
93C2×C137—21no—
94C2×C137—42no—
95C2×C137—81no—
96C2×C137C2×#9911yescharacteristic
97D137—21no—
98D137—42no—
99D137—81no—
100D137#35312yescomplemented
101C137C2×#35311yescharacteristic, complemented, Sylow
102C2×#353—1371nomaximal, Sylow
103C2×#127—1371no—
104C2×#99—1372no—
105#353—1372no—
106#353—1372no—
107C22×D4—2742no—
108C22×D4—1371no—
109C22×D4—1371no—
110#127—2742no—
111#127—1371no—
112#127—1371no—
113#127—1371no—
114#127#1058773911yescharacteristic, complemented
115C2×C22⋊C4—2742no—
116C2×C42—1371no—
117#99—2742no—
118#99—1374no—
119C24—2741no—
120C24—1372no—
121C2×D4—22no—
122C2×D4—2742no—
123C2×D4—2742no—
124C2×D4—2742no—
125C2×D4—2741no—
126C2×D4—5481no—
127C2×D4—5481no—
128C2×D4—5481no—
129C2×D4—5481no—
130C2×D4—5488no—
131C2×D4—1372no—
132C2×D4—1371no—
133C2×D4—1371no—
134C2×D4C2×#1058773911yescharacteristic
135C2×D4C2×#1058773911yescharacteristic, complemented
136C22×C4—2741no—
137C22×C4—5482no—
138C22×C4—1371no—
139C22⋊C4—2744no—
140C22⋊C4—5482no—
141C42—2741no—
142C42—1371no—
143C42C2×#1058773911yescharacteristic, complemented
144C23—21no—
145C23—2742no—
146C23—2742no—
147C23—2741no—
148C23—5484no—
149C23—5484no—
150C23—5481no—
151C23—5481no—
152C23—1371no—
153C23—1372no—
154C23C22⋊#1058773912yes—
155D4—41no—
156D4—41no—
157D4—44no—
158D4—5481no—
159D4—5481no—
160D4—5484no—
161D4—5482no—
162D4—5484no—
163D4—5484no—
164D4—5482no—
165C2×C4—21no—
166C2×C4—2744no—
167C2×C4—2741no—
168C2×C4—2741no—
169C2×C4—2741no—
170C2×C4—5484no—
171C2×C4—10964no—
172C2×C4—1371no—
173C2×C4C22×#1058773911yescharacteristic
174C22—22no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—2742no—
179C22—2742no—
180C22—5482no—
181C22—5482no—
182C22—5484no—
183C22—5481no—
184C22—10962no—
185C22—10962no—
186C22—10962no—
187C22—1371no—
188C22—1371no—
189C22C2×C22⋊#1058773911yescharacteristic
190C4—22no—
191C4—21no—
192C4—2742no—
193C4—2741no—
194C4—10964no—
195C2—21no—
196C2—42no—
197C2—81no—
198C2—2741no—
199C2—5482no—
200C2—10961no—
201C2—1372no—
202C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC2×C137C2×#99
prime radical#127×D137C2
cosocleC2×D548C23
hypercenter#127#10587739
hypocenterC137C2×#353
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. D137⋊#99C2semidirect
2. #22578308C2semidirect
3. #22578307C2semidirect
6. #17984147C22semidirect
9. C2×D4×D137C4semidirect
13. #17984084C4semidirect
15. C42×D137C4semidirect
16. #127×C137C4semidirect
23. C2×D548C2×C4semidirect
37. C4×Dic137C2×C4semidirect
40. C2×D4×C137C2×C4semidirect
41. C42×C137C2×C4semidirect
100. D137#353semidirect
101. C137C2×#353semidirect
114. #127#10587739semidirect
135. C2×D4C2×#10587739semidirect
143. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#992
#225783082
#225783072
#127×D1372
C2×#353137