Subgroups — C₂×C₄₇×#2263 #26073677

Order 12032.

502 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2263	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇×SD₆₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×D₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₄₇	—	2	1	no	—
8	C₂²×D₈×C₄₇	—	2	1	no	—
9	C₄₇×#508	—	2	2	no	—
10	C₄₇×#508	C₂²	1	4	yes	complemented
11	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic, complemented
12	C₂×SD₃₂×C₄₇	—	2	1	no	—
13	C₂×D₁₆×C₄₇	—	2	1	no	—
14	C₂×D₁₆×C₄₇	—	2	2	no	—
15	C₂×D₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂×D₁₆×C₄₇	C₂²	1	4	yes	complemented
17	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×SD₆₄	C₂²	1	8	yes	complemented
20	C₄₇×D₃₂	C₂²	1	8	yes	complemented
21	C₄₇×M₆₄	C₂²	1	4	yes	complemented
22	C₂×C₃₂×C₄₇	C₂²	1	2	yes	complemented
23	C₂×#41×C₄₇	—	4	1	no	—
24	C₂²×D₄×C₄₇	—	4	1	no	—
25	#135×C₄₇	—	2	4	no	—
26	#135×C₄₇	—	4	2	no	—
27	C₂×Q₁₆×C₄₇	—	2	1	no	—
28	C₂×SD₁₆×C₄₇	—	4	1	no	—
29	C₂×D₈×C₄₇	—	2	2	no	—
30	C₂×D₈×C₄₇	—	2	4	no	—
31	C₂×D₈×C₄₇	—	2	1	no	—
32	C₂×D₈×C₄₇	—	4	1	no	—
33	C₂×D₈×C₄₇	—	4	2	no	—
34	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
35	Q₃₂×C₄₇	—	2	1	no	—
36	Q₃₂×C₄₇	C₂³	1	2	yes	complemented
37	SD₃₂×C₄₇	—	2	4	no	—
38	D₁₆×C₄₇	—	2	2	no	—
39	D₁₆×C₄₇	—	2	4	no	—
40	D₁₆×C₄₇	—	2	1	no	—
41	D₁₆×C₄₇	C₂³	1	4	yes	—
42	D₁₆×C₄₇	C₂³	1	2	yes	complemented
43	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
45	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
47	C₃₂×C₄₇	C₂³	1	4	yes	complemented
48	C₂⁴×C₄₇	—	8	1	no	—
49	#41×C₄₇	—	4	4	no	—
50	#41×C₄₇	—	8	2	no	—
51	C₂×Q₈×C₄₇	—	4	1	no	—
52	C₂×D₄×C₄₇	—	4	4	no	—
53	C₂×D₄×C₄₇	—	4	2	no	—
54	C₂×D₄×C₄₇	—	4	1	no	—
55	C₂×D₄×C₄₇	—	8	1	no	—
56	C₂×D₄×C₄₇	—	8	2	no	—
57	C₂×D₄×C₄₇	—	8	1	no	—
58	C₂²×C₄×C₄₇	—	8	1	no	—
59	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
60	Q₁₆×C₄₇	—	2	2	no	—
61	Q₁₆×C₄₇	—	4	1	no	—
62	SD₁₆×C₄₇	—	4	4	no	—
63	D₈×C₄₇	—	2	4	no	—
64	D₈×C₄₇	—	2	2	no	—
65	D₈×C₄₇	—	4	2	no	—
66	D₈×C₄₇	—	4	4	no	—
67	D₈×C₄₇	—	4	1	no	—
68	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
69	C₂×C₈×C₄₇	D₈	1	2	yes	—
70	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
72	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
74	C₁₆×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂³×C₄₇	—	8	4	no	—
76	C₂³×C₄₇	—	8	1	no	—
77	C₂³×C₄₇	—	8	2	no	—
78	C₂³×C₄₇	—	16	4	no	—
79	C₂³×C₄₇	D₁₆	1	1	yes	characteristic
80	Q₈×C₄₇	—	4	2	no	—
81	Q₈×C₄₇	—	8	1	no	—
82	D₄×C₄₇	—	4	4	no	—
83	D₄×C₄₇	—	4	2	no	—
84	D₄×C₄₇	—	8	4	no	—
85	D₄×C₄₇	—	8	4	no	—
86	D₄×C₄₇	—	8	2	no	—
87	D₄×C₄₇	—	8	1	no	—
88	C₂×C₄×C₄₇	—	8	1	no	—
89	C₂×C₄×C₄₇	—	8	4	no	—
90	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
91	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
