Subgroups — C₄₁×#8781 #23784982

Order 10496.

310 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#1726	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1564	C₂	1	2	yes	maximal
3	C₄₁×#1281	C₂	1	2	yes	maximal
4	C₄₁×#1274	C₂	1	2	yes	maximal
5	C₄₁×#431	C₂²	1	4	yes	—
6	C₂×#97×C₄₁	C₄	1	1	yes	characteristic
7	C₂×#97×C₄₁	C₄	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	C₂²	1	1	yes	characteristic
13	C₄₁×#335	—	2	2	no	—
14	C₄₁×#334	—	2	2	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	C₂×C₄	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	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₄₁	—	2	4	no	—
23	C₂²⋊C₈×C₄₁	—	2	4	no	—
24	#97×C₄₁	—	2	2	no	—
25	#97×C₄₁	—	2	2	no	—
26	#97×C₄₁	C₈	1	4	yes	complemented
27	C₄×C₈×C₄₁	—	2	2	no	—
28	C₄×C₈×C₄₁	—	2	2	no	—
29	C₄×C₈×C₄₁	—	2	4	no	—
30	C₄×C₈×C₄₁	D₄	1	2	yes	—
31	C₄×C₈×C₄₁	Q₈	1	2	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	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₁	#41	1	2	yes	—
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	—	4	4	no	—
44	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₄₁	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₄₁	M₁₆	1	2	yes	—
49	C₂×C₈×C₄₁	M₁₆	1	2	yes	—
50	C₄²×C₄₁	—	2	1	no	—
51	C₄²×C₄₁	—	2	1	no	—
52	C₄²×C₄₁	C₂×C₈	1	2	yes	—
53	C₄²×C₄₁	M₁₆	1	2	yes	—
54	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₄₁	#41	1	2	yes	—
57	C₂³×C₄₁	#117	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	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
65	C₂×C₄×C₄₁	#124	1	2	yes	—
66	C₂×C₄×C₄₁	#125	1	2	yes	—
67	C₂×C₄×C₄₁	C₂²×C₈	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	C₂×C₄×C₄₁	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₄₁	—	2	2	no	—
72	C₈×C₄₁	—	2	2	no	—
73	C₈×C₄₁	—	2	2	no	—
74	C₈×C₄₁	—	8	4	no	—
75	C₈×C₄₁	C₄⋊C₈	1	4	yes	—
76	#8781	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₄₁	—	2	2	no	—
78	C₂²×C₄₁	#382	1	1	yes	characteristic
79	C₂²×C₄₁	#431	1	1	yes	characteristic
80	C₂²×C₄₁	#432	1	1	yes	characteristic
81	C₄×C₄₁	—	2	2	no	—
82	C₄×C₄₁	—	2	2	no	—
83	C₄×C₄₁	—	4	2	no	—
84	C₄×C₄₁	—	4	2	no	—
85	C₄×C₄₁	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₄×C₄₁	#423	1	1	yes	characteristic
87	C₄×C₄₁	#431	1	1	yes	characteristic
88	C₄×C₄₁	#432	1	1	yes	characteristic
89	#1726	C₂×C₄₁	1	1	yes	characteristic
90	#1564	C₂×C₄₁	1	2	yes	—
91	#1281	C₂×C₄₁	1	2	yes	—
92	#1274	C₂×C₄₁	1	2	yes	—
93	C₂×C₄₁	—	2	1	no	—
94	C₂×C₄₁	—	2	1	no	—
95	C₂×C₄₁	#1831	1	1	yes	characteristic
96	C₂×C₄₁	#1833	1	2	yes	—
97	#431	C₂²×C₄₁	1	4	yes	—
98	C₂×#97	C₄×C₄₁	1	1	yes	characteristic
99	C₂×#97	C₄×C₄₁	1	1	yes	characteristic
100	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
101	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
102	C₂×C₄×C₈	C₄×C₄₁	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
104	#376	C₂²×C₄₁	1	1	yes	characteristic
105	#335	—	2	2	no	—
106	#334	—	2	2	no	—
107	C₄₁	#8781	1	1	yes	characteristic, direct product factor, Sylow
108	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
110	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₂²⋊C₈	—	2	4	no	—
117	#97	—	2	2	no	—
118	#97	—	2	2	no	—
119	#97	C₈×C₄₁	1	4	yes	complemented
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	D₄×C₄₁	1	2	yes	—
124	C₄×C₈	Q₈×C₄₁	1	2	yes	—
125	#95	—	2	2	no	—
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
129	C₂²×C₄	#41×C₄₁	1	2	yes	—
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	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
138	C₂×C₈	C₄⋊C₄×C₄₁	1	2	yes	—
139	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
140	C₂×C₈	C₂×C₈×C₄₁	1	2	yes	—
141	C₂×C₈	M₁₆×C₄₁	1	2	yes	—
142	C₂×C₈	M₁₆×C₄₁	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂×C₈×C₄₁	1	2	yes	—
146	C₄²	M₁₆×C₄₁	1	2	yes	—
147	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
148	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
149	C₄²	#41×C₄₁	1	2	yes	—
150	C₂³	#117×C₄₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
157	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
158	C₂×C₄	#124×C₄₁	1	2	yes	—
159	C₂×C₄	#125×C₄₁	1	2	yes	—
160	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
161	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
163	C₂×C₄	C₂×M₁₆×C₄₁	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	8	4	no	—
168	C₈	C₄⋊C₈×C₄₁	1	4	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	C₄₁×#382	1	1	yes	characteristic
171	C₂²	C₄₁×#431	1	1	yes	characteristic
172	C₂²	C₄₁×#432	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	C₂×C₄⋊C₈×C₄₁	1	1	yes	characteristic
178	C₄	C₄₁×#423	1	1	yes	characteristic
179	C₄	C₄₁×#431	1	1	yes	characteristic
180	C₄	C₄₁×#432	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	C₄₁×#1831	1	1	yes	characteristic
184	C₂	C₄₁×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	#117
commutator	C₂×C₄	C₂²×C₈×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂²×C₄₁	#382
prime radical	C₂³×C₄₁	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₄₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
26. #97×C₄₁	C₈	semidirect
76. #8781	C₄₁	direct (N × H)
107. C₄₁	#8781	direct (N × H)
119. #97	C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#1726	2
C₄₁×#1564	2
C₄₁×#1281	2
C₄₁×#1274	2
#8781	41

Subgroups — C41×#8781 #23784982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8781 #23784982