Finite Group Databasebeta

← #31779958

Subgroups — #31779958

Order 17536.

194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 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#22578127C211yesmaximal, characteristic
2#22577695C211yesmaximal, characteristic, complemented
3#22577690C211yesmaximal, characteristic
4#22577149C211yesmaximal, characteristic, complemented
5#22577135C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic
7#487×C137C211yesmaximal, characteristic, complemented
8#17984083—22no—
9#17984077—21no—
10#17984068—21no—
11Q8×Dic137—21no—
12C4×Dic274C2211yescharacteristic
13C4×D548C2211yescharacteristic
14#17984046—21no—
15#17984021—21no—
16#17984019—21no—
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984011—21no—
20#17984008—21no—
21C4⋊Q8×C137C2211yescharacteristic
22C4⋊Q8×C137C2211yescharacteristic
23#103×C137—22no—
24C4×C8×C137C2211yescharacteristic
25C2×Dic274D411yescharacteristic
26C2×D548D411yescharacteristic
27C2×C4×D137—21no—
28C2×#11764040—21no—
29C2×#11764040—21no—
30C4⋊Dic137—22no—
31C4⋊Dic137D411yescharacteristic
32C4⋊Dic137D411yescharacteristic
33#14555489—21no—
34#14555487—22no—
35#14555487—22no—
36#14555487—21no—
37C4×Dic137—22no—
38C4×Dic137—21no—
39C2×Q8×C137—21no—
40C2×Q8×C137—21no—
41C2×C8×C137—21no—
42C4⋊C4×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137D412yes—
46C42×C137C2311yescharacteristic
47C22×D137—21no—
48Dic274—21no—
49Dic274—21no—
50D548—21no—
51D548—21no—
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137—21no—
55C4×D137—41no—
56C4×D137—41no—
57#11764040—41no—
58#11764040—41no—
59Q8×C137—42no—
60Q8×C137—41no—
61Q8×C137—41no—
62C2×C4×C137—22no—
63C2×C4×C137—41no—
64C2×C4×C137C2×D411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C2×C4×C137C2×D411yescharacteristic
67C8×C137—41no—
68C22×C137#12011yescharacteristic
69C2×D137—41no—
70C2×D137—41no—
71C2×D137—41no—
72C4×C137—21no—
73C4×C137—42no—
74C4×C137—81no—
75C4×C137#13511yescharacteristic
76C4×C137#13511yescharacteristic
77C4×C137#13711yescharacteristic
78C4×C137#13711yescharacteristic
79Dic137—41no—
80Dic137—41no—
81Dic137—41no—
82C2×C137#45211yescharacteristic
83C2×C137#45211yescharacteristic
84C2×C137#45411yescharacteristic
85D137—81no—
86C137#163811yescharacteristic, complemented, Sylow
87#1638—1371nomaximal, Sylow
88#556—1371no—
89#487D13711yescharacteristic, complemented
90#477—1371no—
91#476—1371no—
92#332—1371no—
93Q8⋊C8—1371no—
94D4⋊C8—1371no—
95C4⋊Q8C2×D13711yescharacteristic
96C4⋊Q8C2×D13711yescharacteristic
97#126—2742no—
98#124—2741no—
99C22⋊Q8—2741no—
100C4×Q8—2741no—
101C4×Q8—1371no—
102C4×D4—1371no—
103#117—2741no—
104#107—2741no—
105#106—2741no—
106C4⋊C8—1371no—
107C4⋊C8—1371no—
108#103—22no—
109#103—2741no—
110#102—2741no—
111C4×C8C2×D13711yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—1371no—
115C2×D4—1371no—
116C22×C4—2741no—
117C2×C8—21no—
118C2×C8—2741no—
119C2×C8—2741no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—2742no—
124C4⋊C4—2741no—
125C4⋊C4—1371no—
126C4⋊C4—1371no—
127C4⋊C4C22⋊D13712yes—
128C22⋊C4—2742no—
129C22⋊C4—2742no—
130C22⋊C4—2741no—
131C42—2742no—
132C42—2741no—
133C42C22×D13711yescharacteristic
134C23—2741no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—2741no—
139Q8—2741no—
140D4—2741no—
141D4—2741no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—2741no—
145C2×C4—2741no—
146C2×C4—2741no—
147C2×C4—5481no—
148C2×C4—5481no—
149C2×C4D4×D13711yescharacteristic
150C2×C4D4×D13711yescharacteristic
151C2×C4C2×C22⋊D13711yescharacteristic
152C8—41no—
153C8—5481no—
154C8—5481no—
155C22—5481no—
156C22—5481no—
157C22—5481no—
158C22#1798405611yescharacteristic
159C4—21no—
160C4—42no—
161C4—81no—
162C4—5481no—
163C4—5481no—
164C4—5481no—
165C4#1798410911yescharacteristic
166C4#1798411111yescharacteristic
167C4#1798411711yescharacteristic
168C4#1798412011yescharacteristic
169C2—10961no—
170C2#2257756211yescharacteristic
171C2#2257756411yescharacteristic
172C2#2257757511yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#487×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#487D137
hypocenterC137#1638
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #22577695C2semidirect
4. #22577149C2semidirect
5. #22577135C2semidirect
7. #487×C137C2semidirect
86. C137#1638semidirect
89. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225781272
#225776952
#225776902
#225771492
#225771352
#225771322
#487×C1372
#1638137