Subgroups — C₂×#131×D₁₈₁ #37548424

Order 23168.

1504 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₈₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153824	C₂	1	3	yes	maximal, complemented
3	C₂×#19153823	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19153822	C₂	1	3	yes	maximal, complemented
5	#131×D₁₈₁	C₂	1	16	yes	maximal, direct product factor
6	C₂×M₁₆×D₁₈₁	C₂	1	3	yes	maximal, complemented
7	C₂²×C₈×D₁₈₁	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668450	C₄	1	1	yes	characteristic
10	C₂×#15668449	C₂²	1	3	yes	—
11	C₂×#15668448	C₄	1	3	yes	—
12	#41×D₁₈₁	C₄	1	8	yes	—
13	#41×D₁₈₁	C₂²	1	8	yes	—
14	C₂×Q₈×D₁₈₁	C₄	1	1	yes	characteristic
15	C₂×D₄×D₁₈₁	C₄	1	3	yes	—
16	C₂²×C₄×D₁₈₁	C₂²	1	3	yes	—
17	#19153824	C₂²	1	24	yes	complemented
18	#19153823	C₂²	1	8	yes	complemented
19	#19153822	C₂²	1	24	yes	complemented
20	C₂×#15668433	C₂²	1	3	yes	complemented
21	C₂×#15668432	C₂²	1	6	yes	complemented
22	M₁₆×D₁₈₁	C₂²	1	24	yes	complemented
23	C₂²×#13089678	C₂²	1	3	yes	complemented
24	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₁₈₁	C₂²	1	3	yes	—
26	C₂×C₈×D₁₈₁	C₂²	1	3	yes	complemented
27	C₂×C₈×D₁₈₁	C₂²	1	3	yes	complemented
28	C₂×C₈×D₁₈₁	C₂²	1	24	yes	complemented
29	C₂×#41×C₁₈₁	C₂²	1	1	yes	characteristic
30	#131×C₁₈₁	C₂²	1	8	yes	complemented
31	C₂×M₁₆×C₁₈₁	C₂²	1	3	yes	complemented
32	C₂²×C₈×C₁₈₁	C₂²	1	3	yes	complemented
33	C₂³×D₁₈₁	C₂×C₄	1	3	yes	—
34	#15668450	C₂×C₄	1	8	yes	—
35	#15668449	C₂×C₄	1	12	yes	—
36	#15668449	C₂³	1	12	yes	—
37	#15668448	C₂×C₄	1	24	yes	—
38	C₂×Dic₃₆₂	C₂×C₄	1	3	yes	—
39	Q₈×D₁₈₁	C₂×C₄	1	8	yes	—
40	C₂×D₇₂₄	C₂×C₄	1	3	yes	—
41	C₂×C₂²⋊D₁₈₁	C₂×C₄	1	6	yes	—
42	D₄×D₁₈₁	C₂×C₄	1	24	yes	—
43	C₂²×Dic₁₈₁	C₂×C₄	1	3	yes	—
44	C₂×C₄×D₁₈₁	C₂×C₄	1	3	yes	—
45	C₂×C₄×D₁₈₁	C₂×C₄	1	12	yes	—
46	C₂×C₄×D₁₈₁	C₂×C₄	1	3	yes	—
47	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₈₁	C₂³	1	12	yes	—
49	C₂×C₄×D₁₈₁	C₂³	1	3	yes	—
50	#15668433	C₂³	1	12	yes	complemented
51	#15668432	C₂³	1	24	yes	complemented
52	C₂×#13089678	C₂³	1	1	yes	characteristic
53	C₂×#13089678	C₂³	1	3	yes	—
54	C₂×#13089678	C₂³	1	12	yes	complemented
55	C₈×D₁₈₁	C₂³	1	4	yes	—
56	C₈×D₁₈₁	C₂³	1	12	yes	—
57	C₈×D₁₈₁	C₂³	1	12	yes	complemented
58	C₈×D₁₈₁	C₂³	1	12	yes	complemented
