Subgroups — C₄₁×#3289 #19107069

Order 5248.

448 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#578	C₂	1	2	yes	maximal, complemented
2	C₄₁×#577	C₂	1	2	yes	maximal, complemented
3	C₄₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#473	C₂	1	2	yes	maximal, complemented
5	C₄₁×#471	C₂	1	2	yes	maximal, complemented
6	C₄₁×#456	C₂	1	2	yes	maximal, complemented
7	C₄₁×#455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#454	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#444	C₂	1	2	yes	maximal, complemented
10	2₋¹⁺⁴×C₄₁	C₂²	1	2	yes	—
11	2₋¹⁺⁴×C₄₁	C₂²	1	1	yes	characteristic, complemented
12	H(5,𝔽₂)×C₄₁	C₂²	1	1	yes	characteristic
13	#137×C₄₁	—	2	2	no	—
14	#137×C₄₁	—	2	2	no	—
15	#137×C₄₁	—	2	2	no	—
16	#137×C₄₁	C₂²	1	2	yes	—
17	#137×C₄₁	C₂²	1	2	yes	—
18	#137×C₄₁	C₂²	1	2	yes	complemented
19	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
20	AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
21	#135×C₄₁	—	2	2	no	—
22	#135×C₄₁	—	2	2	no	—
23	#135×C₄₁	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₄₁	—	2	2	no	—
25	C₂×Q₁₆×C₄₁	C₂²	1	2	yes	complemented
26	C₂×SD₁₆×C₄₁	—	2	2	no	—
27	C₂×SD₁₆×C₄₁	C₂²	1	2	yes	—
28	#131×C₄₁	C₂²	1	2	yes	—
29	#131×C₄₁	C₂²	1	2	yes	complemented
30	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
31	#124×C₄₁	C₂²	1	1	yes	characteristic
32	#108×C₄₁	C₂²	1	2	yes	complemented
33	#104×C₄₁	C₂²	1	2	yes	—
34	#104×C₄₁	C₂²	1	2	yes	complemented
35	#101×C₄₁	C₂²	1	1	yes	characteristic, complemented
36	#101×C₄₁	C₂²	1	2	yes	complemented
37	#100×C₄₁	C₂²	1	1	yes	characteristic
38	#97×C₄₁	C₂²	1	1	yes	characteristic, complemented
39	#41×C₄₁	—	2	2	no	—
40	#41×C₄₁	—	2	2	no	—
41	#41×C₄₁	—	2	2	no	—
42	#41×C₄₁	—	2	2	no	—
43	#41×C₄₁	—	4	2	no	—
44	#41×C₄₁	—	4	1	no	—
45	#41×C₄₁	C₂³	1	2	yes	—
46	#41×C₄₁	C₂³	1	2	yes	—
47	C₂×Q₈×C₄₁	—	2	2	no	—
48	C₂×Q₈×C₄₁	—	2	2	no	—
49	C₂×Q₈×C₄₁	—	2	2	no	—
50	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×Q₈×C₄₁	C₂³	1	2	yes	—
52	C₂×D₄×C₄₁	—	2	2	no	—
53	C₂×D₄×C₄₁	—	4	1	no	—
54	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	Q₁₆×C₄₁	—	2	2	no	—
56	Q₁₆×C₄₁	—	2	2	no	—
57	Q₁₆×C₄₁	—	2	2	no	—
58	Q₁₆×C₄₁	—	2	2	no	—
59	Q₁₆×C₄₁	—	2	2	no	—
60	Q₁₆×C₄₁	D₄	1	2	yes	—
61	SD₁₆×C₄₁	—	2	2	no	—
62	SD₁₆×C₄₁	—	2	2	no	—
63	SD₁₆×C₄₁	—	2	2	no	—
64	SD₁₆×C₄₁	—	2	2	no	—
65	SD₁₆×C₄₁	—	2	2	no	—
66	SD₁₆×C₄₁	—	2	2	no	—
67	SD₁₆×C₄₁	D₄	1	2	yes	—
68	SD₁₆×C₄₁	D₄	1	2	yes	—
69	D₈×C₄₁	—	2	2	no	—
70	D₈×C₄₁	D₄	1	2	yes	—
71	M₁₆×C₄₁	—	2	2	no	—
72	M₁₆×C₄₁	—	2	2	no	—
73	M₁₆×C₄₁	C₂³	1	2	yes	—
74	M₁₆×C₄₁	C₂³	1	2	yes	—
75	C₂×C₈×C₄₁	—	2	2	no	—
76	C₂×C₈×C₄₁	—	2	2	no	—
77	C₂×C₈×C₄₁	C₂³	1	2	yes	—
78	C₄⋊C₄×C₄₁	—	4	1	no	—
79	C₂²⋊C₄×C₄₁	—	4	1	no	—
80	C₄²×C₄₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄₁	—	2	1	no	—
82	C₂³×C₄₁	—	4	1	no	—
83	Q₈×C₄₁	—	2	2	no	—
84	Q₈×C₄₁	—	2	2	no	—
85	Q₈×C₄₁	—	2	2	no	—
86	Q₈×C₄₁	—	4	2	no	—
87	Q₈×C₄₁	—	4	1	no	—
