Subgroups — C₄₁×#21797 #23797998

Order 10496.

1946 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2920	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1965	C₂	1	8	yes	maximal, complemented
4	C₄₁×#1783	C₂	1	4	yes	maximal, complemented
5	C₂³×D₄×C₄₁	C₄	1	4	yes	—
6	C₂³×D₄×C₄₁	C₂²	1	2	yes	—
7	C₂×#127×C₄₁	—	2	4	no	—
8	C₂×#127×C₄₁	—	2	4	no	—
9	C₂×#127×C₄₁	C₄	1	4	yes	—
10	C₂×#127×C₄₁	C₂²	1	4	yes	—
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#432	—	2	4	no	—
13	C₄₁×#431	—	2	4	no	—
14	C₄₁×#423	C₂²	1	4	yes	complemented
15	C₂×#100×C₄₁	C₂²	1	16	yes	complemented
16	C₄₁×#407	C₂²	1	4	yes	—
17	C₄×M₁₆×C₄₁	C₂²	1	4	yes	complemented
18	C₂⁵×C₄₁	C₂×C₄	1	4	yes	—
19	C₂²×D₄×C₄₁	—	2	4	no	—
20	C₂²×D₄×C₄₁	—	2	8	no	—
21	C₂²×D₄×C₄₁	—	2	4	no	—
22	C₂²×D₄×C₄₁	—	2	8	no	—
23	C₂²×D₄×C₄₁	—	4	8	no	—
24	C₂²×D₄×C₄₁	—	4	16	no	—
25	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
26	C₂²×D₄×C₄₁	C₂×C₄	1	16	yes	—
27	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
28	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
31	#127×C₄₁	—	2	4	no	—
32	#127×C₄₁	—	2	8	no	—
33	#127×C₄₁	—	2	4	no	—
34	#127×C₄₁	—	2	4	no	—
35	#127×C₄₁	—	2	4	no	—
36	#127×C₄₁	—	2	8	no	—
37	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₁	D₄	1	4	yes	—
39	C₂×C₄²×C₄₁	D₄	1	4	yes	—
40	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
41	C₄⋊C₈×C₄₁	—	2	8	no	—
42	#100×C₄₁	—	4	16	no	—
43	C₂²⋊C₈×C₄₁	—	2	8	no	—
44	#97×C₄₁	—	2	4	no	—
45	C₄×C₈×C₄₁	—	2	4	no	—
46	C₂⁴×C₄₁	—	2	8	no	—
47	C₂⁴×C₄₁	—	4	8	no	—
48	C₂⁴×C₄₁	—	4	16	no	—
49	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
50	C₂⁴×C₄₁	C₂²×C₄	1	8	yes	—
51	C₂×D₄×C₄₁	—	2	8	no	—
52	C₂×D₄×C₄₁	—	2	16	no	—
53	C₂×D₄×C₄₁	—	2	16	no	—
54	C₂×D₄×C₄₁	—	2	8	no	—
55	C₂×D₄×C₄₁	—	2	16	no	—
56	C₂×D₄×C₄₁	—	4	8	no	—
57	C₂×D₄×C₄₁	—	4	16	no	—
58	C₂×D₄×C₄₁	—	4	32	no	—
59	C₂×D₄×C₄₁	—	4	32	no	—
60	C₂×D₄×C₄₁	—	4	32	no	—
61	C₂×D₄×C₄₁	—	4	16	no	—
62	C₂²×C₄×C₄₁	—	2	8	no	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₄₁	—	2	16	no	—
72	C₂×C₈×C₄₁	—	2	8	no	—
73	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₁	C₂²⋊C₄	1	4	yes	—
75	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₁	C₂×D₄	1	2	yes	—
77	C₄²×C₄₁	C₂×D₄	1	2	yes	—
78	C₄²×C₄₁	C₂×D₄	1	4	yes	—
79	C₂³×C₄₁	—	2	2	no	—
80	C₂³×C₄₁	—	2	4	no	—
81	C₂³×C₄₁	—	2	8	no	—
82	C₂³×C₄₁	—	2	16	no	—
83	C₂³×C₄₁	—	4	16	no	—
84	C₂³×C₄₁	—	4	16	no	—
85	C₂³×C₄₁	—	4	16	no	—
86	C₂³×C₄₁	—	4	16	no	—
87	C₂³×C₄₁	—	8	32	no	—
88	C₂³×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
90	D₄×C₄₁	—	4	16	no	—
91	D₄×C₄₁	—	4	32	no	—
92	D₄×C₄₁	—	4	16	no	—
93	D₄×C₄₁	—	4	32	no	—
