Subgroups — C₄₁×#3335 #19107115

Order 5248.

388 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#547	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#483	C₂	1	2	yes	maximal, complemented
5	C₄₁×#481	C₂	1	2	yes	maximal, complemented
6	C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#465	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#106×C₄₁	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₄₁	—	2	1	no	—
13	C₂×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₁	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	#125×C₄₁	—	2	1	no	—
18	#123×C₄₁	—	2	2	no	—
19	C₂²⋊Q₈×C₄₁	—	2	1	no	—
20	C₂²⋊Q₈×C₄₁	—	2	2	no	—
21	C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
22	#121×C₄₁	C₂²	1	2	yes	—
23	C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₄₁	—	2	1	no	—
25	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
29	#106×C₄₁	C₂²	1	1	yes	characteristic
30	#106×C₄₁	C₂²	1	2	yes	—
31	#106×C₄₁	C₂²	1	2	yes	—
32	#106×C₄₁	C₂²	1	2	yes	—
33	#106×C₄₁	C₂²	1	2	yes	—
34	C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
35	#103×C₄₁	C₂²	1	1	yes	characteristic
36	#103×C₄₁	C₂²	1	2	yes	—
37	#102×C₄₁	C₂²	1	1	yes	characteristic
38	#102×C₄₁	C₂²	1	2	yes	—
39	C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
40	C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
41	#41×C₄₁	—	4	4	no	—
42	C₂×Q₈×C₄₁	—	2	1	no	—
43	C₂×Q₈×C₄₁	—	2	1	no	—
44	C₂×Q₈×C₄₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₁	—	2	2	no	—
46	C₂×D₄×C₄₁	D₄	1	1	yes	characteristic
47	C₂×D₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	—	2	2	no	—
49	C₂²×C₄×C₄₁	—	2	2	no	—
50	C₂²×C₄×C₄₁	—	2	1	no	—
51	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
52	SD₁₆×C₄₁	—	2	2	no	—
53	C₂×C₈×C₄₁	—	2	4	no	—
54	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₄₁	C₂³	1	2	yes	—
56	C₂×C₈×C₄₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₄₁	—	2	2	no	—
58	C₄⋊C₄×C₄₁	—	2	2	no	—
59	C₄⋊C₄×C₄₁	—	2	1	no	—
60	C₄⋊C₄×C₄₁	—	2	1	no	—
61	C₄⋊C₄×C₄₁	—	2	2	no	—
62	C₄⋊C₄×C₄₁	—	2	1	no	—
63	C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
68	C₂²⋊C₄×C₄₁	—	2	2	no	—
69	C₂²⋊C₄×C₄₁	—	2	1	no	—
70	C₂²⋊C₄×C₄₁	—	2	2	no	—
71	C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
72	C₄²×C₄₁	—	2	1	no	—
73	C₄²×C₄₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄₁	—	2	1	no	—
75	C₂³×C₄₁	C₂×D₄	1	2	yes	—
76	Q₈×C₄₁	—	2	1	no	—
77	Q₈×C₄₁	—	2	1	no	—
78	Q₈×C₄₁	—	4	2	no	—
79	Q₈×C₄₁	—	8	1	no	—
80	D₄×C₄₁	—	2	1	no	—
81	D₄×C₄₁	—	2	1	no	—
82	D₄×C₄₁	—	4	4	no	—
83	D₄×C₄₁	SD₁₆	1	4	yes	complemented
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	2	no	—
86	C₂×C₄×C₄₁	—	2	1	no	—
87	C₂×C₄×C₄₁	—	2	1	no	—
88	C₂×C₄×C₄₁	—	2	2	no	—
89	C₂×C₄×C₄₁	—	2	1	no	—
90	C₂×C₄×C₄₁	—	2	2	no	—
91	C₂×C₄×C₄₁	—	4	4	no	—
92	C₂×C₄×C₄₁	—	4	2	no	—
93	C₂×C₄×C₄₁	—	8	2	no	—
94	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
98	C₈×C₄₁	—	2	2	no	—
99	C₈×C₄₁	—	2	1	no	—
100	C₈×C₄₁	#41	1	2	yes	complemented
101	C₂²×C₄₁	—	2	2	no	—
102	C₂²×C₄₁	—	2	2	no	—
103	C₂²×C₄₁	—	4	1	no	—
104	C₂²×C₄₁	—	4	1	no	—
105	C₂²×C₄₁	—	4	1	no	—
106	C₂²×C₄₁	C₂×SD₁₆	1	4	yes	—
107	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₄₁	—	2	2	no	—
109	C₄×C₄₁	—	2	1	no	—
110	C₄×C₄₁	—	4	1	no	—
111	C₄×C₄₁	—	4	1	no	—
112	C₄×C₄₁	—	4	2	no	—
113	C₄×C₄₁	—	4	1	no	—
114	C₄×C₄₁	—	4	2	no	—
115	C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
116	C₄×C₄₁	C₂×#41	1	1	yes	characteristic
117	C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
118	#3335	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₄₁	—	2	4	no	—
120	C₂×C₄₁	—	8	1	no	—
121	C₂×C₄₁	#547	1	1	yes	characteristic
122	C₂×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
123	C₂×C₄₁	#577	1	1	yes	characteristic
124	#554	C₂×C₄₁	1	1	yes	characteristic, complemented
125	#547	C₂×C₄₁	1	1	yes	characteristic
