Finite Group Databasebeta

← #31797909

Subgroups — #31797909

Order 17536.

312 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 71 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#22578516C211yesmaximal, characteristic, complemented
2#22578516C211yesmaximal, characteristic, complemented
3D4⋊#11764049C214yesmaximal, complemented
4C2×D4×#10587739C211yesmaximal, characteristic
5C4⋊#11764049—21no—
6C4⋊#11764049C2212yescomplemented
7C22⋊#11764049—21no—
8C22⋊#11764049—21no—
9C4×#11764049—21no—
10C4×#11764049C2212yescomplemented
11C23×#10587739—21no—
12D4×#10587739—21no—
13D4×#10587739—21no—
14D4×#10587739C412yes—
15D4×#10587739C2212yes—
16C2×C4⋊#10587739C411yescharacteristic
17C2×C22⋊#10587739—21no—
18C2×C4×#10587739C2211yescharacteristic
19C2×D4×D137C411yescharacteristic
20C2×#11764049—22no—
21C2×#11764049—22no—
22C22×#10587739—22no—
23C22×#10587739—22no—
24C22×#10587739—21no—
25C22×#10587739—44no—
26C22×#10587739D411yescharacteristic
27C22×#10587739D411yescharacteristic
28C4⋊#10587739—21no—
29C4⋊#10587739C2×C412yes—
30C22⋊#10587739—22no—
31C22⋊#10587739—22no—
32C4×#10587739C2×C411yescharacteristic
33C4×#10587739D411yescharacteristic
34C4×#10587739D411yescharacteristic
35C4×#10587739C2311yescharacteristic
36C23×D137—21no—
37C2×D548C811yescharacteristic, complemented
38C2×C22⋊D137—21no—
39D4×D137—21no—
40D4×D137—21no—
41D4×D137C812yescomplemented
42D4×D137C2×C412yes—
43C2×C4×D137C2×C411yescharacteristic
44C2×D4×C137C811yescharacteristic, complemented
45#11764049—42no—
46#11764049—42no—
47C2×#10587739—22no—
48C2×#10587739—22no—
49C2×#10587739—41no—
50C2×#10587739—44no—
51C2×#10587739—42no—
52C2×#10587739—44no—
53C2×#10587739C22⋊C411yescharacteristic
54C2×#10587739C22⋊C411yescharacteristic
55C2×#10587739C2×D411yescharacteristic
56C2×#10587739C2×D411yescharacteristic
57C22×D137—21no—
58C22×D137—22no—
59C22×D137—22no—
60C22×D137—42no—
61C22×D137—42no—
62C22×D137C22⋊C411yescharacteristic
63D548—21no—
64D548C2×C812yes—
65C22⋊D137—22no—
66C22⋊D137—22no—
67C2×Dic137M1611yescharacteristic
68C4×D137C22⋊C411yescharacteristic
69C4×D137C2×C811yescharacteristic
70C4×D137M1611yescharacteristic
71C4×D137C22×C411yescharacteristic
72C23×C137—21no—
73D4×C137—21no—
74D4×C137C2×C812yes—
75C2×C4×C137C2×C811yescharacteristic
76C22×C137—22no—
77C22×C137—42no—
78C22×C137C22⋊C811yescharacteristic
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—22no—
82C2×D137—44no—
83C2×D137—42no—
84C2×D137—42no—
85C2×D137—42no—
86C2×D137C22⋊C811yescharacteristic
87C2×D137C2×C22⋊C411yescharacteristic
88#10587739—21no—
89#10587739—21no—
90#10587739—21no—
91#10587739—42no—
92#10587739AGL(1,ℤ/8)12yes—
93C4×C137C22⋊C811yescharacteristic
94C4×C137C22×C811yescharacteristic
95Dic137C22⋊C811yescharacteristic
