Subgroups — C₄₁×#8260 #23784461

Order 10496.

502 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	C₄×C₄⋊C₈×C₄₁	C₂	1	2	yes	maximal
2	C₄²×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#322	C₂	1	2	yes	maximal
4	C₄₁×#1273	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₄₁	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₄₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₄³×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#336	C₄	1	4	yes	—
13	C₄₁×#322	—	2	4	no	—
14	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	—
15	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₄₁	—	2	8	no	—
23	C₄×C₈×C₄₁	—	2	4	no	—
24	C₄×C₈×C₄₁	—	2	4	no	—
25	C₄×C₈×C₄₁	—	2	2	no	—
26	C₄×C₈×C₄₁	—	2	4	no	—
27	C₄×C₈×C₄₁	C₈	1	8	yes	complemented
28	C₄×C₈×C₄₁	D₄	1	2	yes	—
29	C₄×C₈×C₄₁	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₁	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₁	C₄²	1	2	yes	—
32	C₂²×C₄×C₄₁	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₄₁	—	2	2	no	—
36	C₂×C₈×C₄₁	—	2	2	no	—
37	C₂×C₈×C₄₁	—	2	8	no	—
38	C₂×C₈×C₄₁	—	2	8	no	—
39	C₂×C₈×C₄₁	—	2	8	no	—
40	C₂×C₈×C₄₁	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₄₁	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₄₁	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₄₁	M₁₆	1	4	yes	—
44	C₄²×C₄₁	—	2	2	no	—
45	C₄²×C₄₁	—	2	4	no	—
46	C₄²×C₄₁	C₂×C₈	1	4	yes	—
47	C₄²×C₄₁	M₁₆	1	8	yes	—
48	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₄₁	#41	1	2	yes	—
51	C₂³×C₄₁	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₄₁	—	2	2	no	—
53	C₂×C₄×C₄₁	—	2	2	no	—
54	C₂×C₄×C₄₁	—	2	2	no	—
55	C₂×C₄×C₄₁	C₄×C₈	1	2	yes	—
56	C₂×C₄×C₄₁	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₄₁	#97	1	4	yes	—
58	C₂×C₄×C₄₁	#97	1	4	yes	—
59	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
61	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₄₁	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₄₁	C₂×M₁₆	1	4	yes	—
66	C₂×C₄×C₄₁	#131	1	4	yes	—
67	C₈×C₄₁	—	2	4	no	—
68	C₈×C₄₁	—	4	8	no	—
69	C₈×C₄₁	C₄⋊C₈	1	8	yes	complemented
70	#8260	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂²×C₄₁	C₂×#97	1	2	yes	—
74	C₂²×C₄₁	C₄×M₁₆	1	1	yes	characteristic
75	C₂²×C₄₁	#405	1	2	yes	—
76	C₄×C₄₁	—	2	2	no	—
77	C₄×C₄₁	—	2	4	no	—
78	C₄×C₄₁	#322	1	4	yes	complemented
79	C₄×C₄₁	C₂×C₄⋊C₈	1	2	yes	—
80	C₄×C₄₁	#431	1	2	yes	—
81	C₄×C₄₁	#436	1	4	yes	—
82	C₄×C₄₁	#446	1	4	yes	—
83	C₄×C₄⋊C₈	C₂×C₄₁	1	2	yes	—
84	C₄²×C₈	C₂×C₄₁	1	1	yes	characteristic
85	C₂×#322	C₂×C₄₁	1	2	yes	—
86	#1273	C₂×C₄₁	1	2	yes	—
87	C₂×C₄₁	C₂×#322	1	1	yes	characteristic
88	C₂×C₄₁	#1450	1	1	yes	characteristic
89	C₂×C₄₁	#1451	1	2	yes	—
90	C₂×C₄₁	C₄×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₄₁	#1770	1	2	yes	—
92	C₂×C₄⋊C₈	C₄×C₄₁	1	4	yes	—
93	C₂×C₄×C₈	C₄×C₄₁	1	2	yes	—
94	C₂×C₄×C₈	C₄×C₄₁	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
96	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
