Subgroups — C₄₁×#12038 #23788239

Order 10496.

244 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₃₂×C₄₁	—	2	1	no	—
9	C₂²×C₁₆×C₄₁	—	2	1	no	—
10	C₄×D₈×C₄₁	C₄	1	1	yes	characteristic
11	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₈×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄₁×C₂²⋊C₁₆	—	2	1	no	—
16	C₄₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#335	C₄	1	1	yes	characteristic
20	C₄₁×D₄⋊C₈	C₄	1	1	yes	characteristic
21	C₄₁×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×D₈×C₄₁	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₄₁	—	2	1	no	—
25	C₂²×C₈×C₄₁	—	2	1	no	—
26	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₄₁	—	4	2	no	—
29	C₂×C₁₆×C₄₁	—	2	1	no	—
30	C₂×C₁₆×C₄₁	—	2	1	no	—
31	C₂×C₁₆×C₄₁	—	2	1	no	—
32	C₂×C₁₆×C₄₁	—	4	2	no	—
33	C₂×C₁₆×C₄₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₄₁	D₄	1	1	yes	characteristic
35	#107×C₄₁	C₈	1	1	yes	characteristic
36	C₄⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
38	#102×C₄₁	C₈	1	1	yes	characteristic
39	#102×C₄₁	C₈	1	1	yes	characteristic
40	C₂²⋊C₈×C₄₁	—	2	1	no	—
41	C₂²⋊C₈×C₄₁	—	2	1	no	—
42	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₁	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₄₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₁	—	2	1	no	—
48	C₂²×C₄×C₄₁	—	2	1	no	—
49	D₈×C₄₁	—	2	2	no	—
50	C₂×C₈×C₄₁	—	2	1	no	—
51	C₂×C₈×C₄₁	—	2	1	no	—
52	C₂×C₈×C₄₁	—	2	1	no	—
53	C₂×C₈×C₄₁	—	4	1	no	—
54	C₂×C₈×C₄₁	—	4	1	no	—
55	C₂×C₈×C₄₁	—	4	1	no	—
56	C₂×C₈×C₄₁	—	4	1	no	—
57	C₂×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₄₁	C₂×C₈	1	1	yes	characteristic
59	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
62	C₄⋊C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₄₁	—	2	1	no	—
64	C₂²⋊C₄×C₄₁	—	2	1	no	—
65	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
66	C₁₆×C₄₁	—	2	2	no	—
67	C₁₆×C₄₁	—	2	1	no	—
68	C₁₆×C₄₁	—	4	1	no	—
69	C₁₆×C₄₁	—	4	1	no	—
70	C₁₆×C₄₁	D₈	1	2	yes	complemented
71	C₂³×C₄₁	—	2	1	no	—
72	C₂³×C₄₁	—	2	1	no	—
73	D₄×C₄₁	—	2	1	no	—
74	D₄×C₄₁	—	2	1	no	—
75	D₄×C₄₁	—	2	1	no	—
76	D₄×C₄₁	M₃₂	1	2	yes	complemented
77	C₂×C₄×C₄₁	—	2	1	no	—
78	C₂×C₄×C₄₁	—	2	1	no	—
79	C₂×C₄×C₄₁	—	4	1	no	—
80	C₂×C₄×C₄₁	—	4	1	no	—
81	C₂×C₄×C₄₁	—	4	1	no	—
82	C₂×C₄×C₄₁	—	4	1	no	—
83	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₄₁	#131	1	1	yes	characteristic
86	C₈×C₄₁	—	2	1	no	—
87	C₈×C₄₁	—	2	1	no	—
88	C₈×C₄₁	—	2	1	no	—
89	C₈×C₄₁	—	4	1	no	—
90	C₈×C₄₁	—	4	1	no	—
91	C₈×C₄₁	—	4	1	no	—
92	C₈×C₄₁	C₂×D₈	1	1	yes	characteristic
93	C₈×C₄₁	#135	1	1	yes	characteristic
94	#12038	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₄₁	—	2	2	no	—
96	C₂²×C₄₁	—	4	1	no	—
97	C₂²×C₄₁	—	4	1	no	—
98	C₂²×C₄₁	—	4	1	no	—
99	C₂²×C₄₁	—	4	1	no	—
100	C₂²×C₄₁	—	4	1	no	—
101	C₂²×C₄₁	C₈×D₄	1	1	yes	characteristic
102	C₄×C₄₁	—	2	1	no	—
103	C₄×C₄₁	—	4	1	no	—
104	C₄×C₄₁	—	4	1	no	—
105	C₄×C₄₁	C₄×D₈	1	1	yes	characteristic
106	C₄×C₄₁	#443	1	1	yes	characteristic
107	C₄×C₄₁	C₂×M₃₂	1	1	yes	characteristic
108	C₄×C₄₁	#504	1	1	yes	characteristic
109	#2169	C₂×C₄₁	1	1	yes	characteristic, complemented
110	D₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
111	C₈×D₈	C₂×C₄₁	1	1	yes	characteristic
112	#1370	C₂×C₄₁	1	1	yes	characteristic, complemented
113	D₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
114	D₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
115	#1311	C₂×C₄₁	1	1	yes	characteristic, complemented
116	C₂×C₄₁	—	4	2	no	—
117	C₂×C₄₁	—	8	1	no	—
118	C₂×C₄₁	C₈×D₈	1	1	yes	characteristic
119	C₂×C₄₁	#2168	1	1	yes	characteristic
120	C₂×C₄₁	#2171	1	1	yes	characteristic
121	C₂×M₃₂	—	2	1	no	—