88	Q₈×C₄₁	C₂×D₄	1	2	yes	—
89	Q₈×C₄₁	C₂×D₄	1	2	yes	—
90	D₄×C₄₁	—	2	2	no	—
91	D₄×C₄₁	—	2	2	no	—
92	D₄×C₄₁	—	2	2	no	—
93	D₄×C₄₁	—	4	2	no	—
94	D₄×C₄₁	—	4	1	no	—
95	D₄×C₄₁	—	4	2	no	—
96	D₄×C₄₁	C₂×D₄	1	2	yes	—
97	D₄×C₄₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	2	no	—
100	C₂×C₄×C₄₁	—	2	1	no	—
101	C₂×C₄×C₄₁	—	2	1	no	—
102	C₂×C₄×C₄₁	—	2	2	no	—
103	C₂×C₄×C₄₁	—	4	1	no	—
104	C₂×C₄×C₄₁	—	4	2	no	—
105	C₂×C₄×C₄₁	—	4	2	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
108	C₈×C₄₁	—	2	2	no	—
109	C₈×C₄₁	—	2	2	no	—
110	C₈×C₄₁	C₂×D₄	1	2	yes	—
111	C₈×C₄₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄₁	—	2	2	no	—
113	C₂²×C₄₁	—	2	2	no	—
114	C₂²×C₄₁	—	4	1	no	—
115	C₂²×C₄₁	—	8	1	no	—
116	C₂²×C₄₁	—	16	1	no	—
117	C₂²×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
118	C₄×C₄₁	—	2	2	no	—
119	C₄×C₄₁	—	2	2	no	—
120	C₄×C₄₁	—	4	1	no	—
121	C₄×C₄₁	—	4	1	no	—
122	C₄×C₄₁	—	4	2	no	—
123	C₄×C₄₁	C₂²×D₄	1	2	yes	—
124	#3289	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₄₁	—	2	1	no	—
126	C₂×C₄₁	—	4	2	no	—
127	C₂×C₄₁	—	4	2	no	—
128	C₂×C₄₁	—	8	1	no	—
129	C₂×C₄₁	D₄²	1	1	yes	characteristic
130	#578	C₂×C₄₁	1	2	yes	complemented
131	#577	C₂×C₄₁	1	2	yes	complemented
132	#497	C₂×C₄₁	1	1	yes	characteristic, complemented
133	#473	C₂×C₄₁	1	2	yes	complemented
134	#471	C₂×C₄₁	1	2	yes	complemented
135	#456	C₂×C₄₁	1	2	yes	complemented
136	#455	C₂×C₄₁	1	1	yes	characteristic, complemented
137	#454	C₂×C₄₁	1	1	yes	characteristic, complemented
138	#444	C₂×C₄₁	1	2	yes	complemented
139	C₄₁	#3289	1	1	yes	characteristic, direct product factor, Sylow
140	2₋¹⁺⁴	C₂²×C₄₁	1	2	yes	—
141	2₋¹⁺⁴	C₂²×C₄₁	1	1	yes	characteristic, complemented
142	H(5,𝔽₂)	C₂²×C₄₁	1	1	yes	characteristic
143	#137	—	2	2	no	—
144	#137	—	2	2	no	—
145	#137	—	2	2	no	—
146	#137	C₂²×C₄₁	1	2	yes	—
147	#137	C₂²×C₄₁	1	2	yes	—
148	#137	C₂²×C₄₁	1	2	yes	complemented
149	AGL(1,ℤ/8)	—	2	2	no	—
150	AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
151	#135	—	2	2	no	—
152	#135	—	2	2	no	—
153	#135	C₂²×C₄₁	1	2	yes	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	C₂²×C₄₁	1	2	yes	complemented
156	C₂×SD₁₆	—	2	2	no	—
157	C₂×SD₁₆	C₂²×C₄₁	1	2	yes	—
158	#131	C₂²×C₄₁	1	2	yes	—
159	#131	C₂²×C₄₁	1	2	yes	complemented
160	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
161	#124	C₂²×C₄₁	1	1	yes	characteristic
162	#108	C₂²×C₄₁	1	2	yes	complemented
163	#104	C₂²×C₄₁	1	2	yes	—
164	#104	C₂²×C₄₁	1	2	yes	complemented
165	#101	C₂²×C₄₁	1	1	yes	characteristic, complemented
166	#101	C₂²×C₄₁	1	2	yes	complemented
167	#100	C₂²×C₄₁	1	1	yes	characteristic
168	#97	C₂²×C₄₁	1	1	yes	characteristic, complemented
169	#41	—	2	2	no	—
170	#41	—	2	2	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	#41	—	4	2	no	—
174	#41	—	4	1	no	—
175	#41	C₂³×C₄₁	1	2	yes	—
176	#41	C₂³×C₄₁	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
