Subgroups — C₄₁×#12657 #23788858

Order 10496.

334 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2150	C₂	1	2	yes	maximal
2	C₂×C₄₁×#347	C₂	1	2	yes	maximal
3	C₄₁×#1726	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1375	C₂	1	2	yes	maximal
5	C₂×M₃₂×C₄₁	C₂²	1	4	yes	—
6	C₂×#97×C₄₁	C₄	1	1	yes	characteristic
7	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#376	C₄	1	1	yes	characteristic
13	C₄₁×C₄⋊C₁₆	C₄	1	8	yes	complemented
14	C₄₁×#347	—	2	4	no	—
15	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₄₁	—	2	4	no	—
23	M₃₂×C₄₁	—	2	4	no	—
24	C₂×C₁₆×C₄₁	C₂×C₄	1	8	yes	—
25	#97×C₄₁	—	2	2	no	—
26	#97×C₄₁	—	2	2	no	—
27	#97×C₄₁	D₄	1	2	yes	—
28	#97×C₄₁	Q₈	1	2	yes	—
29	C₄×C₈×C₄₁	—	2	2	no	—
30	C₄×C₈×C₄₁	—	2	2	no	—
31	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
32	#95×C₄₁	—	2	2	no	—
33	C₂²×C₄×C₄₁	—	2	1	no	—
34	C₂²×C₄×C₄₁	—	2	1	no	—
35	C₂²×C₄×C₄₁	M₁₆	1	2	yes	—
36	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₁	—	2	1	no	—
38	C₂×C₈×C₄₁	—	2	1	no	—
39	C₂×C₈×C₄₁	—	2	1	no	—
40	C₂×C₈×C₄₁	—	2	1	no	—
41	C₂×C₈×C₄₁	—	2	1	no	—
42	C₂×C₈×C₄₁	—	2	1	no	—
43	C₂×C₈×C₄₁	C₄²	1	2	yes	—
44	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₁	M₁₆	1	2	yes	—
46	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₄₁	#41	1	2	yes	—
50	C₄²×C₄₁	—	2	1	no	—
51	C₄²×C₄₁	—	2	1	no	—
52	C₄²×C₄₁	C₄²	1	2	yes	—
53	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
54	C₄²×C₄₁	M₁₆	1	2	yes	—
55	C₄²×C₄₁	M₁₆	1	2	yes	—
56	C₁₆×C₄₁	—	2	8	no	—
57	C₂³×C₄₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₁	—	2	2	no	—
59	C₂×C₄×C₄₁	—	2	2	no	—
60	C₂×C₄×C₄₁	—	4	2	no	—
61	C₂×C₄×C₄₁	—	4	2	no	—
62	C₂×C₄×C₄₁	—	4	2	no	—
63	C₂×C₄×C₄₁	#97	1	4	yes	—
64	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
67	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₄₁	—	2	2	no	—
71	C₈×C₄₁	—	2	2	no	—
72	C₈×C₄₁	—	2	2	no	—
73	C₈×C₄₁	C₄×D₄	1	2	yes	—
74	C₈×C₄₁	C₄×Q₈	1	2	yes	—
75	#12657	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₄₁	—	2	2	no	—
77	C₂²×C₄₁	C₂×#97	1	1	yes	characteristic
78	C₂²×C₄₁	#423	1	1	yes	characteristic
79	C₂²×C₄₁	#432	1	1	yes	characteristic
80	C₄×C₄₁	—	2	4	no	—
81	C₄×C₄₁	—	4	2	no	—
82	C₄×C₄₁	—	4	2	no	—
83	C₄×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₄₁	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₄₁	#423	1	1	yes	characteristic
86	C₄×C₄₁	#432	1	1	yes	characteristic
87	#2150	C₂×C₄₁	1	2	yes	—
88	C₂×#347	C₂×C₄₁	1	2	yes	—
89	#1726	C₂×C₄₁	1	1	yes	characteristic
90	#1375	C₂×C₄₁	1	2	yes	—
91	C₂×C₄₁	—	2	1	no	—
92	C₂×C₄₁	—	2	1	no	—
93	C₂×C₄₁	#1767	1	1	yes	characteristic
94	C₂×C₄₁	#2110	1	2	yes	—
95	C₂×M₃₂	C₂²×C₄₁	1	4	yes	—
96	C₂×#97	C₄×C₄₁	1	1	yes	characteristic
97	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
98	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
99	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
100	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
102	#376	C₄×C₄₁	1	1	yes	characteristic
103	C₄⋊C₁₆	C₄×C₄₁	1	8	yes	complemented
104	#347	—	2	4	no	—
105	C₄₁	#12657	1	1	yes	characteristic, direct product factor, Sylow
106	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
109	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
113	M₃₂	—	2	4	no	—
114	M₃₂	—	2	4	no	—
115	C₂×C₁₆	C₂×C₄×C₄₁	1	8	yes	—
116	#97	—	2	2	no	—
117	#97	—	2	2	no	—
118	#97	D₄×C₄₁	1	2	yes	—
119	#97	Q₈×C₄₁	1	2	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
123	#95	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	M₁₆×C₄₁	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	C₄²×C₄₁	1	2	yes	—
135	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
136	C₂×C₈	M₁₆×C₄₁	1	2	yes	—
137	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
138	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
140	C₂×C₈	#41×C₄₁	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	—	2	1	no	—
143	C₄²	C₄²×C₄₁	1	2	yes	—
144	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
145	C₄²	M₁₆×C₄₁	1	2	yes	—
146	C₄²	M₁₆×C₄₁	1	2	yes	—
147	C₁₆	—	2	8	no	—
148	C₂³	C₂×M₁₆×C₄₁	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	#97×C₄₁	1	4	yes	—
155	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
157	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
158	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
159	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
160	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	C₄×D₄×C₄₁	1	2	yes	—
165	C₈	C₄×Q₈×C₄₁	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	C₂×#97×C₄₁	1	1	yes	characteristic
168	C₂²	C₄₁×#423	1	1	yes	characteristic
169	C₂²	C₄₁×#432	1	1	yes	characteristic
170	C₄	—	2	4	no	—
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
174	C₄	C₄×M₁₆×C₄₁	1	1	yes	characteristic
175	C₄	C₄₁×#423	1	1	yes	characteristic
176	C₄	C₄₁×#432	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₄₁×#1767	1	1	yes	characteristic
180	C₂	C₄₁×#2110	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×#97
prime radical	C₂³×C₄₁	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄₁×C₄⋊C₁₆	C₄	semidirect
75. #12657	C₄₁	direct (N × H)
103. C₄⋊C₁₆	C₄×C₄₁	semidirect
105. C₄₁	#12657	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2150	2
C₂×C₄₁×#347	2
C₄₁×#1726	2
C₄₁×#1375	2
#12657	41

Subgroups — C41×#12657 #23788858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12657 #23788858