Finite Group Databasebeta

← C2×#22578306 #31797039

Subgroups — C2×#22578306 #31797039

Order 17536.

278 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 58 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×#17984148C211yesmaximal, characteristic
2C2×#17984146C211yesmaximal, characteristic
3#22578306C214yesmaximal, direct product factor
4C2×#17984072C211yesmaximal, characteristic
5C2×#14555519—21no—
6C2×C22⋊#10587739—21no—
7#17984148—21no—
8#17984148C2212yes—
9#17984146—21no—
10#17984146C2212yes—
11C22×C22⋊D137C2211yescharacteristic
12C23×Dic137C411yescharacteristic, complemented
13#17984072—21no—
14#17984072—21no—
15#17984072C412yescomplemented
16#17984072C2212yes—
17C2×C4⋊Dic137—21no—
18C2×#14555487—21no—
19C2×C22⋊C4×C137C411yescharacteristic, complemented
20C22×#10587739—41no—
21#14555519—22no—
22#14555519—41no—
23C2×#11764047—41no—
24C22⋊#10587739—22no—
25C22⋊#10587739—41no—
26C23×D137D411yescharacteristic
27C2×C22⋊D137—41no—
28C2×C22⋊D137—41no—
29C2×C22⋊D137C2×C411yescharacteristic
30C2×C22⋊D137D411yescharacteristic
31C2×C22⋊D137D411yescharacteristic
32C2×C22⋊D137C2311yescharacteristic
33C22×Dic137—21no—
34C22×Dic137—21no—
35C22×Dic137—41no—
36C22×Dic137—41no—
37C22×Dic137C2×C412yescomplemented
38C22×Dic137D411yescharacteristic
39C4⋊Dic137—22no—
40C4⋊Dic137—22no—
41#14555487—22no—
42#14555487—22no—
43C24×C137C2×C411yescharacteristic
44C22×C4×C137—21no—
45C22⋊C4×C137—21no—
46C22⋊C4×C137C2×C412yescomplemented
47C2×#10587739—42no—
48C2×#10587739—82no—
49#11764047—42no—
50C22×D137—21no—
51C22×D137—21no—
52C22×D137—42no—
53C22×D137C22⋊C411yescharacteristic
54C22×D137C2×D411yescharacteristic
55C22⋊D137—41no—
56C22⋊D137—41no—
57C22⋊D137—41no—
58C22⋊D137—41no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137—22no—
62C2×Dic137—42no—
63C2×Dic137—41no—
64C2×Dic137—41no—
65C2×Dic137—42no—
66C2×Dic137C22⋊C411yescharacteristic
67C2×Dic137C2×D411yescharacteristic
68C23×C137—21no—
69C23×C137—21no—
70C23×C137—42no—
71C23×C137C22⋊C411yescharacteristic
72C23×C137C22⋊C411yescharacteristic
73C23×C137C22×C411yescharacteristic
74C2×C4×C137—22no—
75C2×C4×C137—42no—
76C22×C137—22no—
77C22×C137—21no—
78C22×C137—21no—
79C22×C137—41no—
80C22×C137—41no—
81C22×C137—42no—
82C22×C137—42no—
83C22×C137#9911yescharacteristic
84C22×C137#9911yescharacteristic
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—42no—
89C2×D137—41no—
90C2×D137—41no—
91C2×D137—81no—
92#10587739—82no—
93C4×C137—42no—
94Dic137—21no—
95Dic137—21no—
96Dic137—42no—
97C2×C137—21no—
98C2×C137—21no—
99C2×C137—41no—
100C2×C137—41no—
101C2×C137#35212yescomplemented
102C2×C137C2×#9911yescharacteristic
103D137—41no—
104D137—41no—
105C137C2×#35211yescharacteristic, complemented, Sylow
106C2×#352—1371nomaximal, Sylow
107C2×#123—1371no—
108C2×#100—1371no—
109C2×#99—1371no—
110#352—1374no—
111C22×D4—1371no—
112C23×C4—1371no—
113C2×M16—2741no—
114#123—2741no—
115#123—2741no—
116#123—1372no—
117#123—1372no—
118C2×C4⋊C4—2741no—
119C2×C22⋊C4—2741no—
120C2×C22⋊C4—2741no—
121C2×C22⋊C4#1058773911yescharacteristic, complemented
122#100—2741no—
123#100—1372no—
124#99—2741no—
125#99—1372no—
126C24—1371no—
127C24C2×#1058773911yescharacteristic
128C2×D4—5481no—
129C2×D4—5481no—
130C2×D4—1371no—
131C2×D4—1371no—
132C2×D4—1371no—
133C2×D4—1371no—
134C22×C4—21no—
135C22×C4—2741no—
136C22×C4—2741no—
137C22×C4—5481no—
138C22×C4—5481no—
139C22×C4—5481no—
140C22×C4—1371no—
141C22×C4—1372no—
142M16—2742no—
143M16—5481no—
144C2×C8—5481no—
145C4⋊C4—2742no—
146C4⋊C4—2742no—
147C22⋊C4—21no—
148C22⋊C4—2742no—
149C22⋊C4—2742no—
150C22⋊C4—2742no—
151C22⋊C4—5481no—
152C22⋊C4C2×#1058773912yescomplemented
153C23—21no—
154C23—21no—
155C23—42no—
156C23—2741no—
157C23—2741no—
158C23—5482no—
159C23—1371no—
160C23—1371no—
161C23C22⋊#1058773911yescharacteristic
162C23C22⋊#1058773911yescharacteristic
163C23C22×#1058773911yescharacteristic
164D4—5481no—
165D4—5481no—
166D4—5481no—
167D4—5481no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—2741no—
171C2×C4—2741no—
172C2×C4—2742no—
173C2×C4—5482no—
174C2×C4—5481no—
175C2×C4—5481no—
176C2×C4—5482no—
177C2×C4—5482no—
178C2×C4—10962no—
179C2×C4—1371no—
180C2×C4—1371no—
181C8—5482no—
182C22—22no—
183C22—21no—
184C22—21no—
185C22—41no—
186C22—41no—
187C22—42no—
188C22—42no—
189C22—2741no—
190C22—2741no—
191C22—5481no—
192C22—5481no—
193C22—5482no—
194C22—10961no—
195C22#1798414711yescharacteristic
196C22#1798414711yescharacteristic
197C22C2×C22⋊#1058773911yescharacteristic
198C4—42no—
199C4—2741no—
200C4—2741no—
201C4—5482no—
202C4—10962no—
203C2—21no—
204C2—21no—
205C2—41no—
206C2—41no—
207C2—5481no—
208C2—5481no—
209C2#2257830612yesdirect product factor
210C2C2×#1798414711yescharacteristic
subgroupG/H
centerC22#17984147
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137#99
prime radicalC22×C22⋊D137C22
cosocleC2×C22⋊D137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137C2×#352
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #22578306C2direct (N × H)
12. C23×Dic137C4semidirect
15. #17984072C4semidirect
19. C2×C22⋊C4×C137C4semidirect
37. C22×Dic137C2×C4semidirect
46. C22⋊C4×C137C2×C4semidirect
101. C2×C137#352semidirect
105. C137C2×#352semidirect
121. C2×C22⋊C4#10587739semidirect
152. C22⋊C4C2×#10587739semidirect
209. C2#22578306direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841482
C2×#179841462
#225783062
C2×#179840722
C2×#352137