Subgroups — C₂×C₄₇×#2070 #26063319

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#430	C₂	1	3	yes	maximal
2	C₂²×#101×C₄₇	C₂	1	3	yes	maximal
3	C₄₇×#2070	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇×#338	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₄₇	C₂²	1	3	yes	—
7	C₂²×M₁₆×C₄₇	C₂²	1	3	yes	—
8	C₄₇×#430	—	2	6	no	—
9	C₄₇×#430	C₂²	1	12	yes	—
10	C₂×#108×C₄₇	—	2	6	no	—
11	C₂×#101×C₄₇	—	2	6	no	—
12	C₂×#101×C₄₇	—	2	6	no	—
13	C₂×#101×C₄₇	C₂²	1	12	yes	—
14	C₄₇×#338	C₂²	1	4	yes	—
15	C₂²×Q₈×C₄₇	—	2	3	no	—
16	C₂²×Q₈×C₄₇	—	2	3	no	—
17	C₂²×Q₈×C₄₇	—	2	3	no	—
18	C₂²×Q₈×C₄₇	—	2	3	no	—
19	C₂²×Q₈×C₄₇	Q₈	1	2	yes	—
20	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
21	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₄₇	—	2	3	no	—
24	C₂×M₁₆×C₄₇	—	2	3	no	—
25	C₂×M₁₆×C₄₇	—	2	3	no	—
26	C₂×M₁₆×C₄₇	—	2	3	no	—
27	C₂×M₁₆×C₄₇	—	2	3	no	—
28	C₂×M₁₆×C₄₇	—	2	3	no	—
29	C₂×M₁₆×C₄₇	—	2	3	no	—
30	C₂×M₁₆×C₄₇	—	2	3	no	—
31	C₂×M₁₆×C₄₇	D₄	1	6	yes	—
32	C₂×M₁₆×C₄₇	D₄	1	6	yes	—
33	C₂×M₁₆×C₄₇	C₂³	1	6	yes	—
34	C₂×M₁₆×C₄₇	C₂³	1	6	yes	—
35	C₂²×C₈×C₄₇	—	2	3	no	—
36	C₂²×C₈×C₄₇	—	2	3	no	—
37	#108×C₄₇	—	2	24	no	—
38	#101×C₄₇	—	2	24	no	—
39	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×Q₈×C₄₇	—	2	6	no	—
41	C₂×Q₈×C₄₇	—	2	6	no	—
42	C₂×Q₈×C₄₇	—	2	6	no	—
43	C₂×Q₈×C₄₇	—	2	6	no	—
44	C₂×Q₈×C₄₇	—	4	3	no	—
45	C₂×Q₈×C₄₇	—	4	3	no	—
46	C₂×Q₈×C₄₇	—	4	3	no	—
47	C₂×Q₈×C₄₇	—	4	3	no	—
48	C₂×Q₈×C₄₇	—	4	3	no	—
49	C₂×Q₈×C₄₇	—	4	1	no	—
50	C₂²×C₄×C₄₇	—	2	6	no	—
51	C₂²×C₄×C₄₇	—	2	6	no	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	#41	1	3	yes	—
57	C₂²×C₄×C₄₇	#41	1	3	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₄₇	—	2	6	no	—
60	M₁₆×C₄₇	—	2	6	no	—
61	M₁₆×C₄₇	—	2	6	no	—
62	M₁₆×C₄₇	—	2	6	no	—
63	M₁₆×C₄₇	—	2	6	no	—
64	M₁₆×C₄₇	—	4	3	no	—
65	M₁₆×C₄₇	—	4	3	no	—
66	M₁₆×C₄₇	C₂×D₄	1	12	yes	—
67	C₂×C₈×C₄₇	—	2	6	no	—
68	C₂×C₈×C₄₇	—	2	6	no	—
69	C₂×C₈×C₄₇	—	2	6	no	—
70	C₂×C₈×C₄₇	—	2	6	no	—
71	C₂×C₈×C₄₇	—	4	3	no	—
72	C₂×C₈×C₄₇	—	4	3	no	—
73	C₂³×C₄₇	—	4	2	no	—
74	C₂³×C₄₇	C₂²⋊Q₈	1	3	yes	—
75	C₂³×C₄₇	C₂²⋊Q₈	1	3	yes	—
76	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
77	Q₈×C₄₇	—	4	6	no	—
78	Q₈×C₄₇	—	4	2	no	—
79	Q₈×C₄₇	—	4	6	no	—
80	Q₈×C₄₇	—	4	2	no	—
81	C₂×C₄×C₄₇	—	2	12	no	—
82	C₂×C₄×C₄₇	—	4	6	no	—
83	C₂×C₄×C₄₇	—	4	2	no	—
84	C₂×C₄×C₄₇	#121	1	3	yes	—
85	C₂×C₄×C₄₇	#121	1	3	yes	—