94	C₂×C₄×C₄₁	—	2	2	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	8	no	—
97	C₂×C₄×C₄₁	—	2	8	no	—
98	C₂×C₄×C₄₁	—	2	8	no	—
99	C₂×C₄×C₄₁	#100	1	8	yes	complemented
100	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
105	C₈×C₄₁	—	4	8	no	—
106	#21797	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₄₁	—	2	8	no	—
108	C₂²×C₄₁	—	2	2	no	—
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	—	2	2	no	—
111	C₂²×C₄₁	—	4	8	no	—
112	C₂²×C₄₁	—	4	16	no	—
113	C₂²×C₄₁	—	8	16	no	—
114	C₂²×C₄₁	—	8	32	no	—
115	C₂²×C₄₁	C₂×#100	1	4	yes	—
116	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₁	#518	1	2	yes	—
118	C₄×C₄₁	—	2	2	no	—
119	C₄×C₄₁	—	2	4	no	—
120	C₄×C₄₁	C₂×#100	1	8	yes	—
121	C₄×C₄₁	#535	1	4	yes	—
122	C₂²×#127	C₂×C₄₁	1	1	yes	characteristic
123	#2920	C₂×C₄₁	1	2	yes	complemented
124	#1965	C₂×C₄₁	1	8	yes	complemented
125	#1783	C₂×C₄₁	1	4	yes	complemented
126	C₂×C₄₁	—	2	2	no	—
127	C₂×C₄₁	—	2	4	no	—
128	C₂×C₄₁	—	8	8	no	—
129	C₂×C₄₁	#2309	1	1	yes	characteristic
130	C₂×C₄₁	C₂²×#100	1	2	yes	—
131	C₂³×D₄	C₄×C₄₁	1	4	yes	—
132	C₂³×D₄	C₂²×C₄₁	1	2	yes	—
133	C₂×#127	—	2	4	no	—
134	C₂×#127	—	2	4	no	—
135	C₂×#127	C₄×C₄₁	1	4	yes	—
136	C₂×#127	C₂²×C₄₁	1	4	yes	—
137	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
138	#432	—	2	4	no	—
139	#431	—	2	4	no	—
140	#423	C₂²×C₄₁	1	4	yes	complemented
141	C₂×#100	C₂²×C₄₁	1	16	yes	complemented
142	#407	C₂²×C₄₁	1	4	yes	—
143	C₄×M₁₆	C₂²×C₄₁	1	4	yes	complemented
144	C₄₁	#21797	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁵	C₂×C₄×C₄₁	1	4	yes	—
146	C₂²×D₄	—	2	4	no	—
147	C₂²×D₄	—	2	8	no	—
148	C₂²×D₄	—	2	4	no	—
149	C₂²×D₄	—	2	8	no	—
150	C₂²×D₄	—	4	8	no	—
151	C₂²×D₄	—	4	16	no	—
152	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
153	C₂²×D₄	C₂×C₄×C₄₁	1	16	yes	—
154	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
155	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
156	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
157	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
158	#127	—	2	4	no	—
159	#127	—	2	8	no	—
160	#127	—	2	4	no	—
161	#127	—	2	4	no	—
162	#127	—	2	4	no	—
163	#127	—	2	8	no	—
164	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
165	C₂×C₄²	D₄×C₄₁	1	4	yes	—
166	C₂×C₄²	D₄×C₄₁	1	4	yes	—
167	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
168	C₄⋊C₈	—	2	8	no	—
169	#100	—	4	16	no	—
170	C₂²⋊C₈	—	2	8	no	—
171	#97	—	2	4	no	—
172	C₄×C₈	—	2	4	no	—
173	C₂⁴	—	2	8	no	—
174	C₂⁴	—	4	8	no	—
175	C₂⁴	—	4	16	no	—
176	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
177	C₂⁴	C₂²×C₄×C₄₁	1	8	yes	—
178	C₂×D₄	—	2	8	no	—
179	C₂×D₄	—	2	16	no	—
