Subgroups — C₄₁×#9842 #23786043

Order 10496.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1990	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1957	C₂	1	2	yes	maximal
4	C₄₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×SD₁₆×C₄₁	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
9	C₄₁×#492	—	2	2	no	—
10	C₄₁×#492	C₄	1	4	yes	complemented
11	C₂×#107×C₄₁	—	2	2	no	—
12	C₂×#102×C₄₁	—	2	4	no	—
13	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₄₁×#384	C₂²	1	2	yes	—
16	C₄₁×#340	—	2	4	no	—
17	C₄₁×#322	—	2	2	no	—
18	C₂²×Q₈×C₄₁	—	2	1	no	—
19	C₂²×D₄×C₄₁	—	2	1	no	—
20	C₂²×D₄×C₄₁	—	2	2	no	—
21	C₂×SD₁₆×C₄₁	—	2	8	no	—
22	C₂×SD₁₆×C₄₁	—	4	4	no	—
23	C₂×SD₁₆×C₄₁	—	4	4	no	—
24	C₂²×C₈×C₄₁	D₄	1	4	yes	—
25	C₂²×C₈×C₄₁	D₄	1	2	yes	—
26	C₄⋊Q₈×C₄₁	—	2	1	no	—
27	C₄⋊Q₈×C₄₁	—	2	2	no	—
28	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
29	#127×C₄₁	—	2	1	no	—
30	#127×C₄₁	—	2	2	no	—
31	#127×C₄₁	C₂×C₄	1	2	yes	—
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	—	2	2	no	—
36	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
37	#107×C₄₁	—	4	4	no	—
38	#102×C₄₁	—	4	8	no	—
39	C₄×C₈×C₄₁	—	2	2	no	—
40	C₄×C₈×C₄₁	—	2	2	no	—
41	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₄₁	D₄	1	2	yes	—
43	C₂⁴×C₄₁	—	4	1	no	—
44	C₂×Q₈×C₄₁	—	2	2	no	—
45	C₂×Q₈×C₄₁	—	4	1	no	—
46	C₂×Q₈×C₄₁	—	4	2	no	—
47	C₂×Q₈×C₄₁	—	4	2	no	—
48	C₂×D₄×C₄₁	—	2	2	no	—
49	C₂×D₄×C₄₁	—	2	4	no	—
50	C₂×D₄×C₄₁	—	4	2	no	—
51	C₂×D₄×C₄₁	—	4	1	no	—
52	C₂×D₄×C₄₁	—	4	2	no	—
53	C₂×D₄×C₄₁	—	4	2	no	—
54	C₂×D₄×C₄₁	—	4	4	no	—
55	C₂×D₄×C₄₁	—	4	4	no	—
56	C₂²×C₄×C₄₁	—	4	1	no	—
57	C₂²×C₄×C₄₁	—	4	2	no	—
58	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
60	SD₁₆×C₄₁	—	4	16	no	—
61	C₂×C₈×C₄₁	—	2	4	no	—
62	C₂×C₈×C₄₁	—	2	4	no	—
63	C₂×C₈×C₄₁	—	2	4	no	—
64	C₂×C₈×C₄₁	C₂²⋊C₄	1	4	yes	—
65	C₂×C₈×C₄₁	D₈	1	4	yes	—
66	C₂×C₈×C₄₁	SD₁₆	1	4	yes	—
67	C₄⋊C₄×C₄₁	—	2	4	no	—
68	C₄⋊C₄×C₄₁	—	4	2	no	—
69	C₄⋊C₄×C₄₁	—	4	4	no	—
70	C₄⋊C₄×C₄₁	—	4	4	no	—
71	C₄⋊C₄×C₄₁	—	4	4	no	—
72	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₁	#41	1	2	yes	—
75	C₂³×C₄₁	—	4	2	no	—
76	C₂³×C₄₁	—	8	1	no	—
77	C₂³×C₄₁	—	8	1	no	—
78	C₂³×C₄₁	—	8	2	no	—
79	C₂³×C₄₁	—	8	2	no	—
80	C₂³×C₄₁	#127	1	1	yes	characteristic
81	Q₈×C₄₁	—	4	4	no	—
82	Q₈×C₄₁	—	8	2	no	—
83	D₄×C₄₁	—	4	4	no	—
84	D₄×C₄₁	—	4	8	no	—
85	D₄×C₄₁	—	8	4	no	—
86	D₄×C₄₁	—	8	2	no	—
87	C₂×C₄×C₄₁	—	4	2	no	—
88	C₂×C₄×C₄₁	—	8	1	no	—
89	C₂×C₄×C₄₁	—	8	1	no	—
90	C₂×C₄×C₄₁	—	8	2	no	—
91	C₂×C₄×C₄₁	—	8	2	no	—
92	C₂×C₄×C₄₁	—	8	2	no	—
93	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₁	#121	1	2	yes	—
96	C₂×C₄×C₄₁	#124	1	1	yes	characteristic
97	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
99	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
100	C₂×C₄×C₄₁	#137	1	4	yes	—
101	C₈×C₄₁	—	4	4	no	—
102	C₈×C₄₁	#102	1	8	yes	complemented
103	#9842	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	—	8	2	no	—
105	C₂²×C₄₁	—	8	4	no	—
106	C₂²×C₄₁	—	16	2	no	—
107	C₂²×C₄₁	—	16	1	no	—
108	C₂²×C₄₁	—	16	1	no	—
109	C₂²×C₄₁	#390	1	1	yes	characteristic
110	C₂²×C₄₁	#492	1	1	yes	characteristic
111	C₂²×C₄₁	#493	1	1	yes	characteristic
112	C₂²×C₄₁	#496	1	2	yes	—
113	C₂²×C₄₁	#497	1	2	yes	—
114	C₄×C₄₁	—	8	2	no	—
115	C₄×C₄₁	—	16	2	no	—
116	C₄×C₄₁	C₂×#102	1	2	yes	—
117	C₄×C₄₁	#440	1	4	yes	—
118	C₄×C₄₁	#468	1	4	yes	—
119	C₄×C₄₁	#486	1	2	yes	—
120	C₂×#492	C₂×C₄₁	1	1	yes	characteristic
121	#1990	C₂×C₄₁	1	2	yes	complemented
122	#1957	C₂×C₄₁	1	2	yes	—
123	#1845	C₂×C₄₁	1	1	yes	characteristic, complemented
124	C₂×#322	C₂×C₄₁	1	1	yes	characteristic, complemented