98	C₄³	C₂²×C₄₁	1	1	yes	characteristic
99	#336	C₄×C₄₁	1	4	yes	—
100	#322	—	2	4	no	—
101	C₄₁	#8260	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	—
103	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
104	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
105	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
109	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
110	C₄⋊C₈	—	2	8	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	4	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	4	no	—
115	C₄×C₈	C₈×C₄₁	1	8	yes	complemented
116	C₄×C₈	D₄×C₄₁	1	2	yes	—
117	C₄×C₈	Q₈×C₄₁	1	2	yes	—
118	C₂²×C₄	C₄²×C₄₁	1	1	yes	characteristic
119	C₂²×C₄	C₄²×C₄₁	1	2	yes	—
120	C₂²×C₄	C₄²×C₄₁	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	2	8	no	—
128	C₂×C₈	C₄⋊C₄×C₄₁	1	4	yes	—
129	C₂×C₈	C₂×C₈×C₄₁	1	4	yes	—
130	C₂×C₈	C₂×C₈×C₄₁	1	4	yes	—
131	C₂×C₈	M₁₆×C₄₁	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	C₂×C₈×C₄₁	1	4	yes	—
135	C₄²	M₁₆×C₄₁	1	8	yes	—
136	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
137	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
138	C₄²	#41×C₄₁	1	2	yes	—
139	C₂³	C₂×C₄²×C₄₁	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₄×C₈×C₄₁	1	2	yes	—
144	C₂×C₄	C₄×C₈×C₄₁	1	2	yes	—
145	C₂×C₄	#97×C₄₁	1	4	yes	—
146	C₂×C₄	#97×C₄₁	1	4	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
148	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
150	C₂×C₄	C₄×Q₈×C₄₁	1	2	yes	—
151	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
152	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
153	C₂×C₄	C₂×M₁₆×C₄₁	1	4	yes	—
154	C₂×C₄	#131×C₄₁	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	4	8	no	—
157	C₈	C₄⋊C₈×C₄₁	1	8	yes	complemented
158	C₂²	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
159	C₂²	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
160	C₂²	C₂×#97×C₄₁	1	2	yes	—
161	C₂²	C₄×M₁₆×C₄₁	1	1	yes	characteristic
162	C₂²	C₄₁×#405	1	2	yes	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	C₄₁×#322	1	4	yes	complemented
166	C₄	C₂×C₄⋊C₈×C₄₁	1	2	yes	—
167	C₄	C₄₁×#431	1	2	yes	—
168	C₄	C₄₁×#436	1	4	yes	—
169	C₄	C₄₁×#446	1	4	yes	—
170	C₂	C₂×C₄₁×#322	1	1	yes	characteristic
171	C₂	C₄₁×#1450	1	1	yes	characteristic
172	C₂	C₄₁×#1451	1	2	yes	—
173	C₂	C₄×C₄⋊C₈×C₄₁	1	1	yes	characteristic
174	C₂	C₄₁×#1770	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₄₁	C₂³
commutator	C₂²	C₂×C₄×C₈×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₄×C₈×C₄₁	C₈	semidirect
69. C₈×C₄₁	C₄⋊C₈	semidirect
70. #8260	C₄₁	direct (N × H)
78. C₄×C₄₁	#322	semidirect
101. C₄₁	#8260	direct (N × H)
115. C₄×C₈	C₈×C₄₁	semidirect
157. C₈	C₄⋊C₈×C₄₁	semidirect
165. C₄	C₄₁×#322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈×C₄₁	2
C₄²×C₈×C₄₁	2
C₂×C₄₁×#322	2
C₄₁×#1273	2
#8260	41

Subgroups — C41×#8260 #23784461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8260 #23784461