Subgroups — C₂×C₄₇×#2215 #26073295

Order 12032.

806 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₇×#466	C₂	1	2	yes	maximal, complemented
4	C₄₇×#2215	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×#121×C₄₇	—	2	2	no	—
10	C₂×D₁₆×C₄₇	—	2	4	no	—
11	C₂×D₁₆×C₄₇	C₂²	1	4	yes	complemented
12	C₂²×C₁₆×C₄₇	—	2	4	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₄₇	C₂²	1	4	yes	complemented
16	C₄₇×#466	—	2	4	no	—
17	C₄₇×#466	C₂²	1	8	yes	complemented
18	C₂×#107×C₄₇	C₂²	1	2	yes	—
19	C₂×#102×C₄₇	—	2	2	no	—
20	C₄₇×#366	C₂²	1	4	yes	complemented
21	C₄₇×#357	C₂²	1	8	yes	complemented
22	C₂²×D₄×C₄₇	—	2	2	no	—
23	C₂²×D₄×C₄₇	—	4	2	no	—
24	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂×D₈×C₄₇	—	2	4	no	—
26	C₂×D₈×C₄₇	—	2	2	no	—
27	C₂×D₈×C₄₇	—	2	4	no	—
28	C₂×D₈×C₄₇	C₂³	1	4	yes	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	D₄	1	2	yes	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	#121×C₄₇	—	2	8	no	—
36	#121×C₄₇	—	4	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	4	2	no	—
39	D₁₆×C₄₇	—	2	8	no	—
40	C₂×C₁₆×C₄₇	—	2	2	no	—
41	C₂×C₁₆×C₄₇	—	2	8	no	—
42	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
44	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
45	#107×C₄₇	—	2	2	no	—
46	#107×C₄₇	C₂³	1	4	yes	—
47	#102×C₄₇	—	2	8	no	—
48	C₂⁴×C₄₇	—	4	2	no	—
49	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
50	C₂×D₄×C₄₇	—	2	4	no	—
51	C₂×D₄×C₄₇	—	4	8	no	—
52	C₂×D₄×C₄₇	—	4	4	no	—
53	C₂×D₄×C₄₇	—	4	2	no	—
54	C₂×D₄×C₄₇	—	4	4	no	—
55	C₂×D₄×C₄₇	—	8	8	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	D₈	1	2	yes	—
60	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	D₈×C₄₇	—	2	4	no	—
64	D₈×C₄₇	—	2	4	no	—
65	D₈×C₄₇	—	4	4	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₇	#41	1	2	yes	—
74	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	2	no	—
77	C₂²⋊C₄×C₄₇	—	4	8	no	—
78	C₁₆×C₄₇	—	2	2	no	—
79	C₁₆×C₄₇	C₂×D₄	1	4	yes	complemented
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	4	4	no	—
83	C₂³×C₄₇	—	8	4	no	—
84	C₂³×C₄₇	—	8	4	no	—
85	C₂³×C₄₇	—	8	4	no	—
86	C₂³×C₄₇	D₁₆	1	4	yes	complemented
87	C₂³×C₄₇	C₂×D₈	1	2	yes	—
88	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
89	D₄×C₄₇	—	4	4	no	—
90	D₄×C₄₇	—	4	4	no	—
91	D₄×C₄₇	—	8	16	no	—
92	D₄×C₄₇	—	8	4	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	8	4	no	—
99	C₂×C₄×C₄₇	#121	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#135	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₄₇	—	2	2	no	—
105	C₈×C₄₇	#121	1	2	yes	—
106	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×#2215	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₄₇	—	2	2	no	—
110	C₂²×C₄₇	—	2	8	no	—
111	C₂²×C₄₇	—	2	2	no	—
112	C₂²×C₄₇	—	8	4	no	—
113	C₂²×C₄₇	—	8	4	no	—
114	C₂²×C₄₇	—	8	4	no	—
115	C₂²×C₄₇	—	16	8	no	—
116	C₂²×C₄₇	#466	1	2	yes	—
117	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
118	C₂²×C₄₇	C₂×D₁₆	1	4	yes	complemented
119	C₂²×C₄₇	#508	1	2	yes	—
120	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	8	4	no	—
123	C₄×C₄₇	#466	1	2	yes	—
124	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
125	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
126	C₂²×D₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
127	C₂³×C₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
128	C₂×#466	C₂×C₄₇	1	2	yes	complemented
129	#2215	C₂×C₄₇	1	8	yes	direct product factor
130	C₂×#366	C₂×C₄₇	1	1	yes	characteristic, complemented
131	C₂×#357	C₂×C₄₇	1	2	yes	complemented
132	C₂×C₄₇	—	2	4	no	—
133	C₂×C₄₇	—	16	4	no	—
134	C₂×C₄₇	#2215	1	4	yes	direct product factor
135	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
136	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
138	C₂²×D₈	C₂²×C₄₇	1	2	yes	—
139	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
140	C₂×#121	—	2	2	no	—
141	C₂×D₁₆	—	2	4	no	—
142	C₂×D₁₆	C₂²×C₄₇	1	4	yes	complemented
143	C₂²×C₁₆	—	2	4	no	—
144	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
145	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
146	C₂²×C₁₆	C₂²×C₄₇	1	4	yes	complemented
147	#466	—	2	4	no	—
148	#466	C₂²×C₄₇	1	8	yes	complemented