125	C₂×C₄₁	—	16	2	no	—
126	C₂×C₄₁	#1714	1	2	yes	—
127	C₂×C₄₁	#1721	1	1	yes	characteristic
128	C₂×C₄₁	#1722	1	1	yes	characteristic
129	C₂×C₄₁	#1957	1	1	yes	characteristic
130	C₂×C₄₁	#1973	1	2	yes	—
131	C₂²×SD₁₆	—	2	2	no	—
132	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
133	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
134	#492	—	2	2	no	—
135	#492	C₄×C₄₁	1	4	yes	complemented
136	C₂×#107	—	2	2	no	—
137	C₂×#102	—	2	4	no	—
138	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
140	#384	C₂²×C₄₁	1	2	yes	—
141	#340	—	2	4	no	—
142	#322	—	2	2	no	—
143	C₄₁	#9842	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×D₄	—	2	1	no	—
146	C₂²×D₄	—	2	2	no	—
147	C₂×SD₁₆	—	2	8	no	—
148	C₂×SD₁₆	—	4	4	no	—
149	C₂×SD₁₆	—	4	4	no	—
150	C₂²×C₈	D₄×C₄₁	1	4	yes	—
151	C₂²×C₈	D₄×C₄₁	1	2	yes	—
152	C₄⋊Q₈	—	2	1	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
155	#127	—	2	1	no	—
156	#127	—	2	2	no	—
157	#127	C₂×C₄×C₄₁	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
163	#107	—	4	4	no	—
164	#102	—	4	8	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
168	C₄×C₈	D₄×C₄₁	1	2	yes	—
169	C₂⁴	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂²×C₄	—	4	1	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
186	SD₁₆	—	4	16	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂²⋊C₄×C₄₁	1	4	yes	—
191	C₂×C₈	D₈×C₄₁	1	4	yes	—
192	C₂×C₈	SD₁₆×C₄₁	1	4	yes	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
200	C₄²	#41×C₄₁	1	2	yes	—
201	C₂³	—	4	2	no	—
202	C₂³	—	8	1	no	—
203	C₂³	—	8	1	no	—
204	C₂³	—	8	2	no	—
205	C₂³	—	8	2	no	—
206	C₂³	#127×C₄₁	1	1	yes	characteristic
207	Q₈	—	4	4	no	—
208	Q₈	—	8	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	8	no	—
211	D₄	—	8	4	no	—
212	D₄	—	8	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
220	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
221	C₂×C₄	#121×C₄₁	1	2	yes	—
222	C₂×C₄	#124×C₄₁	1	1	yes	characteristic
223	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
224	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
225	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
226	C₂×C₄	#137×C₄₁	1	4	yes	—
227	C₈	—	4	4	no	—
228	C₈	#102×C₄₁	1	8	yes	complemented
229	C₂²	—	8	2	no	—
230	C₂²	—	8	4	no	—
231	C₂²	—	16	2	no	—
232	C₂²	—	16	1	no	—
233	C₂²	—	16	1	no	—
234	C₂²	C₄₁×#390	1	1	yes	characteristic
235	C₂²	C₄₁×#492	1	1	yes	characteristic
236	C₂²	C₄₁×#493	1	1	yes	characteristic
237	C₂²	C₄₁×#496	1	2	yes	—
238	C₂²	C₄₁×#497	1	2	yes	—
239	C₄	—	8	2	no	—
240	C₄	—	16	2	no	—
241	C₄	C₂×#102×C₄₁	1	2	yes	—
242	C₄	C₄₁×#440	1	4	yes	—
243	C₄	C₄₁×#468	1	4	yes	—
244	C₄	C₄₁×#486	1	2	yes	—
245	C₂	—	16	2	no	—
246	C₂	C₄₁×#1714	1	2	yes	—
247	C₂	C₄₁×#1721	1	1	yes	characteristic
248	C₂	C₄₁×#1722	1	1	yes	characteristic
249	C₂	C₄₁×#1957	1	1	yes	characteristic
250	C₂	C₄₁×#1973	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	#127
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₂×#127×C₄₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1990	C₂	semidirect
4. C₄₁×#1845	C₂	semidirect
5. C₂×C₄₁×#322	C₂	semidirect
10. C₄₁×#492	C₄	semidirect
102. C₈×C₄₁	#102	semidirect
103. #9842	C₄₁	direct (N × H)
121. #1990	C₂×C₄₁	semidirect
123. #1845	C₂×C₄₁	semidirect
124. C₂×#322	C₂×C₄₁	semidirect
135. #492	C₄×C₄₁	semidirect
143. C₄₁	#9842	direct (N × H)
228. C₈	#102×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#492	2
C₄₁×#1990	2
C₄₁×#1957	2
C₄₁×#1845	2
C₂×C₄₁×#322	2
#9842	41

Subgroups — C41×#9842 #23786043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#9842 #23786043