Finite Group Databasebeta

← #31796267

Subgroups — #31796267

Order 17536.

218 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578424C211yesmaximal, characteristic, complemented
2C2×#17984143C211yesmaximal, characteristic, complemented
3#22578260C212yesmaximal, complemented
4#22578259C212yesmaximal, complemented
5#22577939C211yesmaximal, characteristic
6C22×#11764047—21no—
7C4⋊#11764047—21no—
8C22⋊#11764047—21no—
9#17984143—21no—
10#17984143C2212yes—
11#17984143C2212yes—
12#17984083—21no—
13#17984079C2211yescharacteristic
14#17984077C2211yescharacteristic
15#17984072—21no—
16#17984062C411yescharacteristic
17Q8×Dic137C411yescharacteristic
18#17984050C411yescharacteristic
19#17984048—21no—
20#17984046—21no—
21C2×C4×Dic137C2211yescharacteristic
22C22⋊Q8×C137C411yescharacteristic
23C2×#11764047—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×D548C2×C411yescharacteristic
27C2×C22⋊D137—21no—
28C22×Dic137D411yescharacteristic
29C22×Dic137D411yescharacteristic
30C2×C4×D137—21no—
31C4⋊Dic137—21no—
32C4⋊Dic137—21no—
33C4⋊Dic137C2×C411yescharacteristic
34#14555489—21no—
35#14555487—21no—
36#14555487—21no—
37#14555487—21no—
38#14555487—21no—
39C4×Dic137—21no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C2×Q8×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C4⋊C4×C137—21no—
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49#11764047—24no—
50#11764047—42no—
51C22×D137—21no—
52D548—41no—
53C22⋊D137—42no—
54C2×Dic137—22no—
55C2×Dic137—21no—
56C2×Dic137—41no—
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C4×D137—42no—
62C23×C137C22⋊C411yescharacteristic
63Q8×C137—41no—
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137—41no—
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C22×C411yescharacteristic
69C22×C137—21no—
70C22×C137—21no—
71C22×C137#10412yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C2×D137—41no—
74C2×D137—41no—
75C2×D137—41no—
76C4×C137—21no—
77C4×C137—21no—
78C4×C137—41no—
79C4×C137—41no—
80Dic137—21no—
81Dic137—21no—
82Dic137—21no—
83Dic137—21no—
84Dic137—41no—
85C2×C137—22no—
86C2×C137#41311yescharacteristic
87C2×C137C2×#10411yescharacteristic
88C2×C137#42111yescharacteristic
89D137—81no—
90C137#153211yescharacteristic, complemented, Sylow
91#1532—1371nomaximal, Sylow
92#529—1371no—
93#432—1371no—
94C2×#97—1371no—
95#330—1372no—
96#329—1372no—
97C22×C8—2741no—
98#126—2741no—
99#125—1371no—
100#124—1371no—
101#123—2741no—
102C22⋊Q8#1058773911yescharacteristic
103#121—1371no—
104C4×Q8—1371no—
105C4×D4—2741no—
106C4×D4—1371no—
107#117—2741no—
108C2×C42—1371no—
109C4⋊C8—2741no—
110C22⋊C8—2741no—
111#97—2741no—
112#97—1372no—
113#97—1372no—
114C2×Q8C2×#1058773911yescharacteristic
115C2×D4—2741no—
116C2×D4—1371no—
117C22×C4—2741no—
118C22×C4—1371no—
119C22×C4—1371no—
120C22×C4C2×#1058773911yescharacteristic
121C2×C8—2744no—
122C2×C8—2742no—
123C2×C8—2742no—
124C4⋊C4—21no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—1371no—
129C4⋊C4C2×#1058773911yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—2741no—
132C22⋊C4—2741no—
133C22⋊C4—2741no—
134C22⋊C4—2741no—
135C42—2741no—
136C42—1371no—
137C42—1371no—
138C42—1371no—
139C42—1371no—
140C23—2741no—
141C23C22⋊#1058773911yescharacteristic
142Q8—41no—
143D4—5481no—
144D4—5482no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—2742no—
149C2×C4—2741no—
150C2×C4—5482no—
151C2×C4—5481no—
152C2×C4—1371no—
153C2×C4—1371no—
154C2×C4—1371no—
155C2×C4—1371no—
156C2×C4C22⋊#1058773911yescharacteristic
157C2×C4C22×#1058773911yescharacteristic
158C8—2744no—
159C8—5482no—
160C22—21no—
161C22—21no—
162C22—5481no—
163C22—5481no—
164C22—5481no—
165C22#1798415312yes—
166C22C2×C22⋊#1058773911yescharacteristic
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—2741no—
172C4—2741no—
173C4—2741no—
174C4—2741no—
175C4—5481no—
176C2—22no—
177C2—10961no—
178C2#2257839011yescharacteristic
179C2C2×#1798415311yescharacteristic
180C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#17984062C4
cosocleC4×Dic137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1532
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22578424C2semidirect
2. C2×#17984143C2semidirect
3. #22578260C2semidirect
4. #22578259C2semidirect
90. C137#1532semidirect

Maximal subgroups

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

subgroupindex
#225784242
C2×#179841432
#225782602
#225782592
#225779392
#1532137