Subgroups — C₂×C₄₇×#1420 #26053322

Order 12032.

286 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#368	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1420	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₄₇	—	2	1	no	—
6	C₂×Q₃₂×C₄₇	—	2	1	no	—
7	C₂×Q₃₂×C₄₇	—	2	1	no	—
8	C₂×Q₃₂×C₄₇	—	2	1	no	—
9	C₂×Q₃₂×C₄₇	C₄	1	2	yes	—
10	C₂×Q₃₂×C₄₇	C₄	1	2	yes	—
11	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₄₇	—	2	1	no	—
14	C₄₇×M₆₄	—	2	1	no	—
15	C₄₇×M₆₄	C₂²	1	2	yes	—
16	C₂×C₃₂×C₄₇	—	2	1	no	—
17	C₄₇×#368	—	2	1	no	—
18	C₄₇×#368	C₂²	1	2	yes	—
19	C₂²×Q₈×C₄₇	—	4	1	no	—
20	C₂×Q₁₆×C₄₇	—	2	2	no	—
21	C₂×Q₁₆×C₄₇	—	2	2	no	—
22	C₂×Q₁₆×C₄₇	—	2	2	no	—
23	C₂×Q₁₆×C₄₇	—	4	1	no	—
24	C₂×Q₁₆×C₄₇	—	4	1	no	—
25	C₂×Q₁₆×C₄₇	—	4	1	no	—
26	C₂×M₁₆×C₄₇	—	4	1	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	Q₃₂×C₄₇	—	2	2	no	—
29	Q₃₂×C₄₇	—	2	2	no	—
30	Q₃₂×C₄₇	—	2	2	no	—
31	Q₃₂×C₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
38	#108×C₄₇	—	2	2	no	—
39	#108×C₄₇	—	4	1	no	—
40	C₃₂×C₄₇	—	2	2	no	—
41	C₂×Q₈×C₄₇	—	4	2	no	—
42	C₂×Q₈×C₄₇	—	4	2	no	—
43	C₂×Q₈×C₄₇	—	4	2	no	—
44	C₂×Q₈×C₄₇	—	8	1	no	—
45	C₂×Q₈×C₄₇	—	8	1	no	—
46	C₂×Q₈×C₄₇	—	8	1	no	—
47	C₂²×C₄×C₄₇	—	8	1	no	—
48	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
49	Q₁₆×C₄₇	—	2	4	no	—
50	Q₁₆×C₄₇	—	4	2	no	—
51	Q₁₆×C₄₇	—	4	2	no	—
52	Q₁₆×C₄₇	—	4	2	no	—
53	M₁₆×C₄₇	—	4	2	no	—
54	M₁₆×C₄₇	—	8	1	no	—
55	C₂×C₈×C₄₇	—	8	1	no	—
56	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
59	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
60	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂³×C₄₇	SD₃₂	1	1	yes	characteristic
67	Q₈×C₄₇	—	4	4	no	—
68	Q₈×C₄₇	—	8	2	no	—
69	Q₈×C₄₇	—	8	2	no	—
70	Q₈×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	—	8	2	no	—
72	C₂×C₄×C₄₇	—	8	2	no	—
73	C₂×C₄×C₄₇	—	8	2	no	—
74	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
80	C₈×C₄₇	—	8	2	no	—
81	C₈×C₄₇	#102	1	1	yes	characteristic
82	C₈×C₄₇	#102	1	1	yes	characteristic
83	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×#1420	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	#357	1	1	yes	characteristic
88	C₂²×C₄₇	#357	1	1	yes	characteristic
89	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
90	C₄×C₄₇	—	8	4	no	—
91	C₄×C₄₇	#357	1	1	yes	characteristic
92	C₄×C₄₇	#357	1	1	yes	characteristic
93	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
94	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
95	C₂²×Q₃₂	C₂×C₄₇	1	1	yes	characteristic
96	C₂×M₆₄	C₂×C₄₇	1	1	yes	characteristic
97	C₂×#368	C₂×C₄₇	1	1	yes	characteristic
98	#1420	C₂×C₄₇	1	4	yes	direct product factor
99	C₂×C₄₇	—	2	1	no	—
100	C₂×C₄₇	—	2	1	no	—
101	C₂×C₄₇	#1420	1	2	yes	direct product factor
102	C₂×C₄₇	C₂×#357	1	1	yes	characteristic
103	C₂²×Q₁₆	—	2	1	no	—
104	C₂×Q₃₂	—	2	1	no	—
105	C₂×Q₃₂	—	2	1	no	—
106	C₂×Q₃₂	—	2	1	no	—
107	C₂×Q₃₂	C₄×C₄₇	1	2	yes	—
108	C₂×Q₃₂	C₄×C₄₇	1	2	yes	—
109	C₂×Q₃₂	C₂²×C₄₇	1	2	yes	—
110	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
111	C₂×#108	—	2	1	no	—
112	M₆₄	—	2	1	no	—
113	M₆₄	C₂²×C₄₇	1	2	yes	—
114	C₂×C₃₂	—	2	1	no	—
115	#368	—	2	1	no	—
116	#368	C₂²×C₄₇	1	2	yes	—
117	C₄₇	C₂×#1420	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×Q₈	—	4	1	no	—
