Finite Group Databasebeta

← #31797901

Subgroups — #31797901

Order 17536.

264 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 69 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#22578516C212yesmaximal, complemented
2#22578489C212yesmaximal, complemented
3#22578488C212yesmaximal
4C2×#17984167C211yesmaximal, characteristic
5C4⋊#11764049—22no—
6C22⋊#11764049C2214yes—
7C4×#11764049—22no—
8#17984167—22no—
9#17984167—22no—
10C2×C4⋊#10587739C411yescharacteristic
11C2×C4⋊#10587739C2211yescharacteristic
12C2×C22⋊#10587739C411yescharacteristic
13C2×C22⋊#10587739C2211yescharacteristic
14C2×C4×#10587739C411yescharacteristic
15C2×C4×#10587739C2211yescharacteristic
16C22×C4×D137C411yescharacteristic
17C2×#11764049—24no—
18C22×#10587739C2×C411yescharacteristic
19C22×#10587739C2×C411yescharacteristic
20C22×#10587739C2×C411yescharacteristic
21C22×#10587739C2311yescharacteristic
22C4⋊#10587739—22no—
23C4⋊#10587739—22no—
24C22⋊#10587739—22no—
25C22⋊#10587739—22no—
26C4×#10587739—21no—
27C4×#10587739—21no—
28C4×#10587739D412yes—
29C4×#10587739D412yes—
30C23×D137C2×C411yescharacteristic
31C22×Dic137C811yescharacteristic, complemented
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—21no—
35C2×C4×D137—21no—
36C2×C4×D137C811yescharacteristic, complemented
37C2×C4×D137C811yescharacteristic, complemented
38C2×C4×D137C2×C411yescharacteristic
39C2×C4×D137C2×C411yescharacteristic
40C22×C4×C137C811yescharacteristic, complemented
41#11764049—44no—
42C2×#10587739—21no—
43C2×#10587739—21no—
44C2×#10587739—41no—
45C2×#10587739—41no—
46C2×#10587739—41no—
47C2×#10587739—41no—
48C2×#10587739C22⋊C412yes—
49C2×#10587739C2×D412yes—
50C22×D137—21no—
51C22×D137—21no—
52C22×D137—41no—
53C22×D137—41no—
54C22×D137C2×C811yescharacteristic
55C22×D137C22×C411yescharacteristic
56C2×Dic137—21no—
57C2×Dic137—21no—
58C2×Dic137C2×C811yescharacteristic
59C2×Dic137C2×C811yescharacteristic
60C4×D137—21no—
61C4×D137—21no—
62C4×D137—21no—
63C4×D137—21no—
64C4×D137—21no—
65C4×D137—21no—
66C4×D137C22⋊C411yescharacteristic
67C4×D137C22⋊C411yescharacteristic
68C4×D137M1611yescharacteristic
69C4×D137M1611yescharacteristic
70C23×C137C2×C811yescharacteristic
71C2×C4×C137—21no—
72C2×C4×C137—21no—
73C2×C4×C137C2×C811yescharacteristic
74C2×C4×C137C2×C811yescharacteristic
75C22×C137—21no—
76C22×C137—41no—
77C22×C137C22×C811yescharacteristic
78C2×D137—21no—
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—42no—
82C2×D137—41no—
83C2×D137—41no—
84C2×D137—41no—
85C2×D137C2×C22⋊C411yescharacteristic
86C2×D137C2×M1611yescharacteristic
87#10587739—22no—
88#10587739—22no—
89#10587739—41no—
90#10587739—41no—
91C4×C137—21no—
92C4×C137C22⋊C811yescharacteristic
93C4×C137C22⋊C811yescharacteristic
94Dic137—21no—
95Dic137C22⋊C811yescharacteristic
96Dic137C22⋊C811yescharacteristic
97C2×C137—21no—
98C2×C137—41no—
99C2×C137C2×C22⋊C811yescharacteristic
100D137—21no—
101D137—41no—
102D137#41011yescharacteristic
103D137#41311yescharacteristic
104C137#145811yescharacteristic, complemented, Sylow
105#1458—1371nomaximal, Sylow
106C2×#117—1371no—
107#431—1372no—
108#324—1372no—
109#323—1372no—
110C23×C4—1371no—
111#117—2742no—
112#117—2742no—
113C2×C4⋊C4—1371no—
114C2×C4⋊C4—1371no—
115C2×C22⋊C4—1371no—
116C2×C22⋊C4—1371no—
117C2×C42—1371no—
118C2×C42—1371no—
119C4⋊C8—2742no—
120C22⋊C8—1374no—
121C4×C8—2742no—
122C24—1371no—
123C22×C4—2741no—
124C22×C4—2741no—
125C22×C4—2741no—
126C22×C4—2741no—
127C22×C4—1371no—
128C22×C4—1371no—
129C22×C4—1371no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4—1371no—
134C22×C4—1371no—
135C22×C4—1371no—
136C22×C4#1176404911yescharacteristic, complemented
137C2×C8—2744no—
138C4⋊C4—2742no—
139C4⋊C4—2742no—
140C22⋊C4—2742no—
141C22⋊C4—2742no—
142C42—2741no—
143C42—2741no—
144C42—1372no—
145C42—1372no—
146C23—2741no—
147C23—2741no—
148C23—5481no—
149C23—5481no—
150C23—1371no—
151C23—1371no—
152C23C2×#1176404911yescharacteristic
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—2741no—
162C2×C4—2741no—
163C2×C4—2741no—
164C2×C4—2741no—
165C2×C4—5481no—
166C2×C4—5481no—
167C2×C4—5481no—
168C2×C4—5481no—
169C2×C4—1372no—
170C2×C4—1372no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4C2×#1176404911yescharacteristic
178C2×C4C2×#1176404911yescharacteristic
179C8—5484no—
180C22—21no—
181C22—41no—
182C22—2741no—
183C22—2741no—
184C22—2741no—
185C22—5481no—
186C22—5482no—
187C22—5481no—
188C22—5481no—
189C22—1371no—
190C22—1371no—
191C22C22×#1176404911yescharacteristic
192C4—21no—
193C4—2742no—
194C4—2742no—
195C4—2741no—
196C4—5481no—
197C4—5481no—
198C4—1371no—
199C4—1371no—
200C4C22⋊#1176404911yescharacteristic
201C4C22⋊#1176404911yescharacteristic
202C2—21no—
203C2—41no—
204C2—2741no—
205C2—5481no—
206C2—1371no—
207C2—1371no—
208C2C2×C22⋊#1176404911yescharacteristic
subgroupG/H
centerC4C22⋊#11764049
commutatorC22×C137C22×C8
FrattiniC22C22×#11764049
FittingC22×C4×C137C8
socleC2×C137C2×C22⋊C8
prime radicalC23×D137C2×C4
cosocleC22×#10587739C23
hypercenterC22×C4#11764049
hypocenterC137#1458
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #22578516C2semidirect
2. #22578489C2semidirect
31. C22×Dic137C8semidirect
36. C2×C4×D137C8semidirect
37. C2×C4×D137C8semidirect
40. C22×C4×C137C8semidirect
104. C137#1458semidirect
136. C22×C4#11764049semidirect

Maximal subgroups

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

subgroupindex
#225785162
#225784892
#225784882
C2×#179841672
#1458137