86	C₂×C₄×C₄₇	#121	1	3	yes	—
87	C₂×C₄×C₄₇	#121	1	3	yes	—
88	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
89	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
90	C₂×C₄×C₄₇	#126	1	3	yes	—
91	C₂×C₄×C₄₇	#126	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
93	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
94	C₈×C₄₇	—	2	12	no	—
95	C₈×C₄₇	—	4	6	no	—
96	C₂×#2070	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₄₇	—	2	6	no	—
98	C₂²×C₄₇	—	4	3	no	—
99	C₂²×C₄₇	—	4	1	no	—
100	C₂²×C₄₇	#396	1	3	yes	—
101	C₂²×C₄₇	#396	1	1	yes	characteristic
102	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	3	yes	—
103	C₄×C₄₇	—	4	4	no	—
104	C₄×C₄₇	#396	1	3	yes	—
105	C₄×C₄₇	#396	1	1	yes	characteristic
106	C₄×C₄₇	C₂×#121	1	3	yes	—
107	C₄×C₄₇	C₂×#126	1	1	yes	characteristic
108	C₂×#430	C₂×C₄₇	1	3	yes	—
109	C₂²×#101	C₂×C₄₇	1	3	yes	—
110	#2070	C₂×C₄₇	1	8	yes	direct product factor
111	C₂×#338	C₂×C₄₇	1	1	yes	characteristic
112	C₂×C₄₇	—	2	3	no	—
113	C₂×C₄₇	—	2	3	no	—
114	C₂×C₄₇	#2070	1	2	yes	direct product factor
115	C₂×C₄₇	C₂×#396	1	1	yes	characteristic
116	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
117	C₂²×M₁₆	C₂²×C₄₇	1	3	yes	—
118	C₂²×M₁₆	C₂²×C₄₇	1	3	yes	—
119	#430	—	2	6	no	—
120	#430	C₂²×C₄₇	1	12	yes	—
121	C₂×#108	—	2	6	no	—
122	C₂×#101	—	2	6	no	—
123	C₂×#101	—	2	6	no	—
124	C₂×#101	C₂²×C₄₇	1	12	yes	—
125	#338	C₂²×C₄₇	1	4	yes	—
126	C₄₇	C₂×#2070	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×Q₈	—	2	3	no	—
128	C₂²×Q₈	—	2	3	no	—
129	C₂²×Q₈	—	2	3	no	—
130	C₂²×Q₈	—	2	3	no	—
131	C₂²×Q₈	Q₈×C₄₇	1	2	yes	—
132	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
133	C₂³×C₄	Q₈×C₄₇	1	2	yes	—
134	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
135	C₂×M₁₆	—	2	3	no	—
136	C₂×M₁₆	—	2	3	no	—
137	C₂×M₁₆	—	2	3	no	—
138	C₂×M₁₆	—	2	3	no	—
139	C₂×M₁₆	—	2	3	no	—
140	C₂×M₁₆	—	2	3	no	—
141	C₂×M₁₆	—	2	3	no	—
142	C₂×M₁₆	—	2	3	no	—
143	C₂×M₁₆	D₄×C₄₇	1	6	yes	—
144	C₂×M₁₆	D₄×C₄₇	1	6	yes	—
145	C₂×M₁₆	C₂³×C₄₇	1	6	yes	—
146	C₂×M₁₆	C₂³×C₄₇	1	6	yes	—
147	C₂²×C₈	—	2	3	no	—
148	C₂²×C₈	—	2	3	no	—
149	#108	—	2	24	no	—
150	#101	—	2	24	no	—
151	C₂⁴	C₂×Q₈×C₄₇	1	1	yes	characteristic
152	C₂×Q₈	—	2	6	no	—
153	C₂×Q₈	—	2	6	no	—
154	C₂×Q₈	—	2	6	no	—
155	C₂×Q₈	—	2	6	no	—
156	C₂×Q₈	—	4	3	no	—
157	C₂×Q₈	—	4	3	no	—
158	C₂×Q₈	—	4	3	no	—
159	C₂×Q₈	—	4	3	no	—