96Dic137C2×M1611yescharacteristic
97C2×C137—21no—
98C2×C137—42no—
99C2×C137C2×C22⋊C811yescharacteristic
100D137—21no—
101D137—42no—
102D137#41811yescharacteristic
103D137#42111yescharacteristic
104C137#147811yescharacteristic, complemented, Sylow
105#1478—1371nomaximal, Sylow
106C2×C4×D4—1371no—
107#431—1371no—
108#431—1371no—
109D4⋊C8—1374no—
110C22×D4—1371no—
111C23×C4—2741no—
112C4×D4—2741no—
113C4×D4—2741no—
114C4×D4—1372no—
115C4×D4—1372no—
116C2×C4⋊C4—1371no—
117C2×C22⋊C4—2741no—
118C2×C42—1371no—
119C4⋊C8—2741no—
120C4⋊C8—1372no—
121C22⋊C8—2741no—
122C22⋊C8—2741no—
123C4×C8—2741no—
124C4×C8—1372no—
125C24—2741no—
126C2×D4—2741no—
127C2×D4—2741no—
128C2×D4—2741no—
129C2×D4—1372no—
130C2×D4—1372no—
131C2×D4—1371no—
132C2×D4#1176404911yescharacteristic, complemented
133C22×C4—2741no—
134C22×C4—2742no—
135C22×C4—2742no—
136C22×C4—5484no—
137C22×C4—1371no—
138C22×C4—1371no—
139C22×C4—1371no—
140C2×C8—2742no—
141C2×C8—2742no—
142C4⋊C4—2741no—
143C4⋊C4—1372no—
144C22⋊C4—2742no—
145C22⋊C4—2742no—
146C42—1371no—
147C42—1371no—
148C42—1371no—
149C42—1371no—
150C23—21no—
151C23—2742no—
152C23—2742no—
153C23—2741no—
154C23—5482no—
155C23—5482no—
156C23—1371no—
157D4—21no—
158D4—2742no—
159D4—2742no—
160D4—2741no—
161D4—1372no—
162D4C2×#1176404912yes—
163C2×C4—2742no—
164C2×C4—2742no—
165C2×C4—5481no—
166C2×C4—5484no—
167C2×C4—5482no—
168C2×C4—5484no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4—1371no—
176C2×C4—1371no—
177C2×C4—1371no—
178C2×C4C2×#1176404911yescharacteristic
179C8—5482no—
180C8—5482no—
181C22—22no—
182C22—42no—
183C22—2741no—
184C22—2741no—
185C22—2742no—
186C22—5482no—
187C22—5484no—
188C22—5482no—
189C22—5482no—
190C22—1371no—
191C22—1371no—
192C22C22⋊#1176404911yescharacteristic
193C4—2741no—
194C4—2741no—
195C4—2741no—
196C4—5482no—
197C4—1372no—
198C4—1371no—
199C4—1371no—
200C4C22⋊#1176404911yescharacteristic
201C4C22×#1176404911yescharacteristic
202C2—21no—
203C2—42no—
204C2—2741no—
205C2—5482no—
206C2—1371no—
207C2—1371no—
208C2C2×C22⋊#1176404911yescharacteristic
subgroupG/H
centerC2C2×C22⋊#11764049
commutatorC4×C137C22×C8
FrattiniC4C22×#11764049
FittingC2×D4×C137C8
socleC2×C137C2×C22⋊C8
prime radicalC2×D4×D137C4
cosocleC4×#10587739C23
hypercenterC2×D4#11764049
hypocenterC137#1478
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #22578516C2semidirect
2. #22578516C2semidirect
3. D4⋊#11764049C2semidirect
6. C4⋊#11764049C22semidirect
10. C4×#11764049C22semidirect
37. C2×D548C8semidirect
41. D4×D137C8semidirect
44. C2×D4×C137C8semidirect
104. C137#1478semidirect
132. C2×D4#11764049semidirect

Maximal subgroups

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

subgroupindex
#225785162
D4⋊#117640492
C2×D4×#105877392
#1478137