Subgroups — C₄₇×#3230 #20240840

Order 6016.

360 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#500	C₂	1	2	yes	maximal, complemented
4	C₄₇×#487	C₂	1	2	yes	maximal, complemented
5	C₄₇×#484	C₂	1	4	yes	maximal
6	C₄₇×#477	C₂	1	4	yes	maximal, complemented
7	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₄₇	—	2	1	no	—
9	C₄⋊Q₈×C₄₇	—	2	1	no	—
10	C₄⋊Q₈×C₄₇	—	2	1	no	—
11	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	#125×C₄₇	—	2	1	no	—
16	C₂²⋊Q₈×C₄₇	—	2	1	no	—
17	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
18	C₄×Q₈×C₄₇	—	2	1	no	—
19	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
20	#117×C₄₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	#107×C₄₇	C₂²	1	8	yes	—
25	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
26	#103×C₄₇	C₂²	1	8	yes	—
27	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	C₂×Q₈×C₄₇	—	2	1	no	—
30	C₂×Q₈×C₄₇	—	2	2	no	—
31	C₂×Q₈×C₄₇	—	2	2	no	—
32	C₂×Q₈×C₄₇	—	4	4	no	—
33	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₇	—	2	2	no	—
35	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂³	1	4	yes	—
39	C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
49	C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₄²×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₄₇	—	2	4	no	—
57	Q₈×C₄₇	—	4	4	no	—
58	Q₈×C₄₇	—	4	2	no	—
59	Q₈×C₄₇	—	4	4	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	4	no	—
65	C₂×C₄×C₄₇	—	4	4	no	—
66	C₂×C₄×C₄₇	Q₁₆	1	4	yes	—
67	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₈×C₄₇	—	2	4	no	—
74	C₂²×C₄₇	—	2	1	no	—
75	C₂²×C₄₇	—	2	1	no	—
76	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
77	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₄₇	—	2	1	no	—
79	C₄×C₄₇	—	2	1	no	—
80	C₄×C₄₇	—	4	2	no	—
81	C₄×C₄₇	—	4	4	no	—
82	C₄×C₄₇	—	4	2	no	—
83	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
84	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
85	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
86	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
87	#3230	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₄₇	—	2	2	no	—
89	C₂×C₄₇	#536	1	1	yes	characteristic
90	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
91	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
92	#533	C₂×C₄₇	1	1	yes	characteristic
93	C₂×C₄⋊Q₈	C₂×C₄₇	1	1	yes	characteristic
94	#500	C₂×C₄₇	1	2	yes	complemented
95	#487	C₂×C₄₇	1	2	yes	complemented
96	#484	C₂×C₄₇	1	4	yes	—
97	#477	C₂×C₄₇	1	4	yes	complemented
98	#431	C₂×C₄₇	1	1	yes	characteristic
99	C₄₇	#3230	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²×Q₈	—	2	1	no	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	2	1	no	—
103	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
104	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
105	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
106	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
107	#125	—	2	1	no	—
108	C₂²⋊Q₈	—	2	1	no	—
109	C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
110	C₄×Q₈	—	2	1	no	—
111	C₄×Q₈	C₂²×C₄₇	1	2	yes	—
112	#117	—	2	1	no	—
113	C₂×C₄⋊C₄	—	2	1	no	—
114	C₂×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
115	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
116	#107	C₂²×C₄₇	1	8	yes	—
117	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
118	#103	C₂²×C₄₇	1	8	yes	—
119	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
120	C₄×C₈	C₂²×C₄₇	1	2	yes	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	—	4	4	no	—
125	C₂×Q₈	C₂³×C₄₇	1	2	yes	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
128	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
129	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
130	C₂×C₈	C₂³×C₄₇	1	4	yes	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
140	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	D₄×C₄₇	1	1	yes	characteristic
144	C₄²	D₄×C₄₇	1	1	yes	characteristic
145	C₄²	C₂³×C₄₇	1	1	yes	characteristic
146	C₄²	C₂³×C₄₇	1	1	yes	characteristic
147	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
148	Q₈	—	2	4	no	—
149	Q₈	—	4	4	no	—
150	Q₈	—	4	2	no	—
151	Q₈	—	4	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	Q₁₆×C₄₇	1	4	yes	—
159	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
160	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
163	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
164	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	C₂×Q₁₆×C₄₇	1	2	yes	—
169	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	4	no	—
174	C₄	—	4	2	no	—
175	C₄	C₂×Q₁₆×C₄₇	1	2	yes	—
176	C₄	C₂×Q₁₆×C₄₇	1	2	yes	—
177	C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
178	C₄	2₋¹⁺⁴×C₄₇	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₄₇×#536	1	1	yes	characteristic
181	C₂	C₂²×Q₁₆×C₄₇	1	1	yes	characteristic
182	C₂	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₇
prime radical	C₂³×C₄₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₇×#500	C₂	semidirect
4. C₄₇×#487	C₂	semidirect
6. C₄₇×#477	C₂	semidirect
87. #3230	C₄₇	direct (N × H)
94. #500	C₂×C₄₇	semidirect
95. #487	C₂×C₄₇	semidirect
97. #477	C₂×C₄₇	semidirect
99. C₄₇	#3230	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#533	2
C₂×C₄⋊Q₈×C₄₇	2
C₄₇×#500	2
C₄₇×#487	2
C₄₇×#484	2
C₄₇×#477	2
C₄₇×#431	2
#3230	47

Subgroups — C47×#3230 #20240840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3230 #20240840