Subgroups — C₄₇×#2985 #20240595

Order 6016.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 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	C₄₇×#446	C₂	1	4	yes	maximal, complemented
3	C₄₇×#446	C₂	1	2	yes	maximal, complemented
4	C₈×Q₈×C₄₇	C₂	1	2	yes	maximal, complemented
5	C₄₇×#433	C₂	1	4	yes	maximal
6	C₄₇×#432	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#431	C₂	1	1	yes	maximal, characteristic
8	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
10	#125×C₄₇	C₄	1	2	yes	—
11	C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
12	C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
13	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
15	#117×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
20	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
21	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
23	#97×C₄₇	C₂²	1	2	yes	—
24	#97×C₄₇	C₂²	1	4	yes	—
25	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂²	1	4	yes	—
28	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂³	1	2	yes	—
34	C₂×C₈×C₄₇	C₂³	1	2	yes	—
35	C₂×C₈×C₄₇	C₂³	1	4	yes	—
36	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
39	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₄²×C₄₇	C₂³	1	4	yes	—
48	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₄₇	—	2	4	no	—
50	Q₈×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₈×C₄₇	—	2	2	no	—
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	4	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	—	2	4	no	—
75	C₄×C₄₇	#131	1	2	yes	—
76	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
77	#2985	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₄₇	—	4	1	no	—
79	C₂×C₄₇	#519	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
81	C₂×C₄₇	#568	1	1	yes	characteristic
82	#533	C₂×C₄₇	1	1	yes	characteristic
83	#446	C₂×C₄₇	1	2	yes	complemented
84	#446	C₂×C₄₇	1	4	yes	complemented
85	C₈×Q₈	C₂×C₄₇	1	2	yes	complemented
86	#433	C₂×C₄₇	1	4	yes	—
87	#432	C₂×C₄₇	1	1	yes	characteristic
88	#431	C₂×C₄₇	1	1	yes	characteristic
89	C₄₇	#2985	1	1	yes	characteristic, direct product factor, Sylow
90	C₄⋊Q₈	C₄×C₄₇	1	1	yes	characteristic
91	C₄⋊Q₈	C₄×C₄₇	1	1	yes	characteristic
92	#125	C₄×C₄₇	1	2	yes	—
93	C₂²⋊Q₈	C₄×C₄₇	1	2	yes	—
94	C₂²⋊Q₈	C₄×C₄₇	1	2	yes	—
95	C₄×Q₈	C₂²×C₄₇	1	2	yes	—
96	C₄×Q₈	C₂²×C₄₇	1	2	yes	—
97	#117	C₂²×C₄₇	1	2	yes	—
98	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
99	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
100	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
101	C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
102	C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
103	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
104	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
105	#97	C₂²×C₄₇	1	2	yes	—
106	#97	C₂²×C₄₇	1	4	yes	—
107	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₄₇	1	4	yes	—
110	C₂×Q₈	C₂×C₄×C₄₇	1	2	yes	—
111	C₂×Q₈	C₂×C₄×C₄₇	1	2	yes	—
112	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
114	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₄₇	1	2	yes	—
116	C₂×C₈	C₂³×C₄₇	1	2	yes	—
117	C₂×C₈	C₂³×C₄₇	1	4	yes	—
118	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
119	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
120	C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
121	C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
122	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
123	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
124	C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
125	C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
126	C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
127	C₄²	C₂³×C₄₇	1	1	yes	characteristic
128	C₄²	C₂³×C₄₇	1	4	yes	—
129	C₄²	C₂³×C₄₇	1	1	yes	characteristic
130	C₂³	C₂²×C₄×C₄₇	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
138	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
140	C₂×C₄	C₂²×C₄×C₄₇	1	4	yes	—
141	C₂×C₄	C₂²×C₄×C₄₇	1	4	yes	—
142	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	2	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂³×C₄×C₄₇	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	4	no	—
157	C₄	#131×C₄₇	1	2	yes	—
158	C₄	2₋¹⁺⁴×C₄₇	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	C₄₇×#519	1	1	yes	characteristic
161	C₂	C₂×#131×C₄₇	1	1	yes	characteristic
162	C₂	C₄₇×#568	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄₇×#446	C₂	semidirect
3. C₄₇×#446	C₂	semidirect
4. C₈×Q₈×C₄₇	C₂	semidirect
77. #2985	C₄₇	direct (N × H)
83. #446	C₂×C₄₇	semidirect
84. #446	C₂×C₄₇	semidirect
85. C₈×Q₈	C₂×C₄₇	semidirect
89. C₄₇	#2985	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
C₄₇×#446	2
C₈×Q₈×C₄₇	2
C₄₇×#433	2
C₄₇×#432	2
C₄₇×#431	2
#2985	47

Subgroups — C47×#2985 #20240595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#2985 #20240595