Subgroups — C₄₁×#17773 #23793974

Order 10496.

1108 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3499	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2991	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1560	C₂	1	3	yes	maximal, complemented
4	C₄₁×#1548	C₂	1	6	yes	maximal, complemented
5	C₄₁×#1543	C₂	1	3	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×C₄₁	C₄	1	2	yes	—
7	D₄×Q₈×C₄₁	—	2	2	no	—
8	C₄₁×#547	—	2	6	no	—
9	C₄₁×#536	C₄	1	6	yes	—
10	C₄₁×#532	C₂²	1	3	yes	—
11	C₂×C₄⋊Q₈×C₄₁	C₂²	1	3	yes	—
12	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₄₁	—	2	2	no	—
14	C₄₁×#436	—	2	6	no	—
15	C₄₁×#431	C₂²	1	6	yes	—
16	C₄₁×#423	C₂²	1	6	yes	—
17	C₄₁×#333	C₂²	1	6	yes	—
18	C₄₁×#333	C₂²	1	6	yes	—
19	C₄₁×#331	C₂²	1	1	yes	characteristic
20	C₄₁×#331	C₂²	1	3	yes	—
21	2₋¹⁺⁴×C₄₁	—	4	8	no	—
22	C₂×#41×C₄₁	—	2	6	no	—
23	C₂×#41×C₄₁	C₂×C₄	1	6	yes	—
24	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Q₈×C₄₁	—	2	2	no	—
26	C₂²×Q₈×C₄₁	C₂×C₄	1	6	yes	—
27	C₂×M₁₆×C₄₁	—	2	6	no	—
28	C₄⋊Q₈×C₄₁	—	2	6	no	—
29	C₄⋊Q₈×C₄₁	C₂×C₄	1	6	yes	—
30	C₄⋊Q₈×C₄₁	C₂³	1	3	yes	—
31	#127×C₄₁	C₂³	1	1	yes	characteristic
32	#124×C₄₁	C₂×C₄	1	6	yes	—
33	#123×C₄₁	—	2	12	no	—
34	C₂²⋊Q₈×C₄₁	—	2	12	no	—
35	#121×C₄₁	—	2	6	no	—
36	C₄×Q₈×C₄₁	D₄	1	2	yes	—
37	C₄×D₄×C₄₁	—	2	6	no	—
38	C₄×D₄×C₄₁	D₄	1	6	yes	—
39	#117×C₄₁	C₂×C₄	1	3	yes	—
40	C₂×C₄⋊C₄×C₄₁	—	2	6	no	—
41	C₂×C₄²×C₄₁	C₂³	1	3	yes	—
42	C₄⋊C₈×C₄₁	—	2	6	no	—
43	C₄⋊C₈×C₄₁	C₂³	1	6	yes	—
44	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
45	C₂²⋊C₈×C₄₁	—	2	6	no	—
46	C₄×C₈×C₄₁	—	2	6	no	—
47	#41×C₄₁	—	2	4	no	—
48	#41×C₄₁	—	4	12	no	—
49	#41×C₄₁	—	4	24	no	—
50	C₂×Q₈×C₄₁	—	2	6	no	—
51	C₂×Q₈×C₄₁	—	4	12	no	—
52	C₂×Q₈×C₄₁	—	4	8	no	—
53	C₂×Q₈×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₄₁	C₂²×C₄	1	6	yes	—
55	C₂×D₄×C₄₁	—	2	6	no	—
56	C₂×D₄×C₄₁	—	2	2	no	—
57	C₂×D₄×C₄₁	C₂²⋊C₄	1	3	yes	—
58	C₂×D₄×C₄₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₁	—	2	6	no	—
60	C₂²×C₄×C₄₁	—	2	6	no	—
61	C₂²×C₄×C₄₁	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₄₁	C₂×D₄	1	6	yes	—
63	M₁₆×C₄₁	—	8	6	no	—
64	C₂×C₈×C₄₁	—	2	2	no	—
65	C₂×C₈×C₄₁	—	2	6	no	—
66	C₄⋊C₄×C₄₁	—	2	12	no	—
67	C₄⋊C₄×C₄₁	—	2	6	no	—
68	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	6	yes	—
69	C₂²⋊C₄×C₄₁	—	2	12	no	—
70	C₂²⋊C₄×C₄₁	C₂²⋊C₄	1	6	yes	—
71	C₄²×C₄₁	C₂²×C₄	1	3	yes	—
72	C₄²×C₄₁	C₂×D₄	1	6	yes	—
73	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
