Subgroups — C₄₁×#12011 #23788212

Order 10496.

220 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#364	C₂	1	2	yes	maximal
2	C₄×M₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1373	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₄₁	—	2	2	no	—
5	C₂×M₃₂×C₄₁	C₄	1	1	yes	characteristic
6	C₂×M₃₂×C₄₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₄₁×#364	—	2	2	no	—
9	C₄₁×#364	C₂²	1	4	yes	—
10	C₄₁×#346	C₄	1	2	yes	—
11	C₄×C₁₆×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
14	C₂×C₄²×C₄₁	Q₈	1	1	yes	characteristic
15	M₃₂×C₄₁	—	2	2	no	—
16	M₃₂×C₄₁	—	2	4	no	—
17	M₃₂×C₄₁	—	4	2	no	—
18	M₃₂×C₄₁	C₈	1	4	yes	—
19	C₂×C₁₆×C₄₁	—	4	2	no	—
20	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₄₁	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₁	—	2	1	no	—
27	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₄₁	M₁₆	1	2	yes	—
32	C₂×C₈×C₄₁	D₈	1	2	yes	—
33	C₂×C₈×C₄₁	Q₁₆	1	2	yes	—
34	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₄₁	—	2	1	no	—
37	C₄²×C₄₁	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₄₁	—	2	2	no	—
40	C₁₆×C₄₁	—	4	4	no	—
41	C₁₆×C₄₁	C₂×C₈	1	4	yes	—
42	C₂³×C₄₁	#108	1	1	yes	characteristic
43	C₂×C₄×C₄₁	—	2	1	no	—
44	C₂×C₄×C₄₁	—	2	1	no	—
45	C₂×C₄×C₄₁	—	4	2	no	—
46	C₂×C₄×C₄₁	C₄⋊C₈	1	2	yes	—
47	C₂×C₄×C₄₁	#107	1	2	yes	—
48	C₂×C₄×C₄₁	#108	1	1	yes	characteristic
49	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₄₁	C₄⋊C₈	1	2	yes	—
51	C₈×C₄₁	#107	1	2	yes	—
52	C₈×C₄₁	C₂²×C₈	1	1	yes	characteristic
53	C₈×C₄₁	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₄₁	C₂×D₈	1	1	yes	characteristic
55	C₈×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
56	#12011	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₄₁	—	4	1	no	—
58	C₂²×C₄₁	#335	1	2	yes	—
59	C₂²×C₄₁	C₂×#108	1	1	yes	characteristic
60	C₄×C₄₁	—	2	2	no	—
61	C₄×C₄₁	—	4	1	no	—
62	C₄×C₄₁	#335	1	2	yes	—
63	C₄×C₄₁	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₄×C₄₁	C₂×#107	1	1	yes	characteristic
65	C₂×#364	C₂×C₄₁	1	2	yes	—
66	C₄×M₃₂	C₂×C₄₁	1	1	yes	characteristic
67	#1373	C₂×C₄₁	1	4	yes	complemented
68	C₂×C₄₁	—	2	1	no	—
69	C₂×C₄₁	—	2	2	no	—
70	C₂×C₄₁	C₂×#335	1	1	yes	characteristic
71	C₂×M₃₂	—	2	2	no	—
72	C₂×M₃₂	C₄×C₄₁	1	1	yes	characteristic
73	C₂×M₃₂	C₄×C₄₁	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
75	#364	—	2	2	no	—
76	#364	C₂²×C₄₁	1	4	yes	—
77	#346	C₄×C₄₁	1	2	yes	—
78	C₄×C₁₆	C₂²×C₄₁	1	2	yes	—
79	C₄₁	#12011	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
81	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
82	C₂×C₄²	Q₈×C₄₁	1	1	yes	characteristic
83	M₃₂	—	2	2	no	—
84	M₃₂	—	2	4	no	—
85	M₃₂	—	4	2	no	—
86	M₃₂	C₈×C₄₁	1	4	yes	—
87	C₂×C₁₆	—	4	2	no	—
88	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
89	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
90	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
91	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
92	C₄×C₈	Q₈×C₄₁	1	1	yes	characteristic
93	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
98	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
99	C₂×C₈	M₁₆×C₄₁	1	2	yes	—
100	C₂×C₈	D₈×C₄₁	1	2	yes	—
101	C₂×C₈	Q₁₆×C₄₁	1	2	yes	—
102	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
103	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	C₄⋊C₄×C₄₁	1	1	yes	characteristic
106	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	—	4	4	no	—
109	C₁₆	C₂×C₈×C₄₁	1	4	yes	—
110	C₂³	#108×C₄₁	1	1	yes	characteristic
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	C₄⋊C₈×C₄₁	1	2	yes	—
115	C₂×C₄	#107×C₄₁	1	2	yes	—
116	C₂×C₄	#108×C₄₁	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
118	C₈	C₄⋊C₈×C₄₁	1	2	yes	—
119	C₈	#107×C₄₁	1	2	yes	—
120	C₈	C₂²×C₈×C₄₁	1	1	yes	characteristic
121	C₈	C₂×M₁₆×C₄₁	1	1	yes	characteristic
122	C₈	C₂×D₈×C₄₁	1	1	yes	characteristic
123	C₈	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
124	C₂²	—	4	1	no	—
125	C₂²	C₄₁×#335	1	2	yes	—
126	C₂²	C₂×#108×C₄₁	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	4	1	no	—
129	C₄	C₄₁×#335	1	2	yes	—
130	C₄	C₂×C₄⋊C₈×C₄₁	1	1	yes	characteristic
131	C₄	C₂×#107×C₄₁	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	2	2	no	—
134	C₂	C₂×C₄₁×#335	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₄₁	C₂×D₈
commutator	C₈	C₂²×C₈×C₄₁
Frattini, cosocle	C₄×C₈	C₂³×C₄₁
socle	C₂×C₄₁	C₂×#335
prime radical	C₂³×C₄₁	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₄₁ ◁ C₂²×C₈×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1373	C₂	semidirect
56. #12011	C₄₁	direct (N × H)
67. #1373	C₂×C₄₁	semidirect
79. C₄₁	#12011	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#364	2
C₄×M₃₂×C₄₁	2
C₄₁×#1373	2
#12011	41

Subgroups — C41×#12011 #23788212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12011 #23788212