Finite Group Databasebeta

← #31796242

Subgroups — #31796242

Order 17536.

234 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C4×#14555519C211yesmaximal, characteristic, complemented
2C2×#17984143C211yesmaximal, characteristic, complemented
3#22578260C212yesmaximal, complemented
4#22578259C212yesmaximal, complemented
5#22577939C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8#17984143—21no—
9#17984143C2212yes—
10#17984143C2212yes—
11C4×#11764047—21no—
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
23#14555519—24no—
24C2×#11764047—24no—
25C2×#11764047—22no—
26C2×#11764047—22no—
27C2×D548C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C22×Dic137D411yescharacteristic
30C22×Dic137D411yescharacteristic
31C2×C4×D137—21no—
32C4⋊Dic137—21no—
33C4⋊Dic137—21no—
34C4⋊Dic137C2×C411yescharacteristic
35#14555489—21no—
36#14555487—21no—
37#14555487—21no—
38#14555487—21no—
39#14555487—21no—
40C4×Dic137—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C2×Q8×C137C2×C411yescharacteristic
46C22×C4×C137C2×C411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C22⋊C4×C137—21no—
50#11764047—24no—
51#11764047—24no—
52C22×D137—21no—
53D548—41no—
54C22⋊D137—42no—
55C2×Dic137—22no—
56C2×Dic137—22no—
57C2×Dic137—21no—
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C4×D137—42no—
63C23×C137C22⋊C411yescharacteristic
64Q8×C137—41no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—41no—
68C2×C4×C137C22⋊C411yescharacteristic
69C2×C4×C137C22×C411yescharacteristic
70C22×C137—21no—
71C22×C137—21no—
72C22×C137#10412yes—
73C22×C137C2×C22⋊C411yescharacteristic
74C2×D137—41no—
75C2×D137—41no—
76C2×D137—41no—
77C4×C137—21no—
78C4×C137—21no—
79C4×C137—41no—
80C4×C137—41no—
81Dic137—21no—
82Dic137—21no—
83Dic137—21no—
84Dic137—41no—
85Dic137#10412yes—
86C2×C137—22no—
87C2×C137#41311yescharacteristic
88C2×C137C2×#10411yescharacteristic
89C2×C137C2×#10411yescharacteristic
90D137—81no—
91C137#152411yescharacteristic, complemented, Sylow
92#1524—1371nomaximal, Sylow
93#529—1371no—
94C4×M16—1371no—
95C2×#97—1371no—
96#330—1372no—
97#329—1372no—
98C2×M16—2741no—
99C22×C8—2741no—
100#126—2741no—
101#125—1371no—
102#124—1371no—
103#123—2741no—
104C22⋊Q8#1058773911yescharacteristic
105#121—1371no—
106C4×Q8—1371no—
107C4×D4—2741no—
108C4×D4—1371no—
109#117—2741no—
110C2×C42—1371no—
111#97—2741no—
112#97—1372no—
113#97—1372no—
114C4×C8—2741no—
115C2×Q8C2×#1058773911yescharacteristic
116C2×D4—2741no—
117C2×D4—1371no—
118C22×C4—2741no—
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4C2×#1058773911yescharacteristic
122M16—2744no—
123C2×C8—2744no—
124C2×C8—2742no—
125C2×C8—2742no—
126C4⋊C4—21no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—1371no—
131C4⋊C4C2×#1058773911yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—2741no—
134C22⋊C4—2741no—
135C22⋊C4—2741no—
136C22⋊C4—2741no—
137C42—2741no—
138C42—1371no—
139C42—1371no—
140C42—1371no—
141C42—1371no—
142C23—2741no—
143C23C22⋊#1058773911yescharacteristic
144Q8—41no—
145D4—5481no—
146D4—5482no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—2742no—
151C2×C4—2742no—
152C2×C4—2741no—
153C2×C4—5482no—
154C2×C4—1371no—
155C2×C4—1371no—
156C2×C4—1371no—
157C2×C4—1371no—
158C2×C4C22⋊#1058773911yescharacteristic
159C2×C4C22×#1058773911yescharacteristic
160C8—2744no—
161C8—2744no—
162C22—21no—
163C22—21no—
164C22—5481no—
165C22—5481no—
166C22—5481no—
167C22#1798415312yes—
168C22C2×C22⋊#1058773911yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—2741no—
174C4—2741no—
175C4—2741no—
176C4—5481no—
177C4—1372no—
178C2—22no—
179C2—10961no—
180C2#2257839011yescharacteristic
181C2D137⋊#10411yescharacteristic
182C2C2×#1798415311yescharacteristic
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#1524
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. C4×#14555519C2semidirect
2. C2×#17984143C2semidirect
3. #22578260C2semidirect
4. #22578259C2semidirect
91. C137#1524semidirect

Maximal subgroups

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

subgroupindex
C4×#145555192
C2×#179841432
#225782602
#225782592
#225779392
#1524137