74	C₂³×C₄₁	—	2	2	no	—
75	C₂³×C₄₁	C₂×C₂²⋊C₄	1	3	yes	—
76	Q₈×C₄₁	—	4	6	no	—
77	Q₈×C₄₁	—	4	12	no	—
78	Q₈×C₄₁	#100	1	4	yes	—
79	D₄×C₄₁	—	2	6	no	—
80	D₄×C₄₁	—	4	2	no	—
81	D₄×C₄₁	—	4	12	no	—
82	D₄×C₄₁	—	4	2	no	—
83	D₄×C₄₁	—	4	2	no	—
84	C₂×C₄×C₄₁	—	2	6	no	—
85	C₂×C₄×C₄₁	—	2	6	no	—
86	C₂×C₄×C₄₁	—	2	6	no	—
87	C₂×C₄×C₄₁	—	2	6	no	—
88	C₂×C₄×C₄₁	—	4	12	no	—
89	C₂×C₄×C₄₁	—	4	12	no	—
90	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	6	yes	—
91	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	3	yes	—
92	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
94	C₈×C₄₁	—	4	6	no	—
95	C₈×C₄₁	—	4	2	no	—
96	#17773	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₄₁	—	2	3	no	—
98	C₂²×C₄₁	—	2	6	no	—
99	C₂²×C₄₁	—	4	2	no	—
100	C₂²×C₄₁	—	4	4	no	—
101	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₄₁	—	2	1	no	—
103	C₄×C₄₁	—	2	6	no	—
104	C₄×C₄₁	—	4	6	no	—
105	C₄×C₄₁	C₂×#100	1	6	yes	—
106	C₄×C₄₁	#577	1	2	yes	—
107	C₄×C₄₁	#578	1	2	yes	—
108	#3499	C₂×C₄₁	1	1	yes	characteristic
109	#2991	C₂×C₄₁	1	2	yes	complemented
110	#1560	C₂×C₄₁	1	3	yes	complemented
111	#1548	C₂×C₄₁	1	6	yes	complemented
112	#1543	C₂×C₄₁	1	3	yes	complemented
113	C₂×C₄₁	—	4	3	no	—
114	C₂×C₄₁	—	8	2	no	—
115	C₂×C₄₁	C₂²×#100	1	1	yes	characteristic
116	C₂×C₄₁	#2898	1	2	yes	—
117	C₂×2₋¹⁺⁴	C₄×C₄₁	1	2	yes	—
118	D₄×Q₈	—	2	2	no	—
119	#547	—	2	6	no	—
120	#536	C₄×C₄₁	1	6	yes	—
121	#532	C₂²×C₄₁	1	3	yes	—
122	C₂×C₄⋊Q₈	C₂²×C₄₁	1	3	yes	—
123	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
124	C₈×Q₈	—	2	2	no	—
125	#436	—	2	6	no	—
126	#431	C₂²×C₄₁	1	6	yes	—
127	#423	C₂²×C₄₁	1	6	yes	—
128	#333	C₂²×C₄₁	1	6	yes	—
129	#333	C₂²×C₄₁	1	6	yes	—
130	#331	C₂²×C₄₁	1	1	yes	characteristic
131	#331	C₂²×C₄₁	1	3	yes	—
132	C₄₁	#17773	1	1	yes	characteristic, direct product factor, Sylow
133	2₋¹⁺⁴	—	4	8	no	—
134	C₂×#41	—	2	6	no	—
135	C₂×#41	C₂×C₄×C₄₁	1	6	yes	—
136	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
137	C₂²×Q₈	—	2	2	no	—
138	C₂²×Q₈	C₂×C₄×C₄₁	1	6	yes	—
139	C₂×M₁₆	—	2	6	no	—
140	C₄⋊Q₈	—	2	6	no	—
141	C₄⋊Q₈	C₂×C₄×C₄₁	1	6	yes	—
142	C₄⋊Q₈	C₂³×C₄₁	1	3	yes	—
143	#127	C₂³×C₄₁	1	1	yes	characteristic
144	#124	C₂×C₄×C₄₁	1	6	yes	—
145	#123	—	2	12	no	—
146	C₂²⋊Q₈	—	2	12	no	—
147	#121	—	2	6	no	—
148	C₄×Q₈	D₄×C₄₁	1	2	yes	—
149	C₄×D₄	—	2	6	no	—
150	C₄×D₄	D₄×C₄₁	1	6	yes	—
151	#117	C₂×C₄×C₄₁	1	3	yes	—
152	C₂×C₄⋊C₄	—	2	6	no	—
153	C₂×C₄²	C₂³×C₄₁	1	3	yes	—
154	C₄⋊C₈	—	2	6	no	—
155	C₄⋊C₈	C₂³×C₄₁	1	6	yes	—
