Finite Group Databasebeta

← #31797069

Subgroups — #31797069

Order 17536.

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

normal subgroups only

This group has 270 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#22578308C212yesmaximal, complemented
3#22578307C212yesmaximal, complemented
4#22577971C211yesmaximal, characteristic
5#17984167—22no—
6#17984147—22no—
7#17984147C2214yescomplemented
8C2×#14555511C2211yescharacteristic
9C2×D4×D137C411yescharacteristic, complemented
10#17984085C2211yescharacteristic
11#17984084C2211yescharacteristic
12#17984077C411yescharacteristic, complemented
13#17984061—22no—
14#17984058—22no—
15#17984045C411yescharacteristic, complemented
16#124×C137C411yescharacteristic, complemented
17C4⋊#10587739—42no—
18C22⋊#10587739—24no—
19C4×#10587739—42no—
20C23×D137—21no—
21#14555511—42no—
22C2×D548—22no—
23C2×D548C2×C411yescharacteristic
24C2×D548D412yes—
25C2×D548C2311yescharacteristic
26C2×C22⋊D137—22no—
27C2×C22⋊D137—21no—
28D4×D137—42no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D412yes—
31#14555489—21no—
32#14555487—21no—
33#14555487—22no—
34C4×Dic137C2×C411yescharacteristic
35C2×Q8×C137C2×C411yescharacteristic
36C2×D4×C137C2×C411yescharacteristic
37C22⋊C4×C137—22no—
38C42×C137C2×C411yescharacteristic
39C2×#10587739—44no—
40C22×D137—21no—
41C22×D137—42no—
42C22×D137—82no—
43C22×D137C22⋊C411yescharacteristic
44C22×D137C2×D412yes—
45D548—42no—
46D548—41no—
47D548—41no—
48D548—82no—
49C22⋊D137—42no—
50C22⋊D137—82no—
51C2×Dic137—21no—
52C2×Dic137C22⋊C411yescharacteristic
53C4×D137—22no—
54C4×D137—22no—
55C4×D137—22no—
56C23×C137—21no—
57Q8×C137—41no—
58D4×C137—41no—
59C2×C4×C137—21no—
60C2×C4×C137C22⋊C412yes—
61C2×C4×C137C22×C411yescharacteristic
62C22×C137—42no—
63C22×C137—41no—
64C22×C137C2×C22⋊C411yescharacteristic
65C2×D137—21no—
66C2×D137—22no—
67C2×D137—42no—
68C2×D137—41no—
69C2×D137—42no—
70C2×D137—41no—
71C2×D137—84no—
72#10587739—84no—
73C4×C137—22no—
74C4×C137—21no—
75C4×C137—41no—
76Dic137—41no—
77Dic137#9912yes—
78C2×C137—21no—
79C2×C137—81no—
80C2×C137C2×#9911yescharacteristic
81D137—42no—
82D137—41no—
83D137—81no—
84C137#212811yescharacteristic, complemented, Sylow
85#2128—1371nomaximal, Sylow
86#535—1371no—
87#410—1372no—
88#353—1372no—
89#353—1372no—
90C2×#41—1371no—
91C22×D4—1371no—
92#127—1371no—
93#127—1371no—
94#124—1371no—
95#124#1058773911yescharacteristic, complemented
96#121—2742no—
97#120—2742no—
98#117—2742no—
99#117—1371no—
100#99—2742no—
101#99—1374no—
102C24—2741no—
103#41—5482no—
104C2×Q8C2×#1058773911yescharacteristic
105C2×D4—2742no—
106C2×D4—2742no—
107C2×D4—2741no—
108C2×D4—5482no—
109C2×D4—1372no—
110C2×D4—1371no—
111C2×D4—1371no—
112C2×D4C2×#1058773911yescharacteristic
113C22×C4—1372no—
114C22×C4—1371no—
115C4⋊C4—2741no—
116C4⋊C4—5482no—
117C22⋊C4—22no—
118C22⋊C4—2741no—
119C22⋊C4—2742no—
120C22⋊C4—2744no—
121C42—5482no—
122C42—1371no—
123C42C2×#1058773911yescharacteristic
124C23—21no—
125C23—2741no—
126C23—5482no—
127C23—10962no—
128C23—1372no—
129C23—1371no—
130Q8—41no—
131D4—41no—
132D4—5482no—
133D4—5481no—
134D4—5481no—
135D4—5482no—
136D4—10962no—
137D4—10962no—
138C2×C4—21no—
139C2×C4—2742no—
140C2×C4—2742no—
141C2×C4—2742no—
142C2×C4—2741no—
143C2×C4—5484no—
144C2×C4—1371no—
145C2×C4C22⋊#1058773912yes—
146C2×C4C22×#1058773911yescharacteristic
147C22—42no—
148C22—41no—
149C22—2742no—
150C22—2741no—
151C22—5482no—
152C22—5481no—
153C22—5482no—
154C22—5481no—
155C22—10964no—
156C22C2×C22⋊#1058773911yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—5481no—
161C4—10964no—
162C4—1372no—
163C2—21no—
164C2—81no—
165C2—5482no—
166C2—5481no—
167C2—10961no—
168C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137C2×#99
prime radical#22577971C2
cosocleC2×D548C23
hypercenter#124#10587739
hypocenterC137#2128
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #22578384C2semidirect
2. #22578308C2semidirect
3. #22578307C2semidirect
7. #17984147C22semidirect
9. C2×D4×D137C4semidirect
12. #17984077C4semidirect
15. #17984045C4semidirect
16. #124×C137C4semidirect
84. C137#2128semidirect
95. #124#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783842
#225783082
#225783072
#225779712
#2128137