Finite Group Databasebeta

← #31796026

Subgroups — #31796026

Order 17536.

332 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 80 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#22578371C212yesmaximal, complemented
2#22578248C212yesmaximal, complemented
3#22578247C212yesmaximal, complemented
4C2×C4×D548C211yesmaximal, characteristic
5C2×#14555518—22no—
6#17984145—22no—
7C22⋊#11764047C2214yes—
8C22×D548C411yescharacteristic
9C22×C4×D137C411yescharacteristic
10C22×C4×D137C2211yescharacteristic
11C4×D548—22no—
12C4×D548—22no—
13C2×C4⋊Dic137C2211yescharacteristic
14C2×#14555487C411yescharacteristic
15C2×#14555487C2211yescharacteristic
16C2×C42×C137C411yescharacteristic
17#14555518—44no—
18C2×#11764047—24no—
19C23×D137C2×C411yescharacteristic
20C23×D137C2×C411yescharacteristic
21C2×D548—22no—
22C2×D548—22no—
23C2×D548C814yescomplemented
24C22×Dic137C2×C411yescharacteristic
25C22×Dic137C2311yescharacteristic
26C2×C4×D137—21no—
27C2×C4×D137—21no—
28C2×C4×D137—42no—
29C2×C4×D137—42no—
30C2×C4×D137D412yes—
31C2×C4×D137D412yes—
32C4⋊Dic137—22no—
33#14555487—22no—
34#14555487—22no—
35C22×C4×C137C2×C411yescharacteristic
36C22×C4×C137C2×C411yescharacteristic
37C22×C4×C137C2×C411yescharacteristic
38C42×C137—21no—
39C42×C137—21no—
40#11764047—44no—
41C22×D137—21no—
42C22×D137—21no—
43C22×D137—41no—
44C22×D137—41no—
45C22×D137—41no—
46C22×D137—41no—
47C22×D137C22⋊C412yes—
48C22×D137C2×C812yes—
49C22×D137C2×C812yescomplemented
50C22×D137M1612yes—
51D548—28no—
52C2×Dic137—21no—
53C2×Dic137—41no—
54C2×Dic137—41no—
55C2×Dic137C2×D412yes—
56C4×D137—24no—
57C4×D137—42no—
58C4×D137—42no—
59C4×D137—42no—
60C23×C137C22×C411yescharacteristic
61C2×C4×C137—21no—
62C2×C4×C137—21no—
63C2×C4×C137—22no—
64C2×C4×C137—21no—
65C2×C4×C137—41no—
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22⋊C411yescharacteristic
68C2×C4×C137C2×C812yescomplemented
69C22×C137—21no—
70C22×C137—21no—
71C22×C137C2×C22⋊C411yescharacteristic
72C22×C137C22×C811yescharacteristic
73C22×C137C2×M1611yescharacteristic
74C2×D137—22no—
75C2×D137—22no—
76C2×D137—22no—
77C2×D137—42no—
78C2×D137—42no—
79C2×D137—42no—
80C2×D137—41no—
81C2×D137—41no—
82C2×D137—42no—
83C2×D137C22⋊C814yes—
84C4×C137—22no—
85C4×C137—21no—
86C4×C137—41no—
87C4×C137#9912yes—
88Dic137—22no—
89Dic137—41no—
90C2×C137—21no—
91C2×C137—21no—
92C2×C137C2×C22⋊C811yescharacteristic
93C2×C137C2×#9911yescharacteristic
94C2×C137#41311yescharacteristic
95D137—24no—
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—1371no—
112C2×C42#1058773911yescharacteristic
113C22⋊C8—2742no—
114C22⋊C8—1374no—
115C24—1371no—
116C24—1371no—
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×C4C2×#1058773911yescharacteristic
129C22×C4C2×#1058773911yescharacteristic
130C22×C4C2×#1058773911yescharacteristic
131C2×C8—2744no—
132C2×C8—5484no—
133C4⋊C4—2742no—
134C22⋊C4—2742no—
135C22⋊C4—2742no—
136C42—21no—
137C42—21no—
138C23—2741no—
139C23—2741no—
140C23—5481no—
141C23—5481no—
142C23—5481no—
143C23—5481no—
144C23—1372no—
145C23—1372no—
146C23—1372no—
147C23—1372no—
148C23C22×#1058773911yescharacteristic
149D4—2748no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—2744no—
156C2×C4—2741no—
157C2×C4—5482no—
158C2×C4—5482no—
159C2×C4—5481no—
160C2×C4—5482no—
161C2×C4—5481no—
162C2×C4—1372no—
163C2×C4C22⋊#1058773911yescharacteristic
164C2×C4C22⋊#1058773911yescharacteristic
165C2×C4#1455551812yescomplemented
166C8—5484no—
167C22—21no—
168C22—21no—
169C22—2742no—
170C22—2742no—
171C22—2742no—
172C22—5482no—
173C22—5482no—
174C22—5482no—
175C22—5482no—
176C22—5481no—
177C22—5481no—
178C22—1374no—
179C22C2×C22⋊#1058773911yescharacteristic
180C22C2×#1455551811yescharacteristic
181C22D137⋊M1611yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—2742no—
186C4—5481no—
187C4#1798414712yes—
188C2—21no—
189C2—21no—
190C2—2744no—
191C2—5482no—
192C2#2257837111yescharacteristic
193C2C2×#1798414711yescharacteristic
194C2#2257838911yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10587739
commutatorC22×C137C22×C8
FrattiniC23C22×#10587739
FittingC2×C42×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D548C4
cosocleC22×Dic137C23
hypercenterC2×C42#10587739
hypocenterC137#1459
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #22578371C2semidirect
2. #22578248C2semidirect
3. #22578247C2semidirect
23. C2×D548C8semidirect
49. C22×D137C2×C8semidirect
68. C2×C4×C137C2×C8semidirect
97. C137#1459semidirect
165. C2×C4#14555518semidirect

Maximal subgroups

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

subgroupindex
#225783712
#225782482
#225782472
C2×C4×D5482
#1459137