Subgroups — C₄₁×#12617 #23788818

Order 10496.

382 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#504	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2115	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2110	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1757	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
7	C₄₁×#504	—	2	4	no	—
8	C₂×M₃₂×C₄₁	C₄	1	2	yes	complemented
9	C₂×M₃₂×C₄₁	C₂²	1	2	yes	—
10	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₄₁	C₄	1	2	yes	complemented
12	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#413	C₄	1	1	yes	characteristic
14	C₄₁×#410	C₄	1	1	yes	characteristic
15	C₄₁×#405	C₂²	1	2	yes	—
16	C₄₁×#349	C₄	1	4	yes	complemented
17	C₄₁×#347	—	2	4	no	—
18	C₄₁×#344	C₄	1	2	yes	—
19	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
20	#131×C₄₁	—	2	2	no	—
21	#131×C₄₁	—	2	2	no	—
22	C₂×M₁₆×C₄₁	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₄₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
27	#117×C₄₁	C₂×C₄	1	2	yes	—
28	M₃₂×C₄₁	—	2	2	no	—
29	M₃₂×C₄₁	—	2	4	no	—
30	M₃₂×C₄₁	C₂×C₄	1	4	yes	—
31	M₃₂×C₄₁	D₄	1	4	yes	—
32	C₂×C₁₆×C₄₁	—	2	2	no	—
33	C₂×C₁₆×C₄₁	—	2	2	no	—
34	C₂×C₁₆×C₄₁	—	2	4	no	—
35	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
38	#101×C₄₁	—	2	1	no	—
39	#100×C₄₁	—	2	1	no	—
40	#99×C₄₁	—	2	2	no	—
41	#97×C₄₁	—	2	2	no	—
42	C₄×C₈×C₄₁	—	2	2	no	—
43	#41×C₄₁	—	2	2	no	—
44	#41×C₄₁	—	2	2	no	—
45	C₂×Q₈×C₄₁	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₄₁	M₁₆	1	2	yes	—
47	C₂×D₄×C₄₁	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₄²	1	2	yes	—
49	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₄₁	—	2	2	no	—
51	M₁₆×C₄₁	—	2	4	no	—
52	M₁₆×C₄₁	—	2	2	no	—
53	M₁₆×C₄₁	—	2	2	no	—
54	C₂×C₈×C₄₁	—	2	2	no	—
55	C₂×C₈×C₄₁	—	2	2	no	—
56	C₂×C₈×C₄₁	—	2	2	no	—
57	C₂×C₈×C₄₁	C₄²	1	2	yes	—
58	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₄₁	#41	1	2	yes	—
62	C₄⋊C₄×C₄₁	—	2	2	no	—
63	C₂²⋊C₄×C₄₁	—	2	2	no	—
64	C₄²×C₄₁	—	2	2	no	—
65	C₁₆×C₄₁	—	2	1	no	—
66	C₁₆×C₄₁	—	2	1	no	—
67	C₁₆×C₄₁	—	2	4	no	—
68	C₁₆×C₄₁	C₂²⋊C₄	1	4	yes	—
69	C₂³×C₄₁	#97	1	2	yes	—
70	C₂³×C₄₁	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₄₁	—	2	2	no	—
72	D₄×C₄₁	—	2	4	no	—
73	D₄×C₄₁	—	2	2	no	—
74	C₂×C₄×C₄₁	—	2	2	no	—
75	C₂×C₄×C₄₁	—	2	2	no	—
76	C₂×C₄×C₄₁	—	4	2	no	—
77	C₂×C₄×C₄₁	#97	1	2	yes	—
78	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₁	#131	1	2	yes	—
82	C₈×C₄₁	—	2	2	no	—
83	C₈×C₄₁	—	4	2	no	—
84	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₈×C₄₁	#117	1	1	yes	characteristic
86	C₈×C₄₁	C₄×D₄	1	2	yes	—
87	#12617	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₄₁	—	2	2	no	—
89	C₂²×C₄₁	—	4	1	no	—
90	C₂²×C₄₁	—	4	2	no	—
91	C₂²×C₄₁	C₂×#97	1	1	yes	characteristic
92	C₂²×C₄₁	#435	1	2	yes	—
93	C₄×C₄₁	—	2	2	no	—
94	C₄×C₄₁	—	8	2	no	—
95	C₄×C₄₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₄₁	#405	1	1	yes	characteristic
97	C₄×C₄₁	#435	1	2	yes	—
98	C₂×#504	C₂×C₄₁	1	1	yes	characteristic
99	#2115	C₂×C₄₁	1	1	yes	characteristic
100	#2110	C₂×C₄₁	1	2	yes	complemented
101	#1757	C₂×C₄₁	1	1	yes	characteristic
102	#1379	C₂×C₄₁	1	2	yes	complemented
103	C₂×C₄₁	—	2	2	no	—
104	C₂×C₄₁	—	2	1	no	—
105	C₂×C₄₁	—	4	2	no	—
106	C₂×C₄₁	#1752	1	1	yes	characteristic
107	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
