Finite Group Databasebeta

← #31796561

Subgroups — #31796561

Order 17536.

324 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#17984151C212yesmaximal
2#22578362C211yesmaximal, characteristic
3C4×C4⋊#10587739C211yesmaximal, characteristic
4#22578276C212yesmaximal
5C4×Q8×D137C211yesmaximal, characteristic
6C2×#14555518C2212yes—
7C2×C4⋊#10587739C2212yes—
8C2×C4×#10587739—21no—
9#17984151C418yescomplemented
10#17984143—21no—
11#17984143—21no—
12(C2×#10587739)⋊C4—21no—
13C2×Q8×D137C2211yescharacteristic
14Q8×Dic137C411yescharacteristic, complemented
15C4×Dic274—21no—
16C4×Dic274C411yescharacteristic, complemented
17#17984053—21no—
18#17984053C411yescharacteristic, complemented
19C4⋊C4×D137—21no—
20C4⋊C4×D137C2211yescharacteristic
21C42×D137—21no—
22C42×D137C2211yescharacteristic
23C4×Q8×C137C411yescharacteristic, complemented
24C22×#10587739—22no—
25#14555518—22no—
26#14555518C2×C414yes—
27C2×#11764047—22no—
28C4⋊#10587739—22no—
29C4⋊#10587739C2×C414yes—
30C4×#10587739—42no—
31C2×Dic274—21no—
32C2×Dic274C2×C411yescharacteristic
33Q8×D137—22no—
34Q8×D137C2×C414yes—
35C2×C4×D137—21no—
36C2×C4×D137—21no—
37C2×C4×D137D411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C2×C4×D137C2311yescharacteristic
40C4⋊Dic137—21no—
41C4⋊Dic137C2×C411yescharacteristic
42#14555489—21no—
43#14555489—21no—
44#14555489—21no—
45C4×Dic137—21no—
46C4×Dic137—21no—
47C4×Dic137—21no—
48C4×Dic137C2×C411yescharacteristic
49C4×Dic137D411yescharacteristic
50C4×Dic137D411yescharacteristic
51C2×Q8×C137C2×C411yescharacteristic
52C4⋊C4×C137—21no—
53C4⋊C4×C137C2×C411yescharacteristic
54C42×C137—21no—
55C42×C137C2×C411yescharacteristic
56C2×#10587739—24no—
57C2×#10587739—42no—
58C2×#10587739—42no—
59#11764047—24no—
60C22×D137C2×D411yescharacteristic
61Dic274—21no—
62Dic274—24no—
63Dic274C4212yescomplemented
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C2×D411yescharacteristic
69C4×D137—24no—
70C4×D137—21no—
71C4×D137—21no—
72C4×D137—22no—
73C4×D137—41no—
74C4×D137—41no—
75C4×D137C22×C412yes—
76C4×D137#4112yes—
77Q8×C137—21no—
78Q8×C137C4212yescomplemented
79C2×C4×C137—21no—
80C2×C4×C137—21no—
81C2×C4×C137C22⋊C411yescharacteristic
82C2×C4×C137C22⋊C411yescharacteristic
83C2×C4×C137C22×C411yescharacteristic
84C22×C137C2×C22⋊C411yescharacteristic
85C2×D137C4×D412yes—
86C2×D137AGL(1,ℤ/8)11yescharacteristic
87C2×D137#13712yes—
88C2×D137#13711yescharacteristic
89#10587739—44no—
90C4×C137—22no—
91C4×C137—21no—
92C4×C137—21no—
93C4×C137—41no—
94C4×C137C2×C4211yescharacteristic
95C4×C137#11711yescharacteristic
96Dic137—22no—
97Dic137—21no—
98Dic137—21no—
99Dic137—41no—
100Dic137C4×D412yes—
101C2×C137C4×C22⋊C411yescharacteristic
102C2×C137#41711yescharacteristic
103C2×C137#41911yescharacteristic