149	C₂×#107	C₂²×C₄₇	1	2	yes	—
150	C₂×#102	—	2	2	no	—
151	#366	C₂²×C₄₇	1	4	yes	complemented
152	#357	C₂²×C₄₇	1	8	yes	complemented
153	C₄₇	C₂×#2215	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×D₄	—	2	2	no	—
155	C₂²×D₄	—	4	2	no	—
156	C₂³×C₄	D₄×C₄₇	1	1	yes	characteristic
157	C₂×D₈	—	2	4	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	—	2	4	no	—
160	C₂×D₈	C₂³×C₄₇	1	4	yes	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
164	C₂²×C₈	D₄×C₄₇	1	2	yes	—
165	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
166	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
167	#121	—	2	8	no	—
168	#121	—	4	4	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	4	2	no	—
171	D₁₆	—	2	8	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	8	no	—
174	C₂×C₁₆	D₄×C₄₇	1	4	yes	complemented
175	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
176	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
177	#107	—	2	2	no	—
178	#107	C₂³×C₄₇	1	4	yes	—
179	#102	—	2	8	no	—
180	C₂⁴	—	4	2	no	—
181	C₂⁴	D₈×C₄₇	1	1	yes	characteristic
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	4	8	no	—
184	C₂×D₄	—	4	4	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	—	8	8	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
192	C₂²×C₄	D₈×C₄₇	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
194	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
195	D₈	—	2	4	no	—
196	D₈	—	2	4	no	—
197	D₈	—	4	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
205	C₂×C₈	#41×C₄₇	1	2	yes	—
206	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	C₂×D₄×C₄₇	1	4	yes	complemented
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	4	no	—
216	C₂³	—	8	4	no	—
217	C₂³	—	8	4	no	—
218	C₂³	D₁₆×C₄₇	1	4	yes	complemented
219	C₂³	C₂×D₈×C₄₇	1	2	yes	—
220	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
221	D₄	—	4	4	no	—
222	D₄	—	4	4	no	—
223	D₄	—	8	16	no	—
224	D₄	—	8	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	#121×C₄₇	1	2	yes	—
232	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
234	C₂×C₄	#135×C₄₇	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#121×C₄₇	1	2	yes	—
238	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
239	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₄₇×#466	1	2	yes	—
248	C₂²	C₂×D₁₆×C₄₇	1	2	yes	—
249	C₂²	C₂×D₁₆×C₄₇	1	4	yes	complemented
250	C₂²	C₄₇×#508	1	2	yes	—
251	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	C₄₇×#466	1	2	yes	—
255	C₄	C₂×#121×C₄₇	1	1	yes	characteristic
256	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	16	4	no	—
259	C₂	C₄₇×#2215	1	4	yes	direct product factor
260	C₂	C₂×C₄₇×#466	1	1	yes	characteristic
261	C₂	C₂²×D₁₆×C₄₇	1	1	yes	characteristic
262	C₂	C₂×C₄₇×#508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ C₂³×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₆×C₄₇	C₂	semidirect
2. C₂³×C₁₆×C₄₇	C₂	semidirect
3. C₂×C₄₇×#466	C₂	semidirect
4. C₄₇×#2215	C₂	direct (N × H)
5. C₂×C₄₇×#366	C₂	semidirect
6. C₂×C₄₇×#357	C₂	semidirect
11. C₂×D₁₆×C₄₇	C₂²	semidirect
15. C₂²×C₁₆×C₄₇	C₂²	semidirect
17. C₄₇×#466	C₂²	semidirect
20. C₄₇×#366	C₂²	semidirect
21. C₄₇×#357	C₂²	semidirect
42. C₂×C₁₆×C₄₇	D₄	semidirect
79. C₁₆×C₄₇	C₂×D₄	semidirect
86. C₂³×C₄₇	D₁₆	semidirect
108. C₂×#2215	C₄₇	direct (N × H)
118. C₂²×C₄₇	C₂×D₁₆	semidirect
126. C₂²×D₁₆	C₂×C₄₇	semidirect
127. C₂³×C₁₆	C₂×C₄₇	semidirect
128. C₂×#466	C₂×C₄₇	semidirect
129. #2215	C₂×C₄₇	direct (N × H)
130. C₂×#366	C₂×C₄₇	semidirect
131. C₂×#357	C₂×C₄₇	semidirect
134. C₂×C₄₇	#2215	direct (N × H)
142. C₂×D₁₆	C₂²×C₄₇	semidirect
146. C₂²×C₁₆	C₂²×C₄₇	semidirect
148. #466	C₂²×C₄₇	semidirect
151. #366	C₂²×C₄₇	semidirect
152. #357	C₂²×C₄₇	semidirect
153. C₄₇	C₂×#2215	direct (N × H)
174. C₂×C₁₆	D₄×C₄₇	semidirect
211. C₁₆	C₂×D₄×C₄₇	semidirect
218. C₂³	D₁₆×C₄₇	semidirect
249. C₂²	C₂×D₁₆×C₄₇	semidirect
259. C₂	C₄₇×#2215	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₄₇	2
C₂³×C₁₆×C₄₇	2
C₂×C₄₇×#466	2
C₄₇×#2215	2
C₂×C₄₇×#366	2
C₂×C₄₇×#357	2
C₂×#2215	47

Subgroups — C2×C47×#2215 #26073295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2215 #26073295