Finite Group Databasebeta

← #31796695

Subgroups — #31796695

Order 17536.

306 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578374C211yesmaximal, characteristic
2C4×C22⋊#10587739C211yesmaximal, characteristic
3#22578273C214yesmaximal, complemented
4C2×#17984046C211yesmaximal, characteristic
5C2×#14555520C2212yes—
6C2×C22⋊#10587739—21no—
7C2×C4×#10587739C2212yes—
8#17984145—21no—
9C22⋊#11764047—21no—
10(C2×#10587739)⋊C4—21no—
11C22×C4×D137C2211yescharacteristic
12#17984046—21no—
13#17984046—21no—
14#17984046C412yescomplemented
15#17984046C2212yes—
16C2×C4⋊Dic137C411yescharacteristic, complemented
17C2×#14555487—21no—
18C2×C4×Dic137—21no—
19C2×C4⋊C4×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555520—24no—
22C2×#11764047—22no—
23C22⋊#10587739—24no—
24C4×#10587739—24no—
25C23×D137D411yescharacteristic
26C22×Dic137—21no—
27C22×Dic137D411yescharacteristic
28C2×C4×D137—24no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×C4×D137C2311yescharacteristic
33C4⋊Dic137—21no—
34C4⋊Dic137C2×C412yes—
35#14555487—22no—
36#14555487—22no—
37C4×Dic137—22no—
38C4×Dic137—22no—
39C22×C4×C137—21no—
40C22×C4×C137C2×C411yescharacteristic
41C4⋊C4×C137—21no—
42C4⋊C4×C137C2×C412yes—
43C2×#10587739—24no—
44C2×#10587739—42no—
45C2×#10587739—42no—
46#11764047—42no—
47C22×D137—22no—
48C22×D137—22no—
49C22×D137—41no—
50C22×D137C22⋊C411yescharacteristic
51C22×D137C2×D411yescharacteristic
52C2×Dic137—22no—
53C2×Dic137—22no—
54C2×Dic137—42no—
55C2×Dic137C22⋊C411yescharacteristic
56C2×Dic137C2×D411yescharacteristic
57C4×D137—22no—
58C4×D137—22no—
59C4×D137—22no—
60C4×D137#4114yes—
61C23×C137C22⋊C411yescharacteristic
62C2×C4×C137—22no—
63C2×C4×C137—22no—
64C2×C4×C137—42no—
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22×C411yescharacteristic
67C22×C137—21no—
68C22×C137—21no—
69C22×C137#10412yes—
70C22×C137C2×C22⋊C411yescharacteristic
71C2×D137—21no—
72C2×D137—21no—
73C2×D137—21no—
74C2×D137—21no—
75C2×D137—21no—
76C2×D137—44no—
77C2×D137#12312yes—
78#10587739—44no—
79C4×C137—21no—
80C4×C137—42no—
81C4×C137#11712yes—
82Dic137—21no—
83Dic137—42no—
84Dic137#12312yes—
85C2×C137—22no—
86C2×C137#38111yescharacteristic
87C2×C137C2×#10411yescharacteristic
88C2×C137#42111yescharacteristic
89D137—22no—
90D137—41no—
91C137#182611yescharacteristic, complemented, Sylow
92#1826—1371nomaximal, Sylow
93C2×#117—1371no—
94#407—1371no—
95C4×C22⋊C4—1371no—
96#339—1374no—
97C23×C4—1371no—
98C2×M16—1372no—
99#117—2741no—
100#117—2741no—
101#117—1372no—
102#117—1372no—
103C2×C4⋊C4—1371no—
104C2×C4⋊C4#1058773911yescharacteristic, complemented
105C2×C22⋊C4—2741no—
106C2×C22⋊C4—2741no—
107C2×C42—2741no—
108C2×C42—1372no—
109C22⋊C8—2741no—
110C22⋊C8—2741no—
111#95—2741no—
112C24—1371no—
113C22×C4—21no—
114C22×C4—2744no—
115C22×C4—2742no—
116C22×C4—2741no—
117C22×C4—1371no—
118C22×C4—1371no—
119C22×C4—1371no—
120C22×C4—1371no—
121C22×C4—1371no—
122C22×C4C2×#1058773911yescharacteristic
123M16—2744no—
124C2×C8—2742no—
125C4⋊C4—21no—
126C4⋊C4—2741no—
127C4⋊C4—1372no—
128C4⋊C4C2×#1058773912yes—
129C22⋊C4—2742no—
130C22⋊C4—2742no—
131C22⋊C4—2744no—
132C42—2742no—
133C42—2742no—
134C42—2744no—
135C23—2742no—
136C23—2742no—
137C23—5481no—
138C23—1371no—
139C23—1371no—
140C23C22⋊#1058773911yescharacteristic
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—2742no—
145C2×C4—2742no—
146C2×C4—2742no—
147C2×C4—2742no—
148C2×C4—2744no—
149C2×C4—2742no—
150C2×C4—5482no—
151C2×C4—5482no—
152C2×C4—5482no—
153C2×C4—1371no—
154C2×C4—1371no—
155C2×C4—1374no—
156C2×C4C22⋊#1058773911yescharacteristic
157C2×C4C22×#1058773911yescharacteristic
158C8—5482no—
159C22—21no—
160C22—21no—
161C22—2741no—
162C22—2741no—
163C22—2741no—
164C22—2741no—
165C22—2741no—
166C22—5484no—
167C22—1372no—
168C22#1798415212yes—
169C22C2×C22⋊#1058773911yescharacteristic
170C4—21no—
171C4—42no—
172C4—2741no—
173C4—5484no—
174C4—5482no—
175C4—1372no—
176C4#1798416712yes—
177C2—22no—
178C2—2742no—
179C2—5481no—
180C2#2257834611yescharacteristic
181C2C2×#1798415211yescharacteristic
182C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1826
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22578273C2semidirect
14. #17984046C4semidirect
16. C2×C4⋊Dic137C4semidirect
19. C2×C4⋊C4×C137C4semidirect
91. C137#1826semidirect
104. C2×C4⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783742
C4×C22⋊#105877392
#225782732
C2×#179840462
#1826137