104D137#44012yes—
105D137#44112yes—
106C137#176311yescharacteristic, complemented, Sylow
107#1763—1371nomaximal, Sylow
108C2×C4×Q8—1371no—
109C2×#103—1372no—
110C2×#97—1371no—
111C4×C4⋊C4—1371no—
112#340—1372no—
113C22×Q8—1371no—
114C22×C8—1372no—
115C4×Q8—2741no—
116C4×Q8—2741no—
117C4×Q8—1371no—
118C4×Q8—1371no—
119C4×Q8—1371no—
120C4×Q8#1058773911yescharacteristic, complemented
121C2×C4⋊C4—2741no—
122C2×C4⋊C4—1372no—
123C2×C4⋊C4—1371no—
124C2×C42—2741no—
125C2×C42—2741no—
126C2×C42—1371no—
127#103—1378no—
128#97—2741no—
129#97—2741no—
130#95—2741no—
131C2×Q8—2741no—
132C2×Q8—2742no—
133C2×Q8—1374no—
134C2×Q8—1371no—
135C2×Q8C2×#1058773911yescharacteristic
136C22×C4—2741no—
137C22×C4—2741no—
138C22×C4—2742no—
139C22×C4—1371no—
140C22×C4—1371no—
141C22×C4—1371no—
142C2×C8—2742no—
143C2×C8—2742no—
144C2×C8—1374no—
145C4⋊C4—21no—
146C4⋊C4—2741no—
147C4⋊C4—2741no—
148C4⋊C4—2741no—
149C4⋊C4—2742no—
150C4⋊C4—2741no—
151C4⋊C4—1374no—
152C4⋊C4—1371no—
153C4⋊C4C2×#1058773911yescharacteristic
154C42—21no—
155C42—2741no—
156C42—2741no—
157C42—2741no—
158C42—5482no—
159C42—1371no—
160C42—1371no—
161C42—1371no—
162C42C2×#1058773911yescharacteristic
163C23—1371no—
164Q8—21no—
165Q8—2744no—
166Q8—2741no—
167Q8—1372no—
168Q8C4×#1058773912yescomplemented
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—2741no—
172C2×C4—2744no—
173C2×C4—2741no—
174C2×C4—2742no—
175C2×C4—2741no—
176C2×C4—2744no—
177C2×C4—2741no—
178C2×C4—5481no—
179C2×C4—5481no—
180C2×C4—5482no—
181C2×C4—5482no—
182C2×C4—1371no—
183C2×C4—1371no—
184C2×C4—1371no—
185C2×C4—1372no—
186C2×C4—1372no—
187C2×C4C22⋊#1058773911yescharacteristic
188C2×C4C22⋊#1058773911yescharacteristic
189C2×C4C22×#1058773911yescharacteristic
190C8—2744no—
191C22—1372no—
192C22—1371no—
193C22—1371no—
194C22—1372no—
195C22C2×C22⋊#1058773911yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—2742no—
201C4—2741no—
202C4—2741no—
203C4—5484no—
204C4—5481no—
205C4—1372no—
206C4C2×C4×#1058773911yescharacteristic
207C4#1798416711yescharacteristic
208C2—1372no—
209C2—1372no—
210C2C4×C22⋊#1058773911yescharacteristic
211C2#2257839411yescharacteristic
212C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC4×C137C2×C42
FrattiniC2×C4C22×#10587739
FittingC4×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4×Q8#10587739
hypocenterC137#1763
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
9. #17984151C4semidirect
14. Q8×Dic137C4semidirect
16. C4×Dic274C4semidirect
18. #17984053C4semidirect
23. C4×Q8×C137C4semidirect
63. Dic274C42semidirect
78. Q8×C137C42semidirect
106. C137#1763semidirect
120. C4×Q8#10587739semidirect
168. Q8C4×#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841512
#225783622
C4×C4⋊#105877392
#225782762
C4×Q8×D1372
#1763137