94	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
95	C₈×C₄₇	C₂×D₈	1	2	yes	—
96	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×#2263	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₄₇	—	2	2	no	—
99	C₂²×C₄₇	—	8	2	no	—
100	C₂²×C₄₇	—	8	4	no	—
101	C₂²×C₄₇	—	16	8	no	—
102	C₂²×C₄₇	—	16	2	no	—
103	C₂²×C₄₇	—	16	4	no	—
104	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
105	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
106	C₄×C₄₇	—	8	2	no	—
107	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
108	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
109	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
110	C₂×#508	C₂×C₄₇	1	1	yes	characteristic, complemented
111	C₂²×D₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
112	#2263	C₂×C₄₇	1	8	yes	direct product factor
113	C₂×SD₆₄	C₂×C₄₇	1	2	yes	complemented
114	C₂×D₃₂	C₂×C₄₇	1	2	yes	complemented
115	C₂×M₆₄	C₂×C₄₇	1	1	yes	characteristic, complemented
116	C₂×C₄₇	—	2	2	no	—
117	C₂×C₄₇	—	16	2	no	—
118	C₂×C₄₇	—	16	4	no	—
119	C₂×C₄₇	#2263	1	2	yes	direct product factor
120	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
121	C₂×#135	—	2	1	no	—
122	C₂²×D₈	—	2	1	no	—
123	#508	—	2	2	no	—
124	#508	C₂²×C₄₇	1	4	yes	complemented
125	C₂×Q₃₂	C₂²×C₄₇	1	1	yes	characteristic, complemented
126	C₂×SD₃₂	—	2	1	no	—
127	C₂×D₁₆	—	2	1	no	—
128	C₂×D₁₆	—	2	2	no	—
129	C₂×D₁₆	C₂²×C₄₇	1	2	yes	—
130	C₂×D₁₆	C₂²×C₄₇	1	4	yes	complemented
131	C₂×D₁₆	C₂²×C₄₇	1	1	yes	characteristic, complemented
132	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
133	SD₆₄	C₂²×C₄₇	1	8	yes	complemented
134	D₃₂	C₂²×C₄₇	1	8	yes	complemented
135	M₆₄	C₂²×C₄₇	1	4	yes	complemented
136	C₂×C₃₂	C₂²×C₄₇	1	2	yes	complemented
137	C₄₇	C₂×#2263	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#41	—	4	1	no	—
139	C₂²×D₄	—	4	1	no	—
140	#135	—	2	4	no	—
141	#135	—	4	2	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×SD₁₆	—	4	1	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	—	4	1	no	—
148	C₂×D₈	—	4	2	no	—
149	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
150	Q₃₂	—	2	1	no	—
151	Q₃₂	C₂³×C₄₇	1	2	yes	complemented
152	SD₃₂	—	2	4	no	—
153	D₁₆	—	2	2	no	—
154	D₁₆	—	2	4	no	—
155	D₁₆	—	2	1	no	—
156	D₁₆	C₂³×C₄₇	1	4	yes	—
157	D₁₆	C₂³×C₄₇	1	2	yes	complemented
158	C₂×C₁₆	D₄×C₄₇	1	1	yes	characteristic
159	C₂×C₁₆	D₄×C₄₇	1	2	yes	—
160	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
162	C₃₂	C₂³×C₄₇	1	4	yes	complemented
163	C₂⁴	—	8	1	no	—
164	#41	—	4	4	no	—
165	#41	—	8	2	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	1	no	—
170	C₂×D₄	—	8	1	no	—
171	C₂×D₄	—	8	1	no	—
172	C₂×D₄	—	8	2	no	—
173	C₂²×C₄	—	8	1	no	—
174	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
