Subgroups — C₄₁×#33255 #23809456

Order 10496.

902 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#487	C₂	1	2	yes	maximal
4	C₄₁×#2862	C₂	1	2	yes	maximal
5	C₄₁×#1971	C₂	1	8	yes	maximal
6	C₄×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
10	C₄₁×#498	—	2	4	no	—
11	C₄₁×#494	—	2	4	no	—
12	C₄₁×#487	—	2	8	no	—
13	C₂×#106×C₄₁	C₂²	1	4	yes	—
14	C₂×#103×C₄₁	C₂²	1	8	yes	—
15	C₂×#97×C₄₁	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#391	—	2	4	no	—
19	C₄₁×#391	C₂²	1	8	yes	—
20	C₄₁×#382	—	2	4	no	—
21	C₄³×C₄₁	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₄₁	—	2	2	no	—
23	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
24	C₂×Q₁₆×C₄₁	—	2	8	no	—
25	C₂×Q₁₆×C₄₁	—	2	16	no	—
26	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
28	C₄⋊Q₈×C₄₁	—	2	4	no	—
29	C₄⋊Q₈×C₄₁	—	2	4	no	—
30	C₄⋊Q₈×C₄₁	—	2	4	no	—
31	C₄⋊Q₈×C₄₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₁	D₄	1	4	yes	—
37	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
38	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
39	#106×C₄₁	—	2	8	no	—
40	#103×C₄₁	—	2	16	no	—
41	#97×C₄₁	—	2	8	no	—
42	C₄×C₈×C₄₁	D₄	1	4	yes	—
43	C₄×C₈×C₄₁	D₄	1	4	yes	—
44	#95×C₄₁	—	2	16	no	—
45	C₂×Q₈×C₄₁	—	2	4	no	—
46	C₂×Q₈×C₄₁	—	2	4	no	—
47	C₂×Q₈×C₄₁	—	2	16	no	—
48	C₂×Q₈×C₄₁	—	4	4	no	—
49	C₂×Q₈×C₄₁	—	4	8	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	2	4	no	—
52	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
55	Q₁₆×C₄₁	—	4	16	no	—
56	C₂×C₈×C₄₁	—	2	4	no	—
57	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
59	C₂×C₈×C₄₁	#41	1	8	yes	—
60	C₄⋊C₄×C₄₁	—	2	4	no	—
61	C₄⋊C₄×C₄₁	—	2	4	no	—
62	C₄⋊C₄×C₄₁	—	4	4	no	—
63	C₄⋊C₄×C₄₁	—	4	4	no	—
64	C₄⋊C₄×C₄₁	—	4	4	no	—
65	C₄²×C₄₁	—	2	8	no	—
66	C₄²×C₄₁	—	2	4	no	—
67	C₄²×C₄₁	C₂×D₄	1	2	yes	—
68	C₄²×C₄₁	C₂×D₄	1	2	yes	—
69	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
70	Q₈×C₄₁	—	4	8	no	—
71	Q₈×C₄₁	—	4	8	no	—
72	Q₈×C₄₁	—	8	8	no	—
73	C₂×C₄×C₄₁	—	2	4	no	—
74	C₂×C₄×C₄₁	—	2	8	no	—
75	C₂×C₄×C₄₁	—	4	4	no	—
76	C₂×C₄×C₄₁	—	4	4	no	—
77	C₂×C₄×C₄₁	—	4	8	no	—
78	C₂×C₄×C₄₁	—	4	8	no	—
79	C₂×C₄×C₄₁	#127	1	4	yes	—
80	C₂×C₄×C₄₁	#137	1	8	yes	—
81	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
84	C₈×C₄₁	—	2	4	no	—
85	C₈×C₄₁	#124	1	8	yes	—
86	#33255	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
88	C₂²×C₄₁	#532	1	2	yes	—
89	C₂²×C₄₁	C₂×#137	1	4	yes	—
90	C₄×C₄₁	—	4	4	no	—
91	C₄×C₄₁	—	8	4	no	—
92	C₄×C₄₁	—	8	4	no	—
93	C₄×C₄₁	#497	1	8	yes	—
94	C₄×C₄₁	C₂×#124	1	2	yes	—
95	C₄×C₄₁	#532	1	2	yes	—
96	C₂×#498	C₂×C₄₁	1	1	yes	characteristic
97	C₂×#494	C₂×C₄₁	1	1	yes	characteristic
98	C₂×#487	C₂×C₄₁	1	2	yes	—
99	#2862	C₂×C₄₁	1	2	yes	—
100	#1971	C₂×C₄₁	1	8	yes	—
101	C₄×#97	C₂×C₄₁	1	1	yes	characteristic
102	C₂×C₄₁	#2858	1	1	yes	characteristic
103	C₂×C₄₁	C₂×#497	1	2	yes	—
104	C₂×C₄₁	#3185	1	4	yes	—