59	#41×C₁₈₁	C₂×C₄	1	4	yes	—
60	#41×C₁₈₁	C₂³	1	4	yes	—
61	C₂×Q₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
62	C₂×D₄×C₁₈₁	C₂×C₄	1	3	yes	—
63	C₂²×C₄×C₁₈₁	C₂³	1	3	yes	—
64	M₁₆×C₁₈₁	C₂³	1	12	yes	complemented
65	C₂×C₈×C₁₈₁	C₂³	1	3	yes	—
66	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₈₁	C₂³	1	12	yes	complemented
68	C₂²×D₁₈₁	—	2	12	no	—
69	C₂²×D₁₈₁	C₂²×C₄	1	12	yes	—
70	C₂²×D₁₈₁	C₂²×C₄	1	3	yes	—
71	C₂²×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
72	Dic₃₆₂	C₂²×C₄	1	12	yes	—
73	D₇₂₄	C₂²×C₄	1	12	yes	—
74	C₂²⋊D₁₈₁	C₂²×C₄	1	24	yes	—
75	C₂×Dic₁₈₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₈₁	C₂²×C₄	1	3	yes	—
77	C₂×Dic₁₈₁	C₂²×C₄	1	12	yes	—
78	C₄×D₁₈₁	C₂²×C₄	1	2	yes	—
79	C₄×D₁₈₁	C₂²×C₄	1	12	yes	—
80	C₄×D₁₈₁	C₂²×C₄	1	12	yes	—
81	C₄×D₁₈₁	C₂²×C₄	1	6	yes	—
82	C₄×D₁₈₁	C₂⁴	1	2	yes	—
83	C₄×D₁₈₁	C₂⁴	1	6	yes	—
84	#13089678	C₂⁴	1	2	yes	—
85	#13089678	C₂⁴	1	6	yes	—
86	C₂³×C₁₈₁	C₂²×C₄	1	3	yes	—
87	Q₈×C₁₈₁	C₂²×C₄	1	4	yes	—
88	D₄×C₁₈₁	C₂²×C₄	1	12	yes	—
89	C₂×C₄×C₁₈₁	C₂²×C₄	1	3	yes	—
90	C₂×C₄×C₁₈₁	C₂²×C₄	1	6	yes	—
91	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₈₁	C₂⁴	1	6	yes	—
93	C₈×C₁₈₁	C₂⁴	1	2	yes	—
94	C₈×C₁₈₁	C₂⁴	1	6	yes	—
95	C₂²×C₁₈₁	—	2	6	no	—
96	C₂²×C₁₈₁	C₂³×C₄	1	6	yes	—
97	C₂²×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₈₁	—	2	6	no	—
99	C₂×D₁₈₁	—	2	24	no	—
100	C₂×D₁₈₁	#131	1	4	yes	direct product factor
101	C₂×D₁₈₁	C₂³×C₄	1	6	yes	—
102	C₂×D₁₈₁	C₂³×C₄	1	2	yes	—
103	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₁₈₁	C₂³×C₄	1	6	yes	—
105	C₄×C₁₈₁	C₂⁵	1	1	yes	characteristic
106	Dic₁₈₁	C₂³×C₄	1	2	yes	—
107	Dic₁₈₁	C₂³×C₄	1	6	yes	—
108	C₂×C₁₈₁	—	2	6	no	—
109	C₂×C₁₈₁	C₂×#131	1	2	yes	complemented
110	C₂×C₁₈₁	C₂⁴×C₄	1	1	yes	characteristic
111	D₁₈₁	—	2	6	no	—
112	D₁₈₁	C₂×#131	1	4	yes	direct product factor
113	C₁₈₁	C₂²×#131	1	1	yes	characteristic, complemented, Sylow
114	C₂²×#131	—	181	1	no	maximal, Sylow
115	C₂²×#41	—	181	1	no	—
116	C₂×#131	—	181	1	no	—
117	C₂×#131	—	181	16	no	—
118	C₂×#131	—	181	3	no	—
119	C₂×#131	—	181	3	no	—
120	C₂×#131	D₁₈₁	1	1	yes	characteristic, direct product factor
121	C₂²×M₁₆	—	181	3	no	—
122	C₂³×C₈	—	181	3	no	—
123	C₂×#41	—	181	8	no	—
124	C₂×#41	—	181	8	no	—
125	C₂×#41	—	181	1	no	—
126	C₂×#41	—	181	3	no	—