181	C₂×Q₈	C₂³×C₄₁	1	2	yes	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	4	1	no	—
184	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
185	Q₁₆	—	2	2	no	—
186	Q₁₆	—	2	2	no	—
187	Q₁₆	—	2	2	no	—
188	Q₁₆	—	2	2	no	—
189	Q₁₆	—	2	2	no	—
190	Q₁₆	D₄×C₄₁	1	2	yes	—
191	SD₁₆	—	2	2	no	—
192	SD₁₆	—	2	2	no	—
193	SD₁₆	—	2	2	no	—
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	2	2	no	—
196	SD₁₆	—	2	2	no	—
197	SD₁₆	D₄×C₄₁	1	2	yes	—
198	SD₁₆	D₄×C₄₁	1	2	yes	—
199	D₈	—	2	2	no	—
200	D₈	D₄×C₄₁	1	2	yes	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	C₂³×C₄₁	1	2	yes	—
204	M₁₆	C₂³×C₄₁	1	2	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂³×C₄₁	1	2	yes	—
208	C₄⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	1	no	—
210	C₄²	C₂³×C₄₁	1	1	yes	characteristic
211	C₂³	—	2	1	no	—
212	C₂³	—	4	1	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	Q₈	—	4	1	no	—
218	Q₈	C₂×D₄×C₄₁	1	2	yes	—
219	Q₈	C₂×D₄×C₄₁	1	2	yes	—
220	D₄	—	2	2	no	—
221	D₄	—	2	2	no	—
222	D₄	—	2	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	1	no	—
225	D₄	—	4	2	no	—
226	D₄	C₂×D₄×C₄₁	1	2	yes	—
227	D₄	C₂×D₄×C₄₁	1	2	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	C₂×D₄×C₄₁	1	2	yes	—
241	C₈	C₂×D₄×C₄₁	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	16	1	no	—
247	C₂²	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂²×D₄×C₄₁	1	2	yes	—
254	C₂	—	2	1	no	—
255	C₂	—	4	2	no	—
256	C₂	—	4	2	no	—
257	C₂	—	8	1	no	—
258	C₂	D₄²×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#578	C₂	semidirect
2. C₄₁×#577	C₂	semidirect
3. C₄₁×#497	C₂	semidirect
4. C₄₁×#473	C₂	semidirect
5. C₄₁×#471	C₂	semidirect
6. C₄₁×#456	C₂	semidirect
7. C₄₁×#455	C₂	semidirect
8. C₄₁×#454	C₂	semidirect
9. C₄₁×#444	C₂	semidirect
11. 2₋¹⁺⁴×C₄₁	C₂²	semidirect
18. #137×C₄₁	C₂²	semidirect
25. C₂×Q₁₆×C₄₁	C₂²	semidirect
29. #131×C₄₁	C₂²	semidirect
30. C₄⋊Q₈×C₄₁	C₂²	semidirect
32. #108×C₄₁	C₂²	semidirect
34. #104×C₄₁	C₂²	semidirect
35. #101×C₄₁	C₂²	semidirect
36. #101×C₄₁	C₂²	semidirect
38. #97×C₄₁	C₂²	semidirect
124. #3289	C₄₁	direct (N × H)
130. #578	C₂×C₄₁	semidirect
131. #577	C₂×C₄₁	semidirect
132. #497	C₂×C₄₁	semidirect
133. #473	C₂×C₄₁	semidirect
134. #471	C₂×C₄₁	semidirect
135. #456	C₂×C₄₁	semidirect
136. #455	C₂×C₄₁	semidirect
137. #454	C₂×C₄₁	semidirect
138. #444	C₂×C₄₁	semidirect
139. C₄₁	#3289	direct (N × H)
141. 2₋¹⁺⁴	C₂²×C₄₁	semidirect
148. #137	C₂²×C₄₁	semidirect
155. C₂×Q₁₆	C₂²×C₄₁	semidirect
159. #131	C₂²×C₄₁	semidirect
160. C₄⋊Q₈	C₂²×C₄₁	semidirect
162. #108	C₂²×C₄₁	semidirect
164. #104	C₂²×C₄₁	semidirect
165. #101	C₂²×C₄₁	semidirect
166. #101	C₂²×C₄₁	semidirect
168. #97	C₂²×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#578	2
C₄₁×#577	2
C₄₁×#497	2
C₄₁×#473	2
C₄₁×#471	2
C₄₁×#456	2
C₄₁×#455	2
C₄₁×#454	2
C₄₁×#444	2
#3289	41

Subgroups — C41×#3289 #19107069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3289 #19107069