Finite Group Databasebeta

← #31796027

Subgroups — #31796027

Order 17536.

332 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×C22⋊#11764047C212yesmaximal, complemented
2#22578249C212yesmaximal, complemented
3#22578247C212yesmaximal, complemented
4C2×#17984048C211yesmaximal, characteristic
5C22×#11764047—22no—
6C22⋊#11764047—22no—
7C22⋊#11764047C2214yes—
8C22×C22⋊D137C411yescharacteristic
9C23×Dic137C2211yescharacteristic
10C22×C4×D137C411yescharacteristic
11#17984048—22no—
12#17984048—22no—
13C2×#14555489C2211yescharacteristic
14C2×#14555487C2211yescharacteristic
15C2×C4×Dic137C411yescharacteristic
16C2×C22⋊C4×C137C411yescharacteristic
17C2×#11764047—24no—
18C2×#11764047—44no—
19C23×D137C2×C411yescharacteristic
20C2×C22⋊D137—22no—
21C2×C22⋊D137—22no—
22C2×C22⋊D137C814yescomplemented
23C22×Dic137—42no—
24C22×Dic137C2×C411yescharacteristic
25C22×Dic137C2×C411yescharacteristic
26C22×Dic137D412yes—
27C22×Dic137D412yes—
28C22×Dic137C2311yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137—21no—
31C2×C4×D137—42no—
32#14555489—22no—
33#14555487—22no—
34C4×Dic137—21no—
35C4×Dic137—21no—
36C24×C137C2×C411yescharacteristic
37C22×C4×C137C2×C411yescharacteristic
38C22×C4×C137C2×C411yescharacteristic
39C22⋊C4×C137—22no—
40#11764047—44no—
41C22×D137—21no—
42C22×D137—21no—
43C22×D137—41no—
44C22×D137—41no—
45C22×D137C2×C812yescomplemented
46C22⋊D137—28no—
47C2×Dic137—21no—
48C2×Dic137—21no—
49C2×Dic137—22no—
50C2×Dic137—24no—
51C2×Dic137—42no—
52C2×Dic137—41no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C2×C812yescomplemented
56C2×Dic137C2×D412yes—
57C4×D137—42no—
58C4×D137—42no—
59C23×C137—41no—
60C23×C137—41no—
61C23×C137C22⋊C412yes—
62C23×C137C2×C812yes—
63C23×C137M1612yes—
64C23×C137C22×C411yescharacteristic
65C2×C4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C22×C137—21no—
70C22×C137—22no—
71C22×C137—21no—
72C22×C137—22no—
73C22×C137—42no—
74C22×C137—42no—
75C22×C137C22⋊C814yes—
76C22×C137C2×C22⋊C411yescharacteristic
77C22×C137C22×C811yescharacteristic
78C22×C137C2×M1611yescharacteristic
79C2×D137—22no—
80C2×D137—42no—
81C2×D137—41no—
82C2×D137—41no—
83C2×D137—42no—
84C4×C137—41no—
85C4×C137—41no—
86Dic137—22no—
87Dic137—21no—
88Dic137—22no—
89Dic137#9912yes—
90C2×C137—21no—
91C2×C137—21no—
92C2×C137—24no—
93C2×C137C2×C22⋊C811yescharacteristic
94C2×C137C2×#9911yescharacteristic
95C2×C137#41311yescharacteristic
96D137—42no—
97C137#145911yescharacteristic, complemented, Sylow
98#1459—1371nomaximal, Sylow
99C2×C4×D4—1371no—
100C2×C22⋊C8—1372no—
101#324—1372no—
102#323—1372no—
103C22×D4—1371no—
104C23×C4—1371no—
105C23×C4—1371no—
106C22×C8—2742no—
107C4×D4—2742no—
108C4×D4—2742no—
109C2×C4⋊C4—1371no—
110C2×C22⋊C4—1371no—
111C2×C22⋊C4#1058773911yescharacteristic
112C2×C42—1371no—
113C22⋊C8—2742no—
114C22⋊C8—1374no—
115C24—1371no—
116C24C2×#1058773911yescharacteristic
117C2×D4—2742no—
118C2×D4—2742no—
119C2×D4—1374no—
120C22×C4—2741no—
121C22×C4—2741no—
122C22×C4—5482no—
123C22×C4—5482no—
124C22×C4—1372no—
125C22×C4—1371no—
126C22×C4—1371no—
127C22×C4—1372no—
128C22×C4—1371no—
129C22×C4C2×#1058773911yescharacteristic
130C22×C4C2×#1058773911yescharacteristic
131C2×C8—2744no—
132C2×C8—5484no—
133C4⋊C4—2742no—
134C22⋊C4—22no—
135C22⋊C4—2742no—
136C42—2741no—
137C42—2741no—
138C23—41no—
139C23—41no—
140C23—2741no—
141C23—2741no—
142C23—5481no—
143C23—5481no—
144C23—1372no—
145C23C22⋊#1058773912yes—
146C23#1455551812yes—
147C23#1455552012yes—
148C23C22×#1058773911yescharacteristic
149D4—2748no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—2741no—
155C2×C4—2741no—
156C2×C4—2744no—
157C2×C4—2742no—
158C2×C4—5482no—
159C2×C4—5482no—
160C2×C4—5481no—
161C2×C4—5482no—
162C2×C4—1371no—
163C2×C4—1372no—
164C2×C4—1372no—
165C2×C4—1371no—
166C8—5484no—
167C22—21no—
168C22—22no—
169C22—21no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22—2742no—
174C22—5482no—
175C22—5482no—
176C22—5481no—
177C22—5481no—
178C22#1798414514yes—
179C22C2×C22⋊#1058773911yescharacteristic
180C22C2×#1455551811yescharacteristic
181C22C2×#1455552011yescharacteristic
182C4—41no—
183C4—41no—
184C4—2742no—
185C4—2742no—
186C4—2741no—
187C4—1372no—
188C2—21no—
189C2—21no—
190C2—24no—
191C2—5482no—
192C2C2×#1798414511yescharacteristic
193C2D137⋊#9911yescharacteristic
194C2#2257839011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC22×C137C22×C8
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×C22⋊D137C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1459
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#11764047C2semidirect
2. #22578249C2semidirect
3. #22578247C2semidirect
22. C2×C22⋊D137C8semidirect
45. C22×D137C2×C8semidirect
55. C2×Dic137C2×C8semidirect
97. C137#1459semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#117640472
#225782492
#225782472
C2×#179840482
#1459137