127	C₂×#41	—	181	3	no	—
128	C₂×#41	C₂×D₁₈₁	1	1	yes	characteristic
129	C₂²×Q₈	—	181	1	no	—
130	C₂²×D₄	—	181	3	no	—
131	C₂³×C₄	—	181	3	no	—
132	#131	—	181	8	no	—
133	#131	—	181	24	no	—
134	#131	—	181	24	no	—
135	#131	C₂×D₁₈₁	1	8	yes	direct product factor
136	C₂×M₁₆	—	181	3	no	—
137	C₂×M₁₆	—	181	24	no	—
138	C₂×M₁₆	—	181	6	no	—
139	C₂×M₁₆	C₂×D₁₈₁	1	3	yes	complemented
140	C₂²×C₈	—	181	3	no	—
141	C₂²×C₈	—	181	3	no	—
142	C₂²×C₈	—	181	3	no	—
143	C₂²×C₈	—	181	1	no	—
144	C₂²×C₈	—	181	3	no	—
145	C₂²×C₈	—	181	24	no	—
146	C₂²×C₈	C₂×D₁₈₁	1	3	yes	complemented
147	C₂⁴	—	181	3	no	—
148	#41	—	181	8	no	—
149	#41	—	181	24	no	—
150	#41	—	181	12	no	—
151	#41	—	181	12	no	—
152	#41	C₄×D₁₈₁	1	4	yes	—
153	#41	C₂²×D₁₈₁	1	4	yes	—
154	C₂×Q₈	—	181	8	no	—
155	C₂×Q₈	—	181	3	no	—
156	C₂×Q₈	C₄×D₁₈₁	1	1	yes	characteristic
157	C₂×D₄	—	181	6	no	—
158	C₂×D₄	—	181	24	no	—
159	C₂×D₄	—	181	3	no	—
160	C₂×D₄	C₄×D₁₈₁	1	3	yes	—
161	C₂²×C₄	—	181	3	no	—
162	C₂²×C₄	—	181	3	no	—
163	C₂²×C₄	—	181	3	no	—
164	C₂²×C₄	—	181	12	no	—
165	C₂²×C₄	—	181	1	no	—
166	C₂²×C₄	—	181	3	no	—
167	C₂²×C₄	—	181	12	no	—
168	C₂²×C₄	C₂²×D₁₈₁	1	3	yes	—
169	M₁₆	—	181	12	no	—
170	M₁₆	—	181	24	no	—
171	M₁₆	C₂²×D₁₈₁	1	12	yes	complemented
172	C₂×C₈	—	181	1	no	—
173	C₂×C₈	—	181	3	no	—
174	C₂×C₈	—	181	12	no	—
175	C₂×C₈	—	181	4	no	—
176	C₂×C₈	—	181	12	no	—
177	C₂×C₈	—	181	12	no	—
178	C₂×C₈	—	181	12	no	—
179	C₂×C₈	C₂²×D₁₈₁	1	3	yes	—
180	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₈₁	1	12	yes	complemented
182	C₂³	—	362	12	no	—
183	C₂³	—	181	12	no	—
184	C₂³	—	181	3	no	—
185	C₂³	—	181	1	no	—
186	C₂³	C₂×C₄×D₁₈₁	1	3	yes	—
187	Q₈	—	181	12	no	—
188	Q₈	C₂×C₄×D₁₈₁	1	4	yes	—
189	D₄	—	181	24	no	—
190	D₄	—	181	12	no	—
191	D₄	C₂×C₄×D₁₈₁	1	12	yes	—
192	C₂×C₄	—	181	1	no	—
193	C₂×C₄	—	181	3	no	—
194	C₂×C₄	—	181	12	no	—
195	C₂×C₄	—	181	2	no	—
196	C₂×C₄	—	181	12	no	—
197	C₂×C₄	—	181	12	no	—
198	C₂×C₄	—	181	6	no	—
199	C₂×C₄	—	181	2	no	—
200	C₂×C₄	—	181	6	no	—
201	C₂×C₄	C₂×C₄×D₁₈₁	1	3	yes	—
202	C₂×C₄	C₂×C₄×D₁₈₁	1	6	yes	—
203	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
204	C₂×C₄	C₂³×D₁₈₁	1	6	yes	—
205	C₈	—	181	2	no	—
206	C₈	—	181	6	no	—
207	C₈	C₂³×D₁₈₁	1	2	yes	—
208	C₈	C₂³×D₁₈₁	1	6	yes	—
209	C₂²	—	2	6	no	—
210	C₂²	—	362	24	no	—
211	C₂²	—	362	6	no	—
