Subgroups — C₄₇×#3341 #20240951

Order 6016.

364 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#555	C₂	1	2	yes	maximal, complemented
2	C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#482	C₂	1	4	yes	maximal, complemented
4	C₄₇×#476	C₂	1	2	yes	maximal, complemented
5	C₄₇×#466	C₂	1	2	yes	maximal, complemented
6	C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#427	C₂	1	2	yes	maximal, complemented
9	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
12	#124×C₄₇	—	2	4	no	—
13	#121×C₄₇	—	2	2	no	—
14	#121×C₄₇	C₂²	1	4	yes	—
15	C₄×D₄×C₄₇	C₂²	1	2	yes	—
16	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
17	#117×C₄₇	—	2	2	no	—
18	#117×C₄₇	C₂²	1	4	yes	—
19	#107×C₄₇	C₂²	1	2	yes	—
20	#107×C₄₇	C₂²	1	2	yes	—
21	#107×C₄₇	C₂²	1	1	yes	characteristic
22	#106×C₄₇	C₂²	1	4	yes	—
23	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
24	#102×C₄₇	C₂²	1	4	yes	—
25	#102×C₄₇	C₂²	1	2	yes	—
26	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	—	2	2	no	—
29	C₂×D₄×C₄₇	—	2	4	no	—
30	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂×D₄×C₄₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
34	D₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	4	no	—
36	C₂×C₈×C₄₇	C₂³	1	2	yes	—
37	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
42	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
43	C₂²⋊C₄×C₄₇	—	2	4	no	—
44	C₂²⋊C₄×C₄₇	—	2	4	no	—
45	C₂²⋊C₄×C₄₇	—	2	2	no	—
46	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
47	C₄²×C₄₇	—	2	4	no	—
48	C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	C₂×D₄	1	2	yes	—
51	Q₈×C₄₇	—	8	2	no	—
52	D₄×C₄₇	—	2	2	no	—
53	D₄×C₄₇	—	2	2	no	—
54	D₄×C₄₇	—	2	2	no	—
55	D₄×C₄₇	—	8	4	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₈×C₄₇	—	2	2	no	—
66	C₈×C₄₇	—	2	1	no	—
67	C₈×C₄₇	#41	1	2	yes	—
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	4	2	no	—
72	C₂²×C₄₇	—	4	2	no	—
73	C₂²×C₄₇	—	4	2	no	—
74	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	4	2	no	—
78	C₄×C₄₇	—	4	4	no	—
79	C₄×C₄₇	#135	1	2	yes	—
80	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
81	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
82	#3341	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₄₇	—	4	2	no	—
84	C₂×C₄₇	—	8	2	no	—
85	C₂×C₄₇	#547	1	1	yes	characteristic
86	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
87	C₂×C₄₇	#576	1	1	yes	characteristic
88	#555	C₂×C₄₇	1	2	yes	complemented
89	#500	C₂×C₄₇	1	1	yes	characteristic, complemented
90	#482	C₂×C₄₇	1	4	yes	complemented
91	#476	C₂×C₄₇	1	2	yes	complemented
92	#466	C₂×C₄₇	1	2	yes	complemented
93	C₄×D₈	C₂×C₄₇	1	1	yes	characteristic, complemented
94	C₈×D₄	C₂×C₄₇	1	1	yes	characteristic, complemented
95	#427	C₂×C₄₇	1	2	yes	complemented
96	C₄₇	#3341	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×D₈	C₂²×C₄₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₄₇	1	2	yes	—
99	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
100	#124	—	2	4	no	—
101	#121	—	2	2	no	—
102	#121	C₂²×C₄₇	1	4	yes	—
103	C₄×D₄	C₂²×C₄₇	1	2	yes	—
104	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	C₂²×C₄₇	1	4	yes	—
107	#107	C₂²×C₄₇	1	1	yes	characteristic
108	#107	C₂²×C₄₇	1	2	yes	—
109	#107	C₂²×C₄₇	1	2	yes	—
110	#106	C₂²×C₄₇	1	4	yes	—
111	C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
112	#102	C₂²×C₄₇	1	2	yes	—
113	#102	C₂²×C₄₇	1	4	yes	—
114	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
115	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
116	C₂×Q₈	—	2	2	no	—
117	C₂×D₄	—	2	4	no	—
118	C₂×D₄	D₄×C₄₇	1	1	yes	characteristic
119	C₂×D₄	C₂³×C₄₇	1	2	yes	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
122	D₈	—	2	2	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
125	C₂×C₈	C₂³×C₄₇	1	2	yes	—
126	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
129	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
130	C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
131	C₂²⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
135	C₄²	—	2	4	no	—
136	C₄²	C₂³×C₄₇	1	1	yes	characteristic
137	C₂³	—	2	2	no	—
138	C₂³	C₂×D₄×C₄₇	1	2	yes	—
139	Q₈	—	8	2	no	—
140	D₄	—	2	2	no	—
141	D₄	—	2	2	no	—
142	D₄	—	2	2	no	—
143	D₄	—	8	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
150	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
152	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	1	no	—
155	C₈	#41×C₄₇	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	#135×C₄₇	1	2	yes	—
168	C₄	C₂×#41×C₄₇	1	1	yes	characteristic
169	C₄	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
170	C₂	—	4	2	no	—
171	C₂	—	8	2	no	—
172	C₂	C₄₇×#547	1	1	yes	characteristic
173	C₂	C₂×#135×C₄₇	1	1	yes	characteristic
174	C₂	C₄₇×#576	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₄₇×#555	C₂	semidirect
2. C₄₇×#500	C₂	semidirect
3. C₄₇×#482	C₂	semidirect
4. C₄₇×#476	C₂	semidirect
5. C₄₇×#466	C₂	semidirect
6. C₄×D₈×C₄₇	C₂	semidirect
7. C₈×D₄×C₄₇	C₂	semidirect
8. C₄₇×#427	C₂	semidirect
82. #3341	C₄₇	direct (N × H)
88. #555	C₂×C₄₇	semidirect
89. #500	C₂×C₄₇	semidirect
90. #482	C₂×C₄₇	semidirect
91. #476	C₂×C₄₇	semidirect
92. #466	C₂×C₄₇	semidirect
93. C₄×D₈	C₂×C₄₇	semidirect
94. C₈×D₄	C₂×C₄₇	semidirect
95. #427	C₂×C₄₇	semidirect
96. C₄₇	#3341	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×#555	2
C₄₇×#500	2
C₄₇×#482	2
C₄₇×#476	2
C₄₇×#466	2
C₄×D₈×C₄₇	2
C₈×D₄×C₄₇	2
C₄₇×#427	2
#3341	47

Subgroups — C47×#3341 #20240951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3341 #20240951