160	C₂×Q₈	—	4	3	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂²×C₄	—	2	6	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	C₂×D₄×C₄₇	1	3	yes	—
165	C₂²×C₄	C₂×D₄×C₄₇	1	3	yes	—
166	C₂²×C₄	C₂×Q₈×C₄₇	1	4	yes	—
167	C₂²×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
168	C₂²×C₄	#41×C₄₇	1	3	yes	—
169	C₂²×C₄	#41×C₄₇	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
171	M₁₆	—	2	6	no	—
172	M₁₆	—	2	6	no	—
173	M₁₆	—	2	6	no	—
174	M₁₆	—	2	6	no	—
175	M₁₆	—	2	6	no	—
176	M₁₆	—	4	3	no	—
177	M₁₆	—	4	3	no	—
178	M₁₆	C₂×D₄×C₄₇	1	12	yes	—
179	C₂×C₈	—	2	6	no	—
180	C₂×C₈	—	2	6	no	—
181	C₂×C₈	—	2	6	no	—
182	C₂×C₈	—	2	6	no	—
183	C₂×C₈	—	4	3	no	—
184	C₂×C₈	—	4	3	no	—
185	C₂³	—	4	2	no	—
186	C₂³	C₂²⋊Q₈×C₄₇	1	3	yes	—
187	C₂³	C₂²⋊Q₈×C₄₇	1	3	yes	—
188	C₂³	C₂²×Q₈×C₄₇	1	1	yes	characteristic
189	Q₈	—	4	6	no	—
190	Q₈	—	4	2	no	—
191	Q₈	—	4	6	no	—
192	Q₈	—	4	2	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	4	6	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#121×C₄₇	1	3	yes	—
197	C₂×C₄	#121×C₄₇	1	3	yes	—
198	C₂×C₄	#121×C₄₇	1	3	yes	—
199	C₂×C₄	#121×C₄₇	1	3	yes	—
200	C₂×C₄	C₂²⋊Q₈×C₄₇	1	3	yes	—
201	C₂×C₄	C₂²⋊Q₈×C₄₇	1	3	yes	—
202	C₂×C₄	#126×C₄₇	1	3	yes	—
203	C₂×C₄	#126×C₄₇	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₄₇	1	3	yes	—
205	C₂×C₄	C₂×#41×C₄₇	1	3	yes	—
206	C₈	—	2	12	no	—
207	C₈	—	4	6	no	—
208	C₂²	—	2	6	no	—
209	C₂²	—	4	3	no	—
210	C₂²	—	4	1	no	—
211	C₂²	C₄₇×#396	1	3	yes	—
212	C₂²	C₄₇×#396	1	1	yes	characteristic
213	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	3	yes	—
214	C₄	—	4	4	no	—
215	C₄	C₄₇×#396	1	3	yes	—
216	C₄	C₄₇×#396	1	1	yes	characteristic
217	C₄	C₂×#121×C₄₇	1	3	yes	—
218	C₄	C₂×#126×C₄₇	1	1	yes	characteristic
219	C₂	—	2	3	no	—
220	C₂	—	2	3	no	—
221	C₂	C₄₇×#2070	1	2	yes	direct product factor
222	C₂	C₂×C₄₇×#396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂⁴×C₄₇	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₇×#2070	C₂	direct (N × H)
96. C₂×#2070	C₄₇	direct (N × H)
110. #2070	C₂×C₄₇	direct (N × H)
114. C₂×C₄₇	#2070	direct (N × H)
126. C₄₇	C₂×#2070	direct (N × H)
221. C₂	C₄₇×#2070	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×#430	2
C₂²×#101×C₄₇	2
C₄₇×#2070	2
C₂×C₄₇×#338	2
C₂×#2070	47

Subgroups — C2×C47×#2070 #26063319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2070 #26063319