212	C₂²	—	181	6	no	—
213	C₂²	—	181	4	no	—
214	C₂²	—	181	2	no	—
215	C₂²	C₂²×C₄×D₁₈₁	1	6	yes	—
216	C₂²	C₂²×C₄×D₁₈₁	1	1	yes	characteristic
217	C₄	—	181	2	no	—
218	C₄	—	181	6	no	—
219	C₄	C₂²×C₄×D₁₈₁	1	1	yes	characteristic
220	C₄	C₂²×C₄×D₁₈₁	1	6	yes	—
221	C₄	C₂⁴×D₁₈₁	1	1	yes	characteristic
222	C₂	—	2	6	no	—
223	C₂	—	362	6	no	—
224	C₂	—	181	4	no	—
225	C₂	#131×D₁₈₁	1	2	yes	direct product factor
226	C₂	C₂³×C₄×D₁₈₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₈₁
commutator	C₂×C₁₈₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₈₁
Fitting	C₂×#131×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂³×C₄
prime radical	C₂×#41×D₁₈₁	C₂
cosocle	C₄×C₁₈₁	C₂⁵
hypercenter	C₂×#131	D₁₈₁
hypocenter	C₁₈₁	C₂²×#131
LCS	G ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#19153824	C₂	semidirect
3. C₂×#19153823	C₂	semidirect
4. C₂×#19153822	C₂	semidirect
5. #131×D₁₈₁	C₂	direct (N × H)
6. C₂×M₁₆×D₁₈₁	C₂	semidirect
7. C₂²×C₈×D₁₈₁	C₂	semidirect
8. C₂×#131×C₁₈₁	C₂	semidirect
17. #19153824	C₂²	semidirect
18. #19153823	C₂²	semidirect
19. #19153822	C₂²	semidirect
20. C₂×#15668433	C₂²	semidirect
21. C₂×#15668432	C₂²	semidirect
22. M₁₆×D₁₈₁	C₂²	semidirect
23. C₂²×#13089678	C₂²	semidirect
26. C₂×C₈×D₁₈₁	C₂²	semidirect
27. C₂×C₈×D₁₈₁	C₂²	semidirect
28. C₂×C₈×D₁₈₁	C₂²	semidirect
30. #131×C₁₈₁	C₂²	semidirect
31. C₂×M₁₆×C₁₈₁	C₂²	semidirect
32. C₂²×C₈×C₁₈₁	C₂²	semidirect
50. #15668433	C₂³	semidirect
51. #15668432	C₂³	semidirect
54. C₂×#13089678	C₂³	semidirect
57. C₈×D₁₈₁	C₂³	semidirect
58. C₈×D₁₈₁	C₂³	semidirect
64. M₁₆×C₁₈₁	C₂³	semidirect
67. C₂×C₈×C₁₈₁	C₂³	semidirect
100. C₂×D₁₈₁	#131	direct (N × H)
109. C₂×C₁₈₁	C₂×#131	semidirect
112. D₁₈₁	C₂×#131	direct (N × H)
113. C₁₈₁	C₂²×#131	semidirect
120. C₂×#131	D₁₈₁	direct (N × H)
135. #131	C₂×D₁₈₁	direct (N × H)
139. C₂×M₁₆	C₂×D₁₈₁	semidirect
146. C₂²×C₈	C₂×D₁₈₁	semidirect
171. M₁₆	C₂²×D₁₈₁	semidirect
181. C₂×C₈	C₂²×D₁₈₁	semidirect
225. C₂	#131×D₁₈₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₈₁	2
C₂×#19153824	2
C₂×#19153823	2
C₂×#19153822	2
#131×D₁₈₁	2
C₂×M₁₆×D₁₈₁	2
C₂²×C₈×D₁₈₁	2
C₂×#131×C₁₈₁	2
C₂²×#131	181

Subgroups — C2×#131×D181 #37548424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#131×D₁₈₁ #37548424