119	C₂×Q₁₆	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	4	1	no	—
123	C₂×Q₁₆	—	4	1	no	—
124	C₂×Q₁₆	—	4	1	no	—
125	C₂×M₁₆	—	4	1	no	—
126	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
127	Q₃₂	—	2	2	no	—
128	Q₃₂	—	2	2	no	—
129	Q₃₂	—	2	2	no	—
130	Q₃₂	C₂×C₄×C₄₇	1	4	yes	—
131	C₂×C₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
132	C₂×C₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic
133	C₂×C₁₆	D₄×C₄₇	1	1	yes	characteristic
134	C₂×C₁₆	D₄×C₄₇	1	1	yes	characteristic
135	C₂×C₁₆	D₄×C₄₇	1	1	yes	characteristic
136	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
137	#108	—	2	2	no	—
138	#108	—	4	1	no	—
139	C₃₂	—	2	2	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	8	1	no	—
144	C₂×Q₈	—	8	1	no	—
145	C₂×Q₈	—	8	1	no	—
146	C₂²×C₄	—	8	1	no	—
147	C₂²×C₄	D₈×C₄₇	1	1	yes	characteristic
148	Q₁₆	—	2	4	no	—
149	Q₁₆	—	4	2	no	—
150	Q₁₆	—	4	2	no	—
151	Q₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	8	1	no	—
154	C₂×C₈	—	8	1	no	—
155	C₂×C₈	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
157	C₂×C₈	D₈×C₄₇	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₄₇	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₄₇	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
161	C₁₆	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
162	C₁₆	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
163	C₁₆	C₂²×C₄×C₄₇	1	1	yes	characteristic
164	C₁₆	C₂×D₄×C₄₇	1	1	yes	characteristic
165	C₂³	SD₃₂×C₄₇	1	1	yes	characteristic
166	Q₈	—	4	4	no	—
167	Q₈	—	8	2	no	—
168	Q₈	—	8	2	no	—
169	Q₈	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	#102×C₄₇	1	1	yes	characteristic
174	C₂×C₄	#102×C₄₇	1	1	yes	characteristic
175	C₂×C₄	D₁₆×C₄₇	1	1	yes	characteristic
176	C₂×C₄	D₁₆×C₄₇	1	1	yes	characteristic
177	C₂×C₄	SD₃₂×C₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
179	C₈	—	8	2	no	—
180	C₈	#102×C₄₇	1	1	yes	characteristic
181	C₈	#102×C₄₇	1	1	yes	characteristic
182	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
183	C₈	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
184	C₂²	—	2	2	no	—
185	C₂²	C₄₇×#357	1	1	yes	characteristic
186	C₂²	C₄₇×#357	1	1	yes	characteristic
187	C₂²	C₂×SD₃₂×C₄₇	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	C₄₇×#357	1	1	yes	characteristic
190	C₄	C₄₇×#357	1	1	yes	characteristic
191	C₄	C₂×#102×C₄₇	1	1	yes	characteristic
192	C₄	C₂×D₁₆×C₄₇	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₄₇×#1420	1	2	yes	direct product factor
196	C₂	C₂×C₄₇×#357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#357
commutator	C₁₆	C₂²×C₄×C₄₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₄₇
prime radical	C₂³×C₄₇	SD₃₂
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
4. C₄₇×#1420	C₂	direct (N × H)
85. C₂×#1420	C₄₇	direct (N × H)
98. #1420	C₂×C₄₇	direct (N × H)
101. C₂×C₄₇	#1420	direct (N × H)
117. C₄₇	C₂×#1420	direct (N × H)
195. C₂	C₄₇×#1420	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×Q₃₂×C₄₇	2
C₂×C₄₇×M₆₄	2
C₂×C₄₇×#368	2
C₄₇×#1420	2
C₂×#1420	47

Subgroups — C2×C47×#1420 #26053322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1420 #26053322