Subgroups — C₄₁×#33256 #23809457

Order 10496.

718 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	C₄₁×#2859	C₂	1	2	yes	maximal
2	C₄₁×#1941	C₂	1	8	yes	maximal
3	C₄₁×#1844	C₂	1	2	yes	maximal
4	C₄₁×#1834	C₂	1	2	yes	maximal
5	C₄²×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₄₁	—	2	2	no	—
7	C₂×#106×C₄₁	C₂²	1	8	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	C₄₁×#389	—	2	4	no	—
12	C₄₁×#389	C₂²	1	8	yes	—
13	C₄₁×#384	C₂²	1	4	yes	—
14	C₄₁×#382	—	2	2	no	—
15	C₄³×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#340	C₂²	1	8	yes	—
17	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
18	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
19	#125×C₄₁	—	4	8	no	—
20	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
22	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
23	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
24	C₂×C₄²×C₄₁	D₄	1	2	yes	—
25	C₂×C₄²×C₄₁	D₄	1	2	yes	—
26	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
27	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
28	#106×C₄₁	—	2	16	no	—
29	C₄×C₈×C₄₁	—	2	8	no	—
30	C₄×C₈×C₄₁	Q₈	1	4	yes	—
31	C₄×C₈×C₄₁	Q₈	1	4	yes	—
32	#95×C₄₁	—	2	8	no	—
33	C₂²×C₄×C₄₁	—	2	8	no	—
34	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
35	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
38	C₂×C₈×C₄₁	—	2	4	no	—
39	C₂×C₈×C₄₁	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₄₁	C₂×Q₈	1	4	yes	—
41	C₂×C₈×C₄₁	#41	1	8	yes	—
42	C₄⋊C₄×C₄₁	—	2	16	no	—
43	C₄⋊C₄×C₄₁	—	4	16	no	—
44	C₄⋊C₄×C₄₁	—	4	8	no	—
45	C₄⋊C₄×C₄₁	—	4	8	no	—
46	C₄²×C₄₁	—	2	4	no	—
47	C₄²×C₄₁	—	2	4	no	—
48	C₄²×C₄₁	SD₁₆	1	8	yes	—
49	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
50	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
51	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₁	—	2	4	no	—
53	C₂×C₄×C₄₁	—	2	4	no	—
54	C₂×C₄×C₄₁	—	4	8	no	—
55	C₂×C₄×C₄₁	—	4	8	no	—
56	C₂×C₄×C₄₁	—	4	8	no	—
57	C₂×C₄×C₄₁	C₄⋊Q₈	1	2	yes	—
58	C₂×C₄×C₄₁	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₄₁	C₂×SD₁₆	1	8	yes	—
60	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
61	C₂×C₄×C₄₁	#135	1	4	yes	—
62	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
65	C₈×C₄₁	—	2	4	no	—
66	C₈×C₄₁	#125	1	8	yes	complemented
67	#33256	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂²×C₄₁	#532	1	2	yes	—
70	C₂²×C₄₁	C₂²×SD₁₆	1	2	yes	—
71	C₂²×C₄₁	C₂×#135	1	2	yes	—
72	C₄×C₄₁	—	2	8	no	—
73	C₄×C₄₁	—	8	8	no	—
74	C₄×C₄₁	#498	1	4	yes	—
75	C₄×C₄₁	#499	1	4	yes	—
76	C₄×C₄₁	C₂×#125	1	2	yes	—
77	C₄×C₄₁	#533	1	2	yes	—
78	#2859	C₂×C₄₁	1	2	yes	—
79	#1941	C₂×C₄₁	1	8	yes	—
80	#1844	C₂×C₄₁	1	2	yes	—
81	#1834	C₂×C₄₁	1	2	yes	—
82	C₄²×C₈	C₂×C₄₁	1	1	yes	characteristic
83	C₂×C₄₁	#2859	1	1	yes	characteristic
84	C₂×C₄₁	C₂×#498	1	1	yes	characteristic
85	C₂×C₄₁	C₂×#499	1	1	yes	characteristic
86	C₂×C₄₁	#3167	1	4	yes	—
87	C₂×#125	—	2	2	no	—
88	C₂×#106	C₂²×C₄₁	1	8	yes	—
89	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
90	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
92	#389	—	2	4	no	—
93	#389	C₂²×C₄₁	1	8	yes	—
94	#384	C₂²×C₄₁	1	4	yes	—
95	#382	—	2	2	no	—
96	C₄³	C₂²×C₄₁	1	1	yes	characteristic
97	#340	C₂²×C₄₁	1	8	yes	—
98	C₄₁	#33256	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
100	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
101	#125	—	4	8	no	—
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄⋊C₄	—	2	8	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
106	C₂×C₄²	D₄×C₄₁	1	2	yes	—
107	C₂×C₄²	D₄×C₄₁	1	2	yes	—
108	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
109	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
110	#106	—	2	16	no	—
111	C₄×C₈	—	2	8	no	—
112	C₄×C₈	Q₈×C₄₁	1	4	yes	—
113	C₄×C₈	Q₈×C₄₁	1	4	yes	—
114	#95	—	2	8	no	—
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
117	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
118	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
119	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	C₂×Q₈×C₄₁	1	4	yes	—
122	C₂×C₈	C₂×Q₈×C₄₁	1	4	yes	—
123	C₂×C₈	#41×C₄₁	1	8	yes	—
124	C₄⋊C₄	—	2	16	no	—
125	C₄⋊C₄	—	4	16	no	—
126	C₄⋊C₄	—	4	8	no	—
127	C₄⋊C₄	—	4	8	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	2	4	no	—
130	C₄²	SD₁₆×C₄₁	1	8	yes	—
131	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
132	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
133	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	—	4	8	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	C₄⋊Q₈×C₄₁	1	2	yes	—
140	C₂×C₄	C₄⋊Q₈×C₄₁	1	2	yes	—
141	C₂×C₄	C₂×SD₁₆×C₄₁	1	8	yes	—
142	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
143	C₂×C₄	#135×C₄₁	1	4	yes	—
144	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
145	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
146	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
147	C₈	—	2	4	no	—
148	C₈	#125×C₄₁	1	8	yes	complemented
149	C₂²	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
150	C₂²	C₄₁×#532	1	2	yes	—
151	C₂²	C₂²×SD₁₆×C₄₁	1	2	yes	—
152	C₂²	C₂×#135×C₄₁	1	2	yes	—
153	C₄	—	2	8	no	—
154	C₄	—	8	8	no	—
155	C₄	C₄₁×#498	1	4	yes	—
156	C₄	C₄₁×#499	1	4	yes	—
157	C₄	C₂×#125×C₄₁	1	2	yes	—
158	C₄	C₄₁×#533	1	2	yes	—
159	C₂	C₄₁×#2859	1	1	yes	characteristic
160	C₂	C₂×C₄₁×#498	1	1	yes	characteristic
161	C₂	C₂×C₄₁×#499	1	1	yes	characteristic
162	C₂	C₄₁×#3167	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
66. C₈×C₄₁	#125	semidirect
67. #33256	C₄₁	direct (N × H)
98. C₄₁	#33256	direct (N × H)
148. C₈	#125×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#2859	2
C₄₁×#1941	2
C₄₁×#1844	2
C₄₁×#1834	2
C₄²×C₈×C₄₁	2
#33256	41

Subgroups — C41×#33256 #23809457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33256 #23809457