Finite Group Databasebeta

← #31796950

Subgroups — #31796950

Order 17536.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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⋊#108C211yesmaximal, characteristic, complemented
2#22578395C212yesmaximal, complemented
3D137⋊#100C212yesmaximal, complemented
4#22578366C211yesmaximal, characteristic, complemented
5C2×D8×D137C211yesmaximal, characteristic
6D137⋊M16C2212yes—
7D137⋊M16C2211yescharacteristic
8#17984167C2211yescharacteristic
9#17984159—21no—
10#17984158—21no—
11#17984150—24no—
12#17984148—24no—
13#17984142—21no—
14C8×#10587739—21no—
15C2×D4×D137C2212yes—
16C2×D1096C411yescharacteristic, complemented
17C2×#14555495C412yescomplemented
18D8×D137—21no—
19D8×D137—21no—
20D8×D137—21no—
21D8×D137—21no—
22C2×C8×D137C2211yescharacteristic
23C2×D8×C137C411yescharacteristic, complemented
24#14555520—22no—
25#14555520—21no—
26#14555519—22no—
27#14555519D412yes—
28#14555518—21no—
29#14555518—22no—
30C4⋊#10587739D412yes—
31C22⋊#10587739—21no—
32C4×#10587739—21no—
33C23×D137—22no—
34C2×D548C2×C412yes—
35C2×C22⋊D137—22no—
36D4×D137—22no—
37D4×D137—22no—
38D4×D137—22no—
39D4×D137—22no—
40C2×C4×D137C2311yescharacteristic
41D1096—21no—
42D1096—21no—
43#14555495—22no—
44#14555495—22no—
45C2×#11764040C2×C411yescharacteristic
46C8×D137D411yescharacteristic
47C8×D137D411yescharacteristic
48C8×D137D411yescharacteristic
49C8×D137D411yescharacteristic
50C2×D4×C137C2×C412yes—
51D8×C137—21no—
52D8×C137—21no—
53C2×C8×C137C2×C411yescharacteristic
54C2×#10587739—22no—
55#11764047—22no—
56#11764047—42no—
57C22×D137—22no—
58C22×D137—42no—
59C22×D137—42no—
60C22×D137—84no—
61C22×D137#4111yescharacteristic
62D548—22no—
63D548—22no—
64C22⋊D137—42no—
65C22⋊D137—42no—
66C2×Dic137C2×D411yescharacteristic
67C4×D137C2×D411yescharacteristic
68C4×D137C2×D411yescharacteristic
69C4×D137C2×D411yescharacteristic
70C4×D137#4111yescharacteristic
71#11764040C22⋊C411yescharacteristic
72#11764040C22⋊C411yescharacteristic
73C23×C137—22no—
74D4×C137—22no—
75D4×C137—22no—
76C2×C4×C137C22×C411yescharacteristic
77C8×C137C22⋊C411yescharacteristic
78C8×C137C22⋊C411yescharacteristic
79C22×C137—42no—
80C22×C137—82no—
81C22×C137C4×D411yescharacteristic
82C2×D137—22no—
83C2×D137—42no—
84C2×D137—84no—
85C2×D137—84no—
86C2×D137—82no—
87C2×D137#12111yescharacteristic
88C2×D137#12411yescharacteristic
89#10587739—42no—
90C4×C137C2×C22⋊C411yescharacteristic
91C4×C137C4×D411yescharacteristic
92Dic137#12111yescharacteristic
93Dic137#12711yescharacteristic
94C2×C137—21no—
95C2×C137—82no—
96C2×C137#39011yescharacteristic
97D137—21no—
98D137—82no—
99D137#47212yes—
100C137#197811yescharacteristic, complemented, Sylow
101#1978—1371nomaximal, Sylow
102C22×D8—1371no—
103C2×#108—1371no—
104#418—1372no—
105C2×#100—1372no—
106#405—1371no—
107C22×D4—1372no—
108C2×D8—2741no—
109C2×D8—2741no—
110C2×D8—2741no—
111C2×D8—2741no—
112C2×D8—1372no—
113C2×D8—1371no—
114C2×D8#1058773911yescharacteristic, complemented
115C2×M16—1372no—
116C2×M16—1371no—
117C22×C8—1371no—
118#117—1371no—
119#108—2741no—
120#108—2741no—
121#102—2744no—
122#100—2744no—
123#97—2741no—
124C4×C8—2741no—
125C24—2742no—
126C2×D4—2742no—
127C2×D4—2742no—
128C2×D4—2742no—
129C2×D4—2742no—
130C2×D4—2742no—
131C2×D4—1372no—
132C2×D4C2×#1058773912yes—
133C22×C4—1371no—
134D8—21no—
135D8—21no—
136D8—2742no—
137D8—2742no—
138D8—2741no—
139D8—2741no—
140M16—2741no—
141M16—2742no—
142M16—2742no—
143M16—1372no—
144C2×C8—2741no—
145C2×C8—2742no—
146C2×C8—1371no—
147C2×C8—1371no—
148C2×C8—1371no—
149C2×C8—1371no—
150C2×C8—1371no—
151C2×C8C2×#1058773911yescharacteristic
152C4⋊C4—1372no—
153C22⋊C4—2741no—
154C42—2741no—
155C23—22no—
156C23—2742no—
157C23—5482no—
158C23—5482no—
159C23—10964no—
160C23—1371no—
161D4—22no—
162D4—22no—
163D4—2742no—
164D4—2742no—
165D4—5482no—
166D4—5482no—
167C2×C4—2742no—
168C2×C4—1371no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4C22×#1058773911yescharacteristic
174C8—2742no—
175C8—5482no—
176C8—1371no—
177C8—1371no—
178C8C22⋊#1058773911yescharacteristic
179C8C22⋊#1058773911yescharacteristic
180C22—42no—
181C22—82no—
182C22—2742no—
183C22—5482no—
184C22—10962no—
185C22—10964no—
186C22—10964no—
187C22—1371no—
188C22—1371no—
189C22D4×#1058773911yescharacteristic
190C4—5482no—
191C4—1371no—
192C4—1371no—
193C4C2×C22⋊#1058773911yescharacteristic
194C4D4×#1058773911yescharacteristic
195C2—21no—
196C2—82no—
197C2—2741no—
198C2—10962no—
199C2—1372no—
200C2#2257835511yescharacteristic
subgroupG/H
centerC2#22578355
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×D8×C137C4
socleC2×C137#390
prime radicalC2×D8×D137C2
cosocleC2×C4×D137C23
hypercenterC2×D8#10587739
hypocenterC137#1978
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. D137⋊#108C2semidirect
2. #22578395C2semidirect
3. D137⋊#100C2semidirect
4. #22578366C2semidirect
16. C2×D1096C4semidirect
17. C2×#14555495C4semidirect
23. C2×D8×C137C4semidirect
100. C137#1978semidirect
114. C2×D8#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#1082
#225783952
D137⋊#1002
#225783662
C2×D8×D1372
#1978137