156	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
157	C₂²⋊C₈	—	2	6	no	—
158	C₄×C₈	—	2	6	no	—
159	#41	—	2	4	no	—
160	#41	—	4	12	no	—
161	#41	—	4	24	no	—
162	C₂×Q₈	—	2	6	no	—
163	C₂×Q₈	—	4	12	no	—
164	C₂×Q₈	—	4	8	no	—
165	C₂×Q₈	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
166	C₂×Q₈	C₂²×C₄×C₄₁	1	6	yes	—
167	C₂×D₄	—	2	6	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	C₂²⋊C₄×C₄₁	1	3	yes	—
170	C₂×D₄	C₂²×C₄×C₄₁	1	2	yes	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	C₂²×C₄×C₄₁	1	3	yes	—
174	C₂²×C₄	C₂×D₄×C₄₁	1	6	yes	—
175	M₁₆	—	8	6	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	6	no	—
178	C₄⋊C₄	—	2	12	no	—
179	C₄⋊C₄	—	2	6	no	—
180	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	6	yes	—
181	C₂²⋊C₄	—	2	12	no	—
182	C₂²⋊C₄	C₂²⋊C₄×C₄₁	1	6	yes	—
183	C₄²	C₂²×C₄×C₄₁	1	3	yes	—
184	C₄²	C₂×D₄×C₄₁	1	6	yes	—
185	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
186	C₂³	—	2	2	no	—
187	C₂³	C₂×C₂²⋊C₄×C₄₁	1	3	yes	—
188	Q₈	—	4	6	no	—
189	Q₈	—	4	12	no	—
190	Q₈	#100×C₄₁	1	4	yes	—
191	D₄	—	2	6	no	—
192	D₄	—	4	2	no	—
193	D₄	—	4	12	no	—
194	D₄	—	4	2	no	—
195	D₄	—	4	2	no	—
196	C₂×C₄	—	2	6	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	4	12	no	—
201	C₂×C₄	—	4	12	no	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	6	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	3	yes	—
204	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
206	C₈	—	4	6	no	—
207	C₈	—	4	2	no	—
208	C₂²	—	2	3	no	—
209	C₂²	—	2	6	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	4	no	—
212	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	6	no	—
215	C₄	—	4	6	no	—
216	C₄	C₂×#100×C₄₁	1	6	yes	—
217	C₄	C₄₁×#577	1	2	yes	—
218	C₄	C₄₁×#578	1	2	yes	—
219	C₂	—	4	3	no	—
220	C₂	—	8	2	no	—
221	C₂	C₂²×#100×C₄₁	1	1	yes	characteristic
222	C₂	C₄₁×#2898	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×#41×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2991	C₂	semidirect
3. C₄₁×#1560	C₂	semidirect
4. C₄₁×#1548	C₂	semidirect
5. C₄₁×#1543	C₂	semidirect
96. #17773	C₄₁	direct (N × H)
109. #2991	C₂×C₄₁	semidirect
110. #1560	C₂×C₄₁	semidirect
111. #1548	C₂×C₄₁	semidirect
112. #1543	C₂×C₄₁	semidirect
132. C₄₁	#17773	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×#3499	2
C₄₁×#2991	2
C₄₁×#1560	2
C₄₁×#1548	2
C₄₁×#1543	2
#17773	41

Subgroups — C41×#17773 #23793974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#17773 #23793974