126	#498	C₂×C₄₁	1	1	yes	characteristic, complemented
127	#483	C₂×C₄₁	1	2	yes	complemented
128	#481	C₂×C₄₁	1	2	yes	complemented
129	#476	C₂×C₄₁	1	1	yes	characteristic, complemented
130	#475	C₂×C₄₁	1	1	yes	characteristic, complemented
131	#465	C₂×C₄₁	1	2	yes	complemented
132	C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
133	C₈×D₄	C₂×C₄₁	1	1	yes	characteristic, complemented
134	C₂×#106	C₂×C₄₁	1	2	yes	complemented
135	C₄₁	#3335	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#41	—	2	1	no	—
137	C₂×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₄₁	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
140	C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
141	#125	—	2	1	no	—
142	#123	—	2	2	no	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
146	#121	C₂²×C₄₁	1	2	yes	—
147	C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
148	C₄×D₄	—	2	1	no	—
149	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
150	C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
153	#106	C₂²×C₄₁	1	1	yes	characteristic
154	#106	C₂²×C₄₁	1	2	yes	—
155	#106	C₂²×C₄₁	1	2	yes	—
156	#106	C₂²×C₄₁	1	2	yes	—
157	#106	C₂²×C₄₁	1	2	yes	—
158	C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
159	#103	C₂²×C₄₁	1	1	yes	characteristic
160	#103	C₂²×C₄₁	1	2	yes	—
161	#102	C₂²×C₄₁	1	1	yes	characteristic
162	#102	C₂²×C₄₁	1	2	yes	—
163	C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
164	C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
165	#41	—	4	4	no	—
166	C₂×Q₈	—	2	1	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	C₂³×C₄₁	1	1	yes	characteristic
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	D₄×C₄₁	1	1	yes	characteristic
171	C₂×D₄	C₂³×C₄₁	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
176	SD₁₆	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₄₁	1	2	yes	—
180	C₂×C₈	C₂³×C₄₁	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
190	C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
196	C₄²	—	2	1	no	—
197	C₄²	C₂³×C₄₁	1	1	yes	characteristic
198	C₂³	—	2	1	no	—
199	C₂³	C₂×D₄×C₄₁	1	2	yes	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	8	1	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	4	4	no	—
207	D₄	SD₁₆×C₄₁	1	4	yes	complemented
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
221	C₂×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	#41×C₄₁	1	2	yes	complemented
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂×SD₁₆×C₄₁	1	4	yes	—
231	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
240	C₄	C₂×#41×C₄₁	1	1	yes	characteristic
241	C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
242	C₂	—	2	4	no	—
243	C₂	—	8	1	no	—
244	C₂	C₄₁×#547	1	1	yes	characteristic
245	C₂	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
246	C₂	C₄₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#554	C₂	semidirect
3. C₄₁×#498	C₂	semidirect
4. C₄₁×#483	C₂	semidirect
5. C₄₁×#481	C₂	semidirect
6. C₄₁×#476	C₂	semidirect
7. C₄₁×#475	C₂	semidirect
8. C₄₁×#465	C₂	semidirect
9. C₄×SD₁₆×C₄₁	C₂	semidirect
10. C₈×D₄×C₄₁	C₂	semidirect
11. C₂×#106×C₄₁	C₂	semidirect
83. D₄×C₄₁	SD₁₆	semidirect
100. C₈×C₄₁	#41	semidirect
118. #3335	C₄₁	direct (N × H)
124. #554	C₂×C₄₁	semidirect
126. #498	C₂×C₄₁	semidirect
127. #483	C₂×C₄₁	semidirect
128. #481	C₂×C₄₁	semidirect
129. #476	C₂×C₄₁	semidirect
130. #475	C₂×C₄₁	semidirect
131. #465	C₂×C₄₁	semidirect
132. C₄×SD₁₆	C₂×C₄₁	semidirect
133. C₈×D₄	C₂×C₄₁	semidirect
134. C₂×#106	C₂×C₄₁	semidirect
135. C₄₁	#3335	direct (N × H)
207. D₄	SD₁₆×C₄₁	semidirect
224. C₈	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#554	2
C₄₁×#547	2
C₄₁×#498	2
C₄₁×#483	2
C₄₁×#481	2
C₄₁×#476	2
C₄₁×#475	2
C₄₁×#465	2
C₄×SD₁₆×C₄₁	2
C₈×D₄×C₄₁	2
C₂×#106×C₄₁	2
#3335	41

Subgroups — C41×#3335 #19107115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#3335 #19107115