Finite Group Databasebeta

← #31796132

Subgroups — #31796132

Order 17536.

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

normal subgroups only

This group has 252 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
1#22578425C211yesmaximal, characteristic
2C4×#14555520C211yesmaximal, characteristic
3#22578253C212yesmaximal, complemented
4Q8⋊#11764047C212yesmaximal, complemented
5C4×Q8×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7C4⋊#11764047C2212yes—
8#17984145—21no—
9#17984143—21no—
10#17984143—21no—
11C4×#11764047C2212yes—
12C2×Q8×D137C411yescharacteristic
13Q8×Dic137C2211yescharacteristic
14C4×Dic274—21no—
15C4×Dic274C411yescharacteristic
16#17984053—21no—
17#17984053C2211yescharacteristic
18C4⋊C4×D137—21no—
19C4⋊C4×D137C411yescharacteristic
20C42×D137—21no—
21C42×D137C2211yescharacteristic
22C4×Q8×C137C411yescharacteristic
23#14555520—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×Dic274—21no—
27C2×Dic274C2×C411yescharacteristic
28Q8×D137—22no—
29Q8×D137—22no—
30C2×C4×D137—21no—
31C2×C4×D137—21no—
32C2×C4×D137C2×C411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38#14555489—21no—
39#14555489—21no—
40C4×Dic137—21no—
41C4×Dic137—21no—
42C4×Dic137—21no—
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137C2311yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C42×C137—21no—
50C42×C137C2×C411yescharacteristic
51#11764047—24no—
52#11764047—42no—
53C22×D137C22⋊C411yescharacteristic
54Dic274—24no—
55Dic274—21no—
56Dic274M1612yescomplemented
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C4×D137—24no—
63C4×D137—22no—
64C4×D137—21no—
65C4×D137—21no—
66C4×D137—22no—
67C4×D137—42no—
68Q8×C137—21no—
69Q8×C137M1612yescomplemented
70C2×C4×C137—21no—
71C2×C4×C137—21no—
72C2×C4×C137C22⋊C411yescharacteristic
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22×C411yescharacteristic
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—21no—
77C2×D137—21no—
78C2×D137#10412yes—
79C4×C137—22no—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—41no—
83C4×C137C2×M1611yescharacteristic
84C4×C137C2×M1611yescharacteristic
85Dic137—22no—
86Dic137—21no—
87Dic137—21no—
88Dic137—41no—
89Dic137#13712yes—
90C2×C137#40711yescharacteristic
91C2×C137#41911yescharacteristic
92C2×C137C2×#10411yescharacteristic
93D137—22no—
94C137#149111yescharacteristic, complemented, Sylow
95#1491—1371nomaximal, Sylow
96C2×C4×Q8—1371no—
97#432—1371no—
98C4×M16—1371no—
99Q8⋊C8—1372no—
100Q8⋊C8—1372no—
101C22×Q8—1371no—
102C2×M16—2741no—
103C4×Q8—2741no—
104C4×Q8—2741no—
105C4×Q8—1371no—
106C4×Q8—1371no—
107C4×Q8—1371no—
108C4×Q8#1058773911yescharacteristic
109C2×C4⋊C4—2741no—
110C2×C4⋊C4—1371no—
111C2×C42—2741no—
112C2×C42—1371no—
113C4⋊C8—1372no—
114C22⋊C8—2741no—
115#97—2741no—
116#97—2741no—
117C4×C8—1372no—
118C2×Q8—2741no—
119C2×Q8—2742no—
120C2×Q8—2742no—
121C2×Q8—1371no—
122C2×Q8C2×#1058773911yescharacteristic
123C22×C4—2741no—
124C22×C4—2741no—
125C22×C4—1371no—
126C22×C4—1371no—
127C22×C4—1371no—
128M16—2744no—
129C2×C8—2742no—
130C2×C8—2742no—
131C4⋊C4—21no—
132C4⋊C4—2741no—
133C4⋊C4—2741no—
134C4⋊C4—2741no—
135C4⋊C4—2741no—
136C4⋊C4—1371no—
137C4⋊C4C2×#1058773911yescharacteristic
138C42—21no—
139C42—2741no—
140C42—2741no—
141C42—2741no—
142C42—1371no—
143C42—1371no—
144C42—1371no—
145C42C2×#1058773911yescharacteristic
146C23—1371no—
147Q8—21no—
148Q8—2744no—
149Q8—2741no—
150Q8—1372no—
151Q8#1455552012yescomplemented
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—2744no—
155C2×C4—2742no—
156C2×C4—2742no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—5482no—
162C2×C4—1371no—
163C2×C4—1371no—
164C2×C4—1371no—
165C2×C4C22⋊#1058773911yescharacteristic
166C2×C4C22⋊#1058773911yescharacteristic
167C2×C4C22×#1058773911yescharacteristic
168C8—2744no—
169C8—5482no—
170C22—2741no—
171C22—2741no—
172C22—1372no—
173C22C2×C22⋊#1058773911yescharacteristic
174C4—22no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—2742no—
179C4—2741no—
180C4—2741no—
181C4—5481no—
182C4—1372no—
183C4D137⋊M1611yescharacteristic
184C4C2×#1455552011yescharacteristic
185C2—2742no—
186C2#2257837411yescharacteristic
187C2#2257839611yescharacteristic
188C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC4×Dic137C23
hypercenterC4×Q8#10587739
hypocenterC137#1491
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #22578253C2semidirect
4. Q8⋊#11764047C2semidirect
56. Dic274M16semidirect
69. Q8×C137M16semidirect
94. C137#1491semidirect
151. Q8#14555520semidirect

Maximal subgroups

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

subgroupindex
#225784252
C4×#145555202
#225782532
Q8⋊#117640472
C4×Q8×D1372
#1491137