Finite Group Databasebeta

← #353×D181 #37535377

Subgroups — #353×D181 #37535377

Order 23168.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D181C211yesmaximal, characteristic
2#99×D181C212yesmaximal, complemented
3#26041072C211yesmaximal, characteristic, complemented
4#26041071C212yesmaximal, complemented
5#353×C181C211yesmaximal, characteristic, complemented
6C2×D4×D181—22no—
7C2×D4×D181C411yescharacteristic, complemented
8C2×D4×D181C2211yescharacteristic
9#19153813C411yescharacteristic, complemented
10#19153812—22no—
11#19153812C411yescharacteristic, complemented
12#19153812C2211yescharacteristic
13C22⋊C4×D181—22no—
14C42×D181C411yescharacteristic, complemented
15#19153731C2212yescomplemented
16#19153730—22no—
17#127×C181C2211yescharacteristic
18#99×C181C2212yescomplemented
19C23×D181—21no—
20C23×D181D412yes—
21C2×D724—22no—
22C2×D724C2×C411yescharacteristic
23C2×D724C2×C411yescharacteristic, complemented
24C2×C22⋊D181—22no—
25C2×C22⋊D181—21no—
26C2×C22⋊D181—22no—
27C2×C22⋊D181D412yes—
28D4×D181—41no—
29D4×D181—41no—
30D4×D181—41no—
31D4×D181—41no—
32D4×D181—48no—
33C2×C4×D181—21no—
34C2×C4×D181—42no—
35C2×C4×D181C2×C411yescharacteristic
36C22⋊Dic181—22no—
37#15668426—42no—
38C4×Dic181—21no—
39C4×Dic181C2×C411yescharacteristic, complemented
40C2×D4×C181—22no—
41C2×D4×C181C2×C411yescharacteristic, complemented
42C2×D4×C181C2311yescharacteristic
43C22⋊C4×C181—22no—
44C42×C181C2×C411yescharacteristic, complemented
45C22×D181—22no—
46C22×D181—22no—
47C22×D181—21no—
48C22×D181—44no—
49C22×D181—44no—
50C22×D181—41no—
51C22×D181—41no—
52C22×D181C22⋊C411yescharacteristic
53C22×D181C22⋊C412yes—
54D724—41no—
55D724—41no—
56D724—44no—
57C22⋊D181—42no—
58C22⋊D181—44no—
59C22⋊D181—44no—
60C22⋊D181—42no—
61C2×Dic181—21no—
62C2×Dic181—42no—
63C2×Dic181C22⋊C411yescharacteristic
64C4×D181—24no—
65C4×D181—21no—
66C4×D181—21no—
67C4×D181—84no—
68C23×C181—21no—
69C23×C181C2×D412yes—
70D4×C181—41no—
71D4×C181—41no—
72D4×C181—44no—
73C2×C4×C181—21no—
74C2×C4×C181—42no—
75C2×C4×C181C22×C411yescharacteristic
76C22×C181—22no—
77C22×C181—42no—
78C22×C181—42no—
79C22×C181—41no—
80C22×C181C2×C22⋊C411yescharacteristic
81C2×D181—22no—
82C2×D181—22no—
83C2×D181—44no—
84C2×D181—42no—
85C2×D181—42no—
86C2×D181—41no—
87C2×D181—82no—
88C2×D181—82no—
89C2×D181—82no—
90C2×D181#9911yescharacteristic
91C2×D181#9911yescharacteristic
92C4×C181—22no—
93C4×C181—21no—
94C4×C181—82no—
95Dic181—22no—
96Dic181—21no—
97Dic181—82no—
98C2×C181—21no—
99C2×C181—42no—
100C2×C181—81no—
101C2×C181C2×#9911yescharacteristic
102D181—21no—
103D181—42no—
104D181—81no—
105D181#35312yesdirect product factor
106C181C2×#35311yescharacteristic, complemented, Sylow
107C2×#353—1811nomaximal, Sylow
108C2×#127—1811no—
109C2×#99—1812no—
110#353—1811no—
111#353—1812no—
112#353D18111yescharacteristic, direct product factor
113C22×D4—3622no—
114C22×D4—1811no—
115C22×D4—1811no—
116#127—3622no—
117#127—1811no—
118#127—1811no—
119#127—1811no—
120#127C2×D18111yescharacteristic
121C2×C22⋊C4—3622no—
122C2×C42—1811no—
123#99—3622no—
124#99—1812no—
125#99C2×D18112yescomplemented
126C24—3621no—
127C24—1812no—
128C2×D4—22no—
129C2×D4—3622no—
130C2×D4—3622no—
131C2×D4—3622no—
132C2×D4—3621no—
133C2×D4—1812no—
134C2×D4—1811no—
135C2×D4—1811no—
136C2×D4—7241no—
137C2×D4—7241no—
138C2×D4—7241no—
139C2×D4—7241no—
140C2×D4—7248no—
141C2×D4C4×D18111yescharacteristic, complemented
142C2×D4C22×D18111yescharacteristic
143C22×C4—3621no—
144C22×C4—1811no—
145C22×C4—7242no—
146C22⋊C4—22no—
147C22⋊C4—3622no—
148C22⋊C4—7242no—
149C42—3621no—
150C42—1811no—
151C42C4×D18111yescharacteristic, complemented
152C23—21no—
153C23—3622no—
154C23—3622no—
155C23—3621no—
156C23—1811no—
157C23—1812no—
158C23—7244no—
159C23—7244no—
160C23—7241no—
161C23—7241no—
162C23D4×D18112yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—7241no—
167D4—7241no—
168D4—7244no—
169D4—7242no—
170D4—7244no—
171D4—7244no—
172D4—7242no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—3624no—
176C2×C4—3621no—
177C2×C4—3621no—
178C2×C4—3621no—
179C2×C4—14484no—
180C2×C4—1811no—
181C2×C4—7242no—
182C2×C4C2×C4×D18111yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—3622no—
188C22—3622no—
189C22—14482no—
190C22—14482no—
191C22—14482no—
192C22—1811no—
193C22—1811no—
194C22—7242no—
195C22—7242no—
196C22—7244no—
197C22—7241no—
198C22C22⋊C4×D18111yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—3622no—
203C4—3621no—
204C4—14482no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—3621no—
209C2—14481no—
210C2—1812no—
211C2—7242no—
212C2#99×D18111yescharacteristic
subgroupG/H
centerC2#99×D181
commutatorC2×C4×C181C22×C4
FrattiniC2×D4C22×D181
Fitting#353×C181C2
socleC2×C181C2×#99
prime radical#127×D181C2
cosocleC2×D4×C181C23
hypercenter#353D181
hypocenterC181C2×#353
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D181C2semidirect
3. #26041072C2semidirect
4. #26041071C2semidirect
5. #353×C181C2semidirect
7. C2×D4×D181C4semidirect
9. #19153813C4semidirect
11. #19153812C4semidirect
14. C42×D181C4semidirect
15. #19153731C22semidirect
18. #99×C181C22semidirect
23. C2×D724C2×C4semidirect
39. C4×Dic181C2×C4semidirect
41. C2×D4×C181C2×C4semidirect
44. C42×C181C2×C4semidirect
105. D181#353direct (N × H)
106. C181C2×#353semidirect
112. #353D181direct (N × H)
125. #99C2×D181semidirect
141. C2×D4C4×D181semidirect
151. C42C4×D181semidirect

Maximal subgroups

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

subgroupindex
#127×D1812
#99×D1812
#260410722
#260410712
#353×C1812
C2×#353181