Finite Group Databasebeta

← #31780121

Subgroups — #31780121

Order 17536.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#22578157C211yesmaximal, characteristic
2#22577771C211yesmaximal, characteristic, complemented
3#22577765C211yesmaximal, characteristic
4#22577145C211yesmaximal, characteristic, complemented
5#22577144C211yesmaximal, characteristic, complemented
6#22577142C211yesmaximal, characteristic
7#501×C137C211yesmaximal, characteristic, complemented
8#17984087—21no—
9#17984083—21no—
10#17984083—21no—
11#17984080C2211yescharacteristic
12#17984076C2211yescharacteristic
13#17984073—21no—
14#17984068—21no—
15#17984066—21no—
16#17984012—21no—
17#17984012—21no—
18#17984011—21no—
19#17984008—21no—
20#17983999C2211yescharacteristic
21#17983999C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23#125×C137C2211yescharacteristic
24#107×C137—21no—
25#106×C137—21no—
26#97×C137C2211yescharacteristic
27C2×Dic274—21no—
28C2×Dic274D411yescharacteristic
29C2×D548D411yescharacteristic
30C2×C4×D137—21no—
31C2×#11764040—21no—
32C2×#11764040—21no—
33C4⋊Dic137—21no—
34C4⋊Dic137D411yescharacteristic
35C4⋊Dic137D411yescharacteristic
36#14555489—21no—
37#14555489—21no—
38#14555489—21no—
39#14555487—21no—
40#14555487—21no—
41#14555487—21no—
42#14555487—21no—
43#14555487—21no—
44C4×Dic137—21no—
45C2×Q8×C137—21no—
46C2×C8×C137—21no—
47C4⋊C4×C137—21no—
48C4⋊C4×C137—21no—
49C4⋊C4×C137—21no—
50C4⋊C4×C137—21no—
51C4⋊C4×C137D411yescharacteristic
52C4⋊C4×C137D411yescharacteristic
53C42×C137C2311yescharacteristic
54C22×D137—21no—
55Dic274—41no—
56Dic274—81no—
57D548—41no—
58C2×Dic137—21no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C4×D137—81no—
62#11764040—41no—
63#11764040—41no—
64Q8×C137—42no—
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—41no—
68C2×C4×C137C2×D411yescharacteristic
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C8×C137—22no—
72C22×C137#12011yescharacteristic
73C2×D137—41no—
74C2×D137—41no—
75C2×D137—41no—
76C4×C137—21no—
77C4×C137—21no—
78C4×C137—21no—
79C4×C137—41no—
80C4×C137—41no—
81C4×C137—81no—
82Dic137—41no—
83Dic137—41no—
84Dic137—41no—
85C2×C137#45411yescharacteristic
86C2×C137#45511yescharacteristic
87C2×C137#45611yescharacteristic
88D137—81no—
89C137#166411yescharacteristic, complemented, Sylow
90#1664—1371nomaximal, Sylow
91#563—1371no—
92#501D13711yescharacteristic, complemented
93#491—1371no—
94#489—1371no—
95#330—1371no—
96#330—1371no—
97#329—1371no—
98C4⋊Q8—2741no—
99C4⋊Q8C2×D13711yescharacteristic
100#126—2741no—
101#126—2741no—
102#125—1371no—
103#125C2×D13711yescharacteristic
104#124—1371no—
105#123—2741no—
106C22⋊Q8—2741no—
107C22⋊Q8—2741no—
108#107—21no—
109#106—21no—
110#103—2741no—
111#103—2741no—
112#103—2741no—
113#102—2741no—
114#97—1371no—
115#97—1371no—
116#97C2×D13711yescharacteristic
117C2×Q8—21no—
118C2×Q8—2741no—
119C2×Q8—1371no—
120C2×D4—1371no—
121C22×C4—2741no—
122C2×C8—21no—
123C2×C8—2741no—
124C2×C8—2741no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—1371no—
134C4⋊C4—1371no—
135C4⋊C4C22⋊D13711yescharacteristic
136C4⋊C4C22⋊D13711yescharacteristic
137C22⋊C4—2741no—
138C22⋊C4—2741no—
139C22⋊C4—2741no—
140C22⋊C4—2741no—
141C22⋊C4—2741no—
142C42—2741no—
143C42C22×D13711yescharacteristic
144C23—2741no—
145Q8—42no—
146Q8—5481no—
147Q8—10961no—
148D4—5481no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—2741no—
153C2×C4—2741no—
154C2×C4—2741no—
155C2×C4—10961no—
156C2×C4D4×D13711yescharacteristic
157C2×C4D4×D13711yescharacteristic
158C2×C4C2×C22⋊D13711yescharacteristic
159C8—22no—
160C8—5481no—
161C8—5481no—
162C22—5481no—
163C22—5481no—
164C22—5481no—
165C22#1798405611yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—5481no—
173C4—5481no—
174C4—5481no—
175C2—10961no—
176C2#2257757411yescharacteristic
177C2#2257757911yescharacteristic
178C2#2257758511yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#501×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#501D137
hypocenterC137#1664
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #22577771C2semidirect
4. #22577145C2semidirect
5. #22577144C2semidirect
7. #501×C137C2semidirect
89. C137#1664semidirect
92. #501D137semidirect

Maximal subgroups

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

subgroupindex
#225781572
#225777712
#225777652
#225771452
#225771442
#225771422
#501×C1372
#1664137