Finite Group Databasebeta

← #31779960

Subgroups — #31779960

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#22578126C211yesmaximal, characteristic
2#22577753C211yesmaximal, characteristic, complemented
3#22577690C211yesmaximal, characteristic
4#22577148C211yesmaximal, characteristic, complemented
5Q8⋊#11764040C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic
7#477×C137C211yesmaximal, characteristic, complemented
8#17984088C2211yescharacteristic
9#17984083—22no—
10#17984076—21no—
11#17984064—21no—
12C4×Dic274—21no—
13C4×D548C2211yescharacteristic
14#17984046—21no—
15#17984019—21no—
16C4⋊#11764040C2211yescharacteristic
17#17984012—22no—
18#17984008—21no—
19C4×#11764040C2211yescharacteristic
20C4⋊Q8×C137C2211yescharacteristic
21C4×Q8×C137C2211yescharacteristic
22#107×C137—21no—
23C4⋊C8×C137C2211yescharacteristic
24#103×C137—21no—
25C2×Dic274—21no—
26C2×D548D411yescharacteristic
27C2×C4×D137—21no—
28C2×#11764040—21no—
29C2×#11764040—21no—
30C4⋊Dic137—21no—
31C4⋊Dic137D412yes—
32C4⋊Dic137D411yescharacteristic
33#14555489—22no—
34#14555487—22no—
35#14555487—22no—
36#14555487—21no—
37C4×Dic137—22no—
38C2×Q8×C137—21no—
39C2×Q8×C137D411yescharacteristic
40C2×C8×C137—21no—
41C4⋊C4×C137—21no—
42C4⋊C4×C137—21no—
43C4⋊C4×C137—21no—
44C4⋊C4×C137D411yescharacteristic
45C42×C137—21no—
46C42×C137C2311yescharacteristic
47C22×D137—21no—
48Dic274—41no—
49Dic274—41no—
50D548—21no—
51D548—21no—
52C2×Dic137—21no—
53C2×Dic137—22no—
54C4×D137—41no—
55C4×D137—41no—
56#11764040—41no—
57#11764040—41no—
58Q8×C137—21no—
59Q8×C137—21no—
60Q8×C137—42no—
61C2×C4×C137—21no—
62C2×C4×C137—21no—
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—41no—
74C4×C137—41no—
75C4×C137—81no—
76C4×C137#13511yescharacteristic
77C4×C137#13511yescharacteristic
78C4×C137#13711yescharacteristic
79C4×C137#13711yescharacteristic
80Dic137—41no—
81Dic137—42no—
82C2×C137#45211yescharacteristic
83C2×C137#45211yescharacteristic
84C2×C137#45411yescharacteristic
85D137—81no—
86C137#163811yescharacteristic, complemented, Sylow
87#1638—1371nomaximal, Sylow
88#556—1371no—
89#487—1371no—
90#477D13711yescharacteristic, complemented
91#476—1371no—
92#332—1371no—
93Q8⋊C8—1371no—
94D4⋊C8—1371no—
95C4⋊Q8—1371no—
96C4⋊Q8C2×D13711yescharacteristic
97#126—2742no—
98#124—2741no—
99C22⋊Q8—2741no—
100C4×Q8—2741no—
101C4×Q8C2×D13711yescharacteristic
102C4×D4—1371no—
103#117—2741no—
104#107—21no—
105#106—2741no—
106C4⋊C8—1371no—
107C4⋊C8C2×D13711yescharacteristic
108#103—21no—
109#103—2742no—
110#102—2741no—
111C4×C8—1371no—
112C2×Q8—21no—
113C2×Q8—2741no—
114C2×Q8C22⋊D13711yescharacteristic
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—1372no—
126C4⋊C4—1371no—
127C4⋊C4C22⋊D13711yescharacteristic
128C22⋊C4—2742no—
129C22⋊C4—2742no—
130C22⋊C4—2741no—
131C42—21no—
132C42—2742no—
133C42C22×D13711yescharacteristic
134C23—2741no—
135Q8—21no—
136Q8—21no—
137Q8—42no—
138Q8—5481no—
139Q8—5481no—
140D4—2741no—
141D4—2741no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—2741no—
146C2×C4—2742no—
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—41no—
161C4—41no—
162C4—81no—
163C4—5481no—
164C4—5482no—
165C4#1798410811yescharacteristic
166C4#1798411111yescharacteristic
167C4#1798411911yescharacteristic
168C4#1798412011yescharacteristic
169C2—10961no—
170C2#2257756311yescharacteristic
171C2#2257756411yescharacteristic
172C2#2257757411yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#477×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenter#477D137
hypocenterC137#1638
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #22577753C2semidirect
4. #22577148C2semidirect
5. Q8⋊#11764040C2semidirect
7. #477×C137C2semidirect
86. C137#1638semidirect
90. #477D137semidirect

Maximal subgroups

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

subgroupindex
#225781262
#225777532
#225776902
#225771482
Q8⋊#117640402
#225771302
#477×C1372
#1638137