Finite Group Databasebeta

← #31797072

Subgroups — #31797072

Order 17536.

238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 52 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#22578384C212yesmaximal
2#22578310C212yesmaximal, complemented
3#22578309C212yesmaximal, complemented
4#22577978C211yesmaximal, characteristic
5#17984167—22no—
6#17984147—22no—
7#17984147C2214yes—
8C2×#14555511C2211yescharacteristic
9C2×Q8×D137C411yescharacteristic, complemented
10#17984086C411yescharacteristic, complemented
11#17984077C2211yescharacteristic
12#17984076C2211yescharacteristic
13#17984073—22no—
14#17984066—22no—
15#17984045C411yescharacteristic, complemented
16C4⋊Q8×C137C411yescharacteristic, complemented
17C4⋊#10587739—42no—
18C22⋊#10587739—24no—
19C4×#10587739—42no—
20#14555511—42no—
21C2×Dic274—21no—
22C2×Dic274C2×C411yescharacteristic
23Q8×D137—42no—
24C2×D548D412yes—
25C2×D548C2311yescharacteristic
26C2×C4×D137—21no—
27C2×C4×D137C2×C411yescharacteristic
28C2×C4×D137D412yes—
29#14555489—21no—
30#14555489—22no—
31#14555487—21no—
32#14555487—22no—
33#14555487—22no—
34C4×Dic137C2×C411yescharacteristic
35C2×Q8×C137C2×C411yescharacteristic
36C2×Q8×C137C2×C411yescharacteristic
37C4⋊C4×C137—22no—
38C42×C137C2×C411yescharacteristic
39C2×#10587739—44no—
40C22×D137C22⋊C411yescharacteristic
41C22×D137C2×D412yes—
42Dic274—42no—
43Dic274—41no—
44D548—42no—
45D548—41no—
46C2×Dic137—21no—
47C2×Dic137—21no—
48C2×Dic137C22⋊C411yescharacteristic
49C4×D137—22no—
50C4×D137—22no—
51C4×D137—22no—
52C4×D137—42no—
53Q8×C137—41no—
54Q8×C137—41no—
55C2×C4×C137—21no—
56C2×C4×C137—21no—
57C2×C4×C137C22⋊C412yes—
58C2×C4×C137C22×C411yescharacteristic
59C22×C137C2×C22⋊C411yescharacteristic
60C2×D137—21no—
61C2×D137—22no—
62C2×D137—41no—
63C2×D137—42no—
64#10587739—84no—
65C4×C137—22no—
66C4×C137—21no—
67C4×C137—41no—
68C4×C137—41no—
69Dic137—41no—
70Dic137—41no—
71Dic137#9912yes—
72C2×C137—21no—
73C2×C137C2×#9911yescharacteristic
74D137—42no—
75D137—41no—
76C137#212911yescharacteristic, complemented, Sylow
77#2129—1371nomaximal, Sylow
78#536—1371no—
79#410—1372no—
80#354—1372no—
81#354—1372no—
82C2×#41—1371no—
83C22×Q8—1371no—
84C4⋊Q8—1371no—
85C4⋊Q8#1058773911yescharacteristic, complemented
86#124—1371no—
87#124—1371no—
88#123—2742no—
89C22⋊Q8—2742no—
90#117—2742no—
91#117—1371no—
92#99—2742no—
93#99—1374no—
94#41—5482no—
95C2×Q8—2741no—
96C2×Q8—5482no—
97C2×Q8—1371no—
98C2×Q8C2×#1058773911yescharacteristic
99C2×Q8C2×#1058773911yescharacteristic
100C2×D4—1372no—
101C2×D4—1371no—
102C22×C4—2741no—
103C22×C4—1372no—
104C22×C4—1371no—
105C4⋊C4—22no—
106C4⋊C4—2741no—
107C4⋊C4—2742no—
108C4⋊C4—5482no—
109C22⋊C4—2741no—
110C22⋊C4—2742no—
111C22⋊C4—2742no—
112C22⋊C4—2744no—
113C42—5482no—
114C42—1371no—
115C42C2×#1058773911yescharacteristic
116C23—1372no—
117C23—1371no—
118Q8—41no—
119Q8—41no—
120Q8—5481no—
121Q8—5482no—
122D4—5482no—
123D4—5481no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—2742no—
127C2×C4—2742no—
128C2×C4—2742no—
129C2×C4—2741no—
130C2×C4—2741no—
131C2×C4—5482no—
132C2×C4—5484no—
133C2×C4—1371no—
134C2×C4C22⋊#1058773912yes—
135C2×C4C22×#1058773911yescharacteristic
136C22—2742no—
137C22—2741no—
138C22—5482no—
139C22—5481no—
140C22C2×C22⋊#1058773911yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—5481no—
146C4—5481no—
147C4—10964no—
148C4—1372no—
149C2—21no—
150C2—5482no—
151C2—5481no—
152C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555511C22
cosocleC2×D548C23
hypercenterC4⋊Q8#10587739
hypocenterC137#2129
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22578310C2semidirect
3. #22578309C2semidirect
9. C2×Q8×D137C4semidirect
10. #17984086C4semidirect
15. #17984045C4semidirect
16. C4⋊Q8×C137C4semidirect
76. C137#2129semidirect
85. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783842
#225783102
#225783092
#225779782
#2129137