Finite Group Databasebeta

← #31797063

Subgroups — #31797063

Order 17536.

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

normal subgroups only

This group has 258 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, complemented
2#22578309C212yesmaximal, complemented
3#22578308C212yesmaximal, complemented
4#22577954C211yesmaximal, characteristic
5#17984167—22no—
6#17984147—22no—
7#17984147C2214yes—
8C2×#14555511C411yescharacteristic, complemented
9C2×#14555511C2211yescharacteristic
10#17984085C2211yescharacteristic
11#17984077C411yescharacteristic, complemented
12#17984077C2211yescharacteristic
13#17984063—22no—
14#17984050—22no—
15C42×D137C411yescharacteristic, complemented
16C4⋊Q8×C137C411yescharacteristic, complemented
17C4⋊#10587739—42no—
18C22⋊#10587739—24no—
19C4×#10587739—42no—
20#14555511—41no—
21#14555511—41no—
22#14555511—41no—
23#14555511—41no—
24C2×D548—21no—
25C2×D548—22no—
26C2×D548C2×C411yescharacteristic
27C2×D548D412yes—
28C2×D548C2311yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137C2×C411yescharacteristic
32C2×C4×D137D412yes—
33#14555487—22no—
34#14555487—22no—
35C4×Dic137—21no—
36C4×Dic137C2×C411yescharacteristic
37C2×Q8×C137C2×C411yescharacteristic
38C2×Q8×C137C2×C411yescharacteristic
39C4⋊C4×C137—22no—
40C42×C137C2×C411yescharacteristic
41C2×#10587739—44no—
42C22×D137—21no—
43C22×D137C22⋊C411yescharacteristic
44C22×D137C2×D412yes—
45D548—42no—
46D548—41no—
47D548—41no—
48D548—44no—
49D548—42no—
50C2×Dic137—21no—
51C2×Dic137C22⋊C411yescharacteristic
52C4×D137—21no—
53C4×D137—21no—
54C4×D137—22no—
55C4×D137—22no—
56C4×D137—42no—
57C4×D137—41no—
58C4×D137—41no—
59Q8×C137—41no—
60Q8×C137—41no—
61C2×C4×C137—21no—
62C2×C4×C137—21no—
63C2×C4×C137C22⋊C412yes—
64C2×C4×C137C22×C411yescharacteristic
65C22×C137C2×C22⋊C411yescharacteristic
66C2×D137—21no—
67C2×D137—22no—
68C2×D137—41no—
69C2×D137—42no—
70C2×D137—42no—
71C2×D137—41no—
72#10587739—84no—
73C4×C137—22no—
74C4×C137—22no—
75C4×C137—21no—
76C4×C137—41no—
77Dic137—41no—
78Dic137#9911yescharacteristic
79Dic137#9911yescharacteristic
80C2×C137—21no—
81C2×C137C2×#9911yescharacteristic
82D137—42no—
83D137—41no—
84D137—81no—
85C137#212611yescharacteristic, complemented, Sylow
86#2126—1371nomaximal, Sylow
87#532—1371no—
88#410—1372no—
89#354—1372no—
90#353—1372no—
91C2×#41—1371no—
92C2×#41—1371no—
93C4⋊Q8#1058773911yescharacteristic, complemented
94#127—1371no—
95#124—1371no—
96#124—1371no—
97#121—2742no—
98C4×D4—2742no—
99#117—2742no—
100C2×C42—1371no—
101#99—2742no—
102#99—1374no—
103#41—5481no—
104#41—5481no—
105#41—5481no—
106#41—5481no—
107C2×Q8C2×#1058773911yescharacteristic
108C2×Q8C2×#1058773911yescharacteristic
109C2×D4—2742no—
110C2×D4—2741no—
111C2×D4—1372no—
112C2×D4—1371no—
113C2×D4—1371no—
114C22×C4—2741no—
115C22×C4—2741no—
116C22×C4—1372no—
117C22×C4—1371no—
118C4⋊C4—22no—
119C4⋊C4—5482no—
120C22⋊C4—2742no—
121C22⋊C4—2742no—
122C22⋊C4—2744no—
123C42—2741no—
124C42—5482no—
125C42—1371no—
126C42C2×#1058773911yescharacteristic
127C23—2741no—
128C23—1372no—
129C23—1371no—
130Q8—41no—
131Q8—41no—
132D4—5482no—
133D4—5481no—
134D4—5481no—
135D4—5484no—
136D4—5482no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—2742no—
140C2×C4—2742no—
141C2×C4—2741no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—5482no—
145C2×C4—5481no—
146C2×C4—5481no—
147C2×C4—5484no—
148C2×C4—1371no—
149C2×C4C22⋊#1058773912yes—
150C2×C4C22×#1058773911yescharacteristic
151C22—2742no—
152C22—2741no—
153C22—5481no—
154C22—5482no—
155C22—5482no—
156C22—5481no—
157C22C2×C22⋊#1058773911yescharacteristic
158C4—22no—
159C4—22no—
160C4—21no—
161C4—41no—
162C4—5481no—
163C4—10964no—
164C4—1371no—
165C4—1371no—
166C2—21no—
167C2—5482no—
168C2—5481no—
169C2—10961no—
170C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC2×C137C2×#99
prime radical#22577954C2
cosocleC2×D548C23
hypercenterC4⋊Q8#10587739
hypocenterC137#2126
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
1. #22578384C2semidirect
2. #22578309C2semidirect
3. #22578308C2semidirect
8. C2×#14555511C4semidirect
11. #17984077C4semidirect
15. C42×D137C4semidirect
16. C4⋊Q8×C137C4semidirect
85. C137#2126semidirect
93. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783842
#225783092
#225783082
#225779542
#2126137