180	C₂×D₄	—	2	16	no	—
181	C₂×D₄	—	2	8	no	—
182	C₂×D₄	—	2	16	no	—
183	C₂×D₄	—	4	8	no	—
184	C₂×D₄	—	4	16	no	—
185	C₂×D₄	—	4	32	no	—
186	C₂×D₄	—	4	32	no	—
187	C₂×D₄	—	4	32	no	—
188	C₂×D₄	—	4	16	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
196	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
197	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
198	M₁₆	—	2	16	no	—
199	C₂×C₈	—	2	8	no	—
200	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
201	C₄²	C₂²⋊C₄×C₄₁	1	4	yes	—
202	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
203	C₄²	C₂×D₄×C₄₁	1	2	yes	—
204	C₄²	C₂×D₄×C₄₁	1	2	yes	—
205	C₄²	C₂×D₄×C₄₁	1	4	yes	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	4	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	16	no	—
210	C₂³	—	4	16	no	—
211	C₂³	—	4	16	no	—
212	C₂³	—	4	16	no	—
213	C₂³	—	4	16	no	—
214	C₂³	—	8	32	no	—
215	C₂³	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
216	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
217	D₄	—	4	16	no	—
218	D₄	—	4	32	no	—
219	D₄	—	4	16	no	—
220	D₄	—	4	32	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	#100×C₄₁	1	8	yes	complemented
227	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
231	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
232	C₈	—	4	8	no	—
233	C₂²	—	2	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	8	no	—
238	C₂²	—	4	16	no	—
239	C₂²	—	8	16	no	—
240	C₂²	—	8	32	no	—
241	C₂²	C₂×#100×C₄₁	1	4	yes	—
242	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
243	C₂²	C₄₁×#518	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	2	4	no	—
246	C₄	C₂×#100×C₄₁	1	8	yes	—
247	C₄	C₄₁×#535	1	4	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	4	no	—
250	C₂	—	8	8	no	—
251	C₂	C₄₁×#2309	1	1	yes	characteristic
252	C₂	C₂²×#100×C₄₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×#127×C₄₁	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2920	C₂	semidirect
3. C₄₁×#1965	C₂	semidirect
4. C₄₁×#1783	C₂	semidirect
14. C₄₁×#423	C₂²	semidirect
15. C₂×#100×C₄₁	C₂²	semidirect
17. C₄×M₁₆×C₄₁	C₂²	semidirect
99. C₂×C₄×C₄₁	#100	semidirect
106. #21797	C₄₁	direct (N × H)
123. #2920	C₂×C₄₁	semidirect
124. #1965	C₂×C₄₁	semidirect
125. #1783	C₂×C₄₁	semidirect
140. #423	C₂²×C₄₁	semidirect
141. C₂×#100	C₂²×C₄₁	semidirect
143. C₄×M₁₆	C₂²×C₄₁	semidirect
144. C₄₁	#21797	direct (N × H)
226. C₂×C₄	#100×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#127×C₄₁	2
C₄₁×#2920	2
C₄₁×#1965	2
C₄₁×#1783	2
#21797	41

Subgroups — C41×#21797 #23797998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21797 #23797998