105	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
106	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
107	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
108	#498	—	2	4	no	—
109	#494	—	2	4	no	—
110	#487	—	2	8	no	—
111	C₂×#106	C₂²×C₄₁	1	4	yes	—
112	C₂×#103	C₂²×C₄₁	1	8	yes	—
113	C₂×#97	C₂²×C₄₁	1	4	yes	—
114	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
116	#391	—	2	4	no	—
117	#391	C₂²×C₄₁	1	8	yes	—
118	#382	—	2	4	no	—
119	C₄³	C₂²×C₄₁	1	1	yes	characteristic
120	C₄₁	#33255	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
123	C₂×Q₁₆	—	2	8	no	—
124	C₂×Q₁₆	—	2	16	no	—
125	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
126	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	2	4	no	—
129	C₄⋊Q₈	—	2	4	no	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
135	C₂×C₄²	D₄×C₄₁	1	4	yes	—
136	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
137	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
138	#106	—	2	8	no	—
139	#103	—	2	16	no	—
140	#97	—	2	8	no	—
141	C₄×C₈	D₄×C₄₁	1	4	yes	—
142	C₄×C₈	D₄×C₄₁	1	4	yes	—
143	#95	—	2	16	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	2	16	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂×Q₈	—	4	8	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
152	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
153	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
154	Q₁₆	—	4	16	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
157	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
158	C₂×C₈	#41×C₄₁	1	8	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄²	—	2	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	C₂×D₄×C₄₁	1	2	yes	—
167	C₄²	C₂×D₄×C₄₁	1	2	yes	—
168	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	4	8	no	—
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	—	4	8	no	—
178	C₂×C₄	#127×C₄₁	1	4	yes	—
179	C₂×C₄	#137×C₄₁	1	8	yes	—
180	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
181	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
182	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
183	C₈	—	2	4	no	—
184	C₈	#124×C₄₁	1	8	yes	—
185	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
186	C₂²	C₄₁×#532	1	2	yes	—
187	C₂²	C₂×#137×C₄₁	1	4	yes	—
188	C₄	—	4	4	no	—
189	C₄	—	8	4	no	—
190	C₄	—	8	4	no	—
191	C₄	C₄₁×#497	1	8	yes	—
192	C₄	C₂×#124×C₄₁	1	2	yes	—
193	C₄	C₄₁×#532	1	2	yes	—
194	C₂	C₄₁×#2858	1	1	yes	characteristic
195	C₂	C₂×C₄₁×#497	1	2	yes	—
196	C₂	C₄₁×#3185	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
86. #33255	C₄₁	direct (N × H)
120. C₄₁	#33255	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×#498	2
C₂×C₄₁×#494	2
C₂×C₄₁×#487	2
C₄₁×#2862	2
C₄₁×#1971	2
C₄×#97×C₄₁	2
#33255	41

Subgroups — C41×#33255 #23809456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33255 #23809456