Finite Group Databasebeta

← #31797051

Subgroups — #31797051

Order 17536.

234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 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#22578389C211yesmaximal, characteristic
2#22578382C211yesmaximal, characteristic
3#22578305C214yesmaximal, complemented
4#22577977C211yesmaximal, characteristic
5D137⋊M16—21no—
6#17984167—21no—
7#17984149—21no—
8#17984148C2212yes—
9#17984147—21no—
10#17984146C2212yes—
11C2×#14555510C2211yescharacteristic
12#17984086—21no—
13#17984077—21no—
14#17984071C412yescomplemented
15#17984071C2212yes—
16#17984064—21no—
17#17984064—21no—
18#17984047C411yescharacteristic, complemented
19C22×Q8×C137C411yescharacteristic, complemented
20#14555520—41no—
21#14555519—22no—
22#14555518—41no—
23C4⋊#10587739—41no—
24C22⋊#10587739—22no—
25C4×#10587739—41no—
26#14555510—41no—
27#14555510—41no—
28C2×Dic274D411yescharacteristic
29C2×D548D411yescharacteristic
30C2×C22⋊D137C2×C411yescharacteristic
31C2×C22⋊D137C2311yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C4⋊Dic137—21no—
35#14555489—22no—
36#14555489—22no—
37C22⋊Dic137C2×C412yes—
38#14555487—22no—
39#14555487—22no—
40C4×Dic137—21no—
41C2×Q8×C137—21no—
42C2×Q8×C137—21no—
43C2×Q8×C137—41no—
44C2×Q8×C137—41no—
45C22×C4×C137C2×C412yes—
46C22×C4×C137C2×C411yescharacteristic
47C2×#10587739—42no—
48#11764047—42no—
49C22×D137C22⋊C411yescharacteristic
50C22×D137C2×D411yescharacteristic
51Dic274—41no—
52D548—41no—
53C22⋊D137—41no—
54C22⋊D137—41no—
55C2×Dic137—22no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C2×D411yescharacteristic
58C4×D137—21no—
59C4×D137—21no—
60C4×D137—21no—
61C4×D137—21no—
62C23×C137C22×C411yescharacteristic
63Q8×C137—42no—
64Q8×C137—41no—
65Q8×C137—41no—
66C2×C4×C137—21no—
67C2×C4×C137—21no—
68C2×C4×C137—22no—
69C2×C4×C137—42no—
70C2×C4×C137C22⋊C411yescharacteristic
71C2×C4×C137C22⋊C411yescharacteristic
72C22×C137—21no—
73C22×C137—41no—
74C22×C137C2×C22⋊C411yescharacteristic
75C2×D137—21no—
76C2×D137—21no—
77C2×D137—41no—
78C2×D137—41no—
79#10587739—82no—
80C4×C137—21no—
81C4×C137—42no—
82C4×C137#9911yescharacteristic
83C4×C137#9911yescharacteristic
84Dic137—21no—
85Dic137—21no—
86Dic137—42no—
87C2×C137—21no—
88C2×C137—41no—
89C2×C137C2×#9911yescharacteristic
90D137—41no—
91D137—41no—
92C137#212311yescharacteristic, complemented, Sylow
93#2123—1371nomaximal, Sylow
94#536—1371no—
95#413—1371no—
96#410—1371no—
97#352—1374no—
98C2×#41—1371no—
99C22×Q8#1058773911yescharacteristic, complemented
100C2×M16—2741no—
101C4⋊Q8—2741no—
102#124—2741no—
103#123—1372no—
104#123—1372no—
105C22⋊Q8—2741no—
106C22⋊Q8—2741no—
107#117—2741no—
108#117—1371no—
109#101—2741no—
110#100—1372no—
111#99—2741no—
112#99—1372no—
113#41—5481no—
114#41—5481no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—1371no—
120C2×D4—1371no—
121C2×D4—1371no—
122C2×D4—1371no—
123C22×C4—1371no—
124C22×C4—1371no—
125C22×C4C2×#1058773912yes—
126C22×C4C2×#1058773911yescharacteristic
127M16—2742no—
128M16—5481no—
129C2×C8—5481no—
130C4⋊C4—2742no—
131C4⋊C4—2742no—
132C4⋊C4—2741no—
133C4⋊C4—5481no—
134C22⋊C4—2742no—
135C22⋊C4—2742no—
136C22⋊C4—2742no—
137C22⋊C4—1372no—
138C42—2741no—
139C42—5481no—
140C23—1371no—
141C23—1371no—
142C23C22×#1058773911yescharacteristic
143Q8—42no—
144Q8—41no—
145Q8—41no—
146Q8—5481no—
147D4—5481no—
148D4—5481no—
149D4—5481no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2742no—
159C2×C4—5482no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4C22⋊#1058773911yescharacteristic
163C2×C4C22⋊#1058773911yescharacteristic
164C8—5482no—
165C22—21no—
166C22—41no—
167C22—2741no—
168C22—2741no—
169C22—5481no—
170C22—5481no—
171C22C2×C22⋊#1058773911yescharacteristic
172C4—21no—
173C4—42no—
174C4—2741no—
175C4—2741no—
176C4—5482no—
177C4—10962no—
178C4#1798414611yescharacteristic
179C4#1798414611yescharacteristic
180C2—21no—
181C2—41no—
182C2—5481no—
183C2—5481no—
184C2C2×#1798414611yescharacteristic
subgroupG/H
centerC2C2×#17984146
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC22×Q8×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555510C22
cosocleC2×C22⋊D137C23
hypercenterC22×Q8#10587739
hypocenterC137#2123
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #22578305C2semidirect
14. #17984071C4semidirect
18. #17984047C4semidirect
19. C22×Q8×C137C4semidirect
92. C137#2123semidirect
99. C22×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783892
#225783822
#225783052
#225779772
#2123137