122	C₂²×C₁₆	—	2	1	no	—
123	C₄×D₈	C₄×C₄₁	1	1	yes	characteristic
124	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
125	C₈×D₄	C₂²×C₄₁	1	1	yes	characteristic
126	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
127	C₄⋊C₁₆	C₂²×C₄₁	1	1	yes	characteristic
128	C₂²⋊C₁₆	—	2	1	no	—
129	C₂²⋊C₁₆	—	2	1	no	—
130	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
131	C₄×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
132	#335	C₄×C₄₁	1	1	yes	characteristic
133	D₄⋊C₈	C₄×C₄₁	1	1	yes	characteristic
134	D₄⋊C₈	C₄×C₄₁	1	1	yes	characteristic
135	C₈²	C₂²×C₄₁	1	1	yes	characteristic
136	C₄₁	#12038	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×D₈	C₈×C₄₁	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
141	C₄×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
142	M₃₂	—	4	2	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	4	2	no	—
147	C₂×C₁₆	D₄×C₄₁	1	1	yes	characteristic
148	C₂×C₁₆	D₄×C₄₁	1	1	yes	characteristic
149	#107	C₈×C₄₁	1	1	yes	characteristic
150	C₄⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
152	#102	C₈×C₄₁	1	1	yes	characteristic
153	#102	C₈×C₄₁	1	1	yes	characteristic
154	C₂²⋊C₈	—	2	1	no	—
155	C₂²⋊C₈	—	2	1	no	—
156	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
157	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
159	C₂×D₄	C₂×C₈×C₄₁	1	1	yes	characteristic
160	C₂×D₄	C₂×C₈×C₄₁	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	D₈	—	2	2	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
172	C₂×C₈	C₂×C₈×C₄₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
174	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
175	C₄⋊C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
180	C₁₆	—	2	2	no	—
181	C₁₆	—	2	1	no	—
182	C₁₆	—	4	1	no	—
183	C₁₆	—	4	1	no	—
184	C₁₆	D₈×C₄₁	1	2	yes	complemented
185	C₂³	—	2	1	no	—
186	C₂³	—	2	1	no	—
187	D₄	—	2	1	no	—
188	D₄	—	2	1	no	—
189	D₄	—	2	1	no	—
190	D₄	M₃₂×C₄₁	1	2	yes	complemented
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
198	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
199	C₂×C₄	#131×C₄₁	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	1	no	—
204	C₈	—	4	1	no	—
205	C₈	—	4	1	no	—
206	C₈	C₂×D₈×C₄₁	1	1	yes	characteristic
207	C₈	#135×C₄₁	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₈×D₄×C₄₁	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	C₄×D₈×C₄₁	1	1	yes	characteristic
219	C₄	C₄₁×#443	1	1	yes	characteristic
220	C₄	C₂×M₃₂×C₄₁	1	1	yes	characteristic
221	C₄	C₄₁×#504	1	1	yes	characteristic
222	C₂	—	4	2	no	—
223	C₂	—	8	1	no	—
224	C₂	C₈×D₈×C₄₁	1	1	yes	characteristic
225	C₂	C₄₁×#2168	1	1	yes	characteristic
226	C₂	C₄₁×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₄₁
Frattini, cosocle	C₄×C₈	C₂³×C₄₁
socle	C₂²×C₄₁	C₈×D₄
prime radical	C₂×D₈×C₄₁	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₄₁ ◁ C₄×C₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#2169	C₂	semidirect
2. D₄×C₁₆×C₄₁	C₂	semidirect
4. C₄₁×#1370	C₂	semidirect
5. C₄₁×D₄⋊C₁₆	C₂	semidirect
6. C₄₁×D₄⋊C₁₆	C₂	semidirect
7. C₄₁×#1311	C₂	semidirect
70. C₁₆×C₄₁	D₈	semidirect
76. D₄×C₄₁	M₃₂	semidirect
94. #12038	C₄₁	direct (N × H)
109. #2169	C₂×C₄₁	semidirect
110. D₄×C₁₆	C₂×C₄₁	semidirect
112. #1370	C₂×C₄₁	semidirect
113. D₄⋊C₁₆	C₂×C₄₁	semidirect
114. D₄⋊C₁₆	C₂×C₄₁	semidirect
115. #1311	C₂×C₄₁	semidirect
136. C₄₁	#12038	direct (N × H)
184. C₁₆	D₈×C₄₁	semidirect
190. D₄	M₃₂×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#2169	2
D₄×C₁₆×C₄₁	2
C₈×D₈×C₄₁	2
C₄₁×#1370	2
C₄₁×D₄⋊C₁₆	2
C₄₁×#1311	2
#12038	41

Subgroups — C41×#12038 #23788239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12038 #23788239