175	Q₁₆	—	2	2	no	—
176	Q₁₆	—	4	1	no	—
177	SD₁₆	—	4	4	no	—
178	D₈	—	2	4	no	—
179	D₈	—	2	2	no	—
180	D₈	—	4	2	no	—
181	D₈	—	4	4	no	—
182	D₈	—	4	1	no	—
183	C₂×C₈	D₈×C₄₇	1	1	yes	characteristic
184	C₂×C₈	D₈×C₄₇	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
187	C₁₆	C₂×D₄×C₄₇	1	1	yes	characteristic
188	C₁₆	C₂×D₄×C₄₇	1	2	yes	—
189	C₁₆	C₂⁴×C₄₇	1	1	yes	characteristic
190	C₂³	—	8	1	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	16	4	no	—
194	C₂³	D₁₆×C₄₇	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	D₁₆×C₄₇	1	1	yes	characteristic
206	C₂×C₄	D₁₆×C₄₇	1	2	yes	—
207	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
209	C₈	C₂×D₈×C₄₇	1	1	yes	characteristic
210	C₈	C₂×D₈×C₄₇	1	2	yes	—
211	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	4	no	—
215	C₂²	—	16	8	no	—
216	C₂²	—	16	4	no	—
217	C₂²	—	16	2	no	—
218	C₂²	C₂×D₁₆×C₄₇	1	2	yes	—
219	C₂²	C₂×D₁₆×C₄₇	1	1	yes	characteristic
220	C₄	—	8	2	no	—
221	C₄	C₂×D₁₆×C₄₇	1	2	yes	—
222	C₄	C₂×D₁₆×C₄₇	1	1	yes	characteristic
223	C₄	C₂²×D₈×C₄₇	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	16	4	no	—
226	C₂	—	16	2	no	—
227	C₂	C₄₇×#2263	1	2	yes	direct product factor
228	C₂	C₂²×D₁₆×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₄₇
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂²×C₈×C₄₇ ◁ C₂²×C₁₆×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#508	C₂	semidirect
2. C₂²×D₁₆×C₄₇	C₂	semidirect
3. C₄₇×#2263	C₂	direct (N × H)
4. C₂×C₄₇×SD₆₄	C₂	semidirect
5. C₂×C₄₇×D₃₂	C₂	semidirect
6. C₂×C₄₇×M₆₄	C₂	semidirect
10. C₄₇×#508	C₂²	semidirect
11. C₂×Q₃₂×C₄₇	C₂²	semidirect
16. C₂×D₁₆×C₄₇	C₂²	semidirect
17. C₂×D₁₆×C₄₇	C₂²	semidirect
19. C₄₇×SD₆₄	C₂²	semidirect
20. C₄₇×D₃₂	C₂²	semidirect
21. C₄₇×M₆₄	C₂²	semidirect
22. C₂×C₃₂×C₄₇	C₂²	semidirect
36. Q₃₂×C₄₇	C₂³	semidirect
42. D₁₆×C₄₇	C₂³	semidirect
47. C₃₂×C₄₇	C₂³	semidirect
97. C₂×#2263	C₄₇	direct (N × H)
110. C₂×#508	C₂×C₄₇	semidirect
111. C₂²×D₁₆	C₂×C₄₇	semidirect
112. #2263	C₂×C₄₇	direct (N × H)
113. C₂×SD₆₄	C₂×C₄₇	semidirect
114. C₂×D₃₂	C₂×C₄₇	semidirect
115. C₂×M₆₄	C₂×C₄₇	semidirect
119. C₂×C₄₇	#2263	direct (N × H)
124. #508	C₂²×C₄₇	semidirect
125. C₂×Q₃₂	C₂²×C₄₇	semidirect
130. C₂×D₁₆	C₂²×C₄₇	semidirect
131. C₂×D₁₆	C₂²×C₄₇	semidirect
133. SD₆₄	C₂²×C₄₇	semidirect
134. D₃₂	C₂²×C₄₇	semidirect
135. M₆₄	C₂²×C₄₇	semidirect
136. C₂×C₃₂	C₂²×C₄₇	semidirect
137. C₄₇	C₂×#2263	direct (N × H)
151. Q₃₂	C₂³×C₄₇	semidirect
157. D₁₆	C₂³×C₄₇	semidirect
162. C₃₂	C₂³×C₄₇	semidirect
227. C₂	C₄₇×#2263	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×C₄₇×#508	2
C₂²×D₁₆×C₄₇	2
C₄₇×#2263	2
C₂×C₄₇×SD₆₄	2
C₂×C₄₇×D₃₂	2
C₂×C₄₇×M₆₄	2
C₂×#2263	47

Subgroups — C2×C47×#2263 #26073677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2263 #26073677