108	#504	—	2	4	no	—
109	C₂×M₃₂	C₄×C₄₁	1	2	yes	complemented
110	C₂×M₃₂	C₂²×C₄₁	1	2	yes	—
111	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
112	C₂²×C₁₆	C₄×C₄₁	1	2	yes	complemented
113	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
114	#413	C₄×C₄₁	1	1	yes	characteristic
115	#410	C₄×C₄₁	1	1	yes	characteristic
116	#405	C₂²×C₄₁	1	2	yes	—
117	#349	C₄×C₄₁	1	4	yes	complemented
118	#347	—	2	4	no	—
119	#344	C₄×C₄₁	1	2	yes	—
120	C₄₁	#12617	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	C₂×C₄×C₄₁	1	2	yes	—
125	C₂×M₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
126	C₂×M₁₆	C₂×C₄×C₄₁	1	2	yes	—
127	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
128	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
129	#117	C₂×C₄×C₄₁	1	2	yes	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	2	4	no	—
132	M₃₂	C₂×C₄×C₄₁	1	4	yes	—
133	M₃₂	D₄×C₄₁	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
138	C₂×C₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
139	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
140	#101	—	2	1	no	—
141	#100	—	2	1	no	—
142	#99	—	2	2	no	—
143	#97	—	2	2	no	—
144	C₄×C₈	—	2	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	M₁₆×C₄₁	1	1	yes	characteristic
148	C₂×D₄	M₁₆×C₄₁	1	2	yes	—
149	C₂×D₄	M₁₆×C₄₁	1	1	yes	characteristic
150	C₂²×C₄	C₄²×C₄₁	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	C₄²×C₄₁	1	2	yes	—
160	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
163	C₂×C₈	#41×C₄₁	1	2	yes	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	2	4	no	—
170	C₁₆	C₂²⋊C₄×C₄₁	1	4	yes	—
171	C₂³	#97×C₄₁	1	2	yes	—
172	C₂³	C₂×M₁₆×C₄₁	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#97×C₄₁	1	2	yes	—
180	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
181	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
182	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
183	C₂×C₄	#131×C₄₁	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	4	2	no	—
186	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
187	C₈	#117×C₄₁	1	1	yes	characteristic
188	C₈	C₄×D₄×C₄₁	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#97×C₄₁	1	1	yes	characteristic
193	C₂²	C₄₁×#435	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	8	2	no	—
196	C₄	C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
197	C₄	C₄₁×#405	1	1	yes	characteristic
198	C₄	C₄₁×#435	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	C₄₁×#1752	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₄₁	#117
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle	C₂×C₄₁	#1752
prime radical	C₂×#41×C₄₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₄₁ ◁ C₂×M₃₂×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#2110	C₂	semidirect
5. C₄₁×#1379	C₂	semidirect
8. C₂×M₃₂×C₄₁	C₄	semidirect
11. C₂²×C₁₆×C₄₁	C₄	semidirect
16. C₄₁×#349	C₄	semidirect
87. #12617	C₄₁	direct (N × H)
100. #2110	C₂×C₄₁	semidirect
102. #1379	C₂×C₄₁	semidirect
109. C₂×M₃₂	C₄×C₄₁	semidirect
112. C₂²×C₁₆	C₄×C₄₁	semidirect
117. #349	C₄×C₄₁	semidirect
120. C₄₁	#12617	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×#504	2
C₄₁×#2115	2
C₄₁×#2110	2
C₄₁×#1757	2
C₄₁×#1379	2
#12617	41

Subgroups — C41×#12617 #23788818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12617 #23788818