Subgroups — C₄₁×#21691 #23797892

Order 10496.

1062 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#441	C₂	1	4	yes	maximal
3	C₄×C₄⋊Q₈×C₄₁	C₂	1	2	yes	maximal
4	C₄₁×#1971	C₂	1	4	yes	maximal
5	C₄₁×#1956	C₂	1	2	yes	maximal
6	C₄₁×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₄₁	C₂²	1	4	yes	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₄₁	C₂²	1	4	yes	—
11	C₄₁×#494	C₄	1	8	yes	complemented
12	C₄₁×#441	—	2	16	no	—
13	C₂×#106×C₄₁	C₂²	1	4	yes	—
14	C₂×#103×C₄₁	C₂²	1	8	yes	—
15	C₂×#97×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#391	C₂²	1	4	yes	—
20	C₄₁×#389	—	2	4	no	—
21	C₄₁×#384	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₄₁	—	2	4	no	—
23	C₄³×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
25	C₂×Q₁₆×C₄₁	—	2	8	no	—
26	C₂×Q₁₆×C₄₁	—	2	8	no	—
27	C₂×Q₁₆×C₄₁	C₂×C₄	1	16	yes	—
28	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₄₁	—	2	4	no	—
31	C₄⋊Q₈×C₄₁	C₂×C₄	1	8	yes	—
32	C₄×Q₈×C₄₁	—	2	16	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
37	C₂×C₄²×C₄₁	—	2	4	no	—
38	C₂×C₄²×C₄₁	D₄	1	4	yes	—
39	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
42	#106×C₄₁	—	2	8	no	—
43	#103×C₄₁	—	2	16	no	—
44	#97×C₄₁	—	2	4	no	—
45	#97×C₄₁	D₄	1	8	yes	—
46	C₄×C₈×C₄₁	—	2	2	no	—
47	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
48	#95×C₄₁	—	2	8	no	—
49	C₂×Q₈×C₄₁	—	2	4	no	—
50	C₂×Q₈×C₄₁	—	2	8	no	—
51	C₂×Q₈×C₄₁	—	2	8	no	—
52	C₂×Q₈×C₄₁	C₂²×C₄	1	8	yes	—
53	C₂²×C₄×C₄₁	—	2	4	no	—
54	C₂²×C₄×C₄₁	—	2	4	no	—
55	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₄₁	—	2	32	no	—
60	C₂×C₈×C₄₁	—	2	2	no	—
61	C₂×C₈×C₄₁	—	2	2	no	—
62	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₄₁	#41	1	4	yes	—
66	C₄⋊C₄×C₄₁	—	2	16	no	—
67	C₄⋊C₄×C₄₁	—	2	8	no	—
68	C₄⋊C₄×C₄₁	—	4	8	no	—
69	C₄⋊C₄×C₄₁	—	4	8	no	—
70	C₄²×C₄₁	—	2	8	no	—
71	C₄²×C₄₁	—	2	2	no	—
72	C₄²×C₄₁	—	4	8	no	—
73	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₁	C₂×D₄	1	4	yes	—
75	C₄²×C₄₁	#41	1	2	yes	—
76	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₄₁	—	2	16	no	—
78	Q₈×C₄₁	—	4	8	no	—
79	C₂×C₄×C₄₁	—	2	4	no	—
80	C₂×C₄×C₄₁	—	2	8	no	—
81	C₂×C₄×C₄₁	—	2	2	no	—
82	C₂×C₄×C₄₁	—	2	4	no	—
83	C₂×C₄×C₄₁	—	4	4	no	—
84	C₂×C₄×C₄₁	—	4	4	no	—
85	C₂×C₄×C₄₁	—	4	4	no	—
86	C₂×C₄×C₄₁	—	4	8	no	—
87	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₄₁	#127	1	4	yes	—
89	C₂×C₄×C₄₁	#137	1	8	yes	—
90	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
93	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
94	C₈×C₄₁	—	2	4	no	—
95	C₈×C₄₁	C₄×D₄	1	8	yes	—
96	#21691	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
99	C₂²×C₄₁	#532	1	1	yes	characteristic
100	C₂²×C₄₁	C₂×#137	1	4	yes	—
101	C₄×C₄₁	—	2	4	no	—
102	C₄×C₄₁	—	4	8	no	—
103	C₄×C₄₁	—	4	2	no	—
104	C₄×C₄₁	—	8	4	no	—
105	C₄×C₄₁	#441	1	8	yes	complemented
106	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
107	C₄×C₄₁	#532	1	2	yes	—
108	C₂×#494	C₂×C₄₁	1	1	yes	characteristic
109	C₂×#441	C₂×C₄₁	1	4	yes	—
110	C₄×C₄⋊Q₈	C₂×C₄₁	1	2	yes	—
111	#1971	C₂×C₄₁	1	4	yes	—
112	#1956	C₂×C₄₁	1	2	yes	—
113	#1844	C₂×C₄₁	1	1	yes	characteristic
114	C₄×#97	C₂×C₄₁	1	1	yes	characteristic
115	C₂×C₄₁	C₄×#127	1	1	yes	characteristic
116	C₂×C₄₁	C₂×#441	1	2	yes	—
117	C₂×C₄₁	#3152	1	2	yes	—
118	C₂×C₄₁	#3185	1	2	yes	—
119	C₂²×Q₁₆	C₂²×C₄₁	1	4	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
121	C₂×C₄×Q₈	C₂²×C₄₁	1	4	yes	—
122	#494	C₄×C₄₁	1	8	yes	complemented
123	#441	—	2	16	no	—
124	C₂×#106	C₂²×C₄₁	1	4	yes	—
125	C₂×#103	C₂²×C₄₁	1	8	yes	—
126	C₂×#97	C₂²×C₄₁	1	2	yes	—
127	C₂×#97	C₂²×C₄₁	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
130	#391	C₂²×C₄₁	1	4	yes	—
131	#389	—	2	4	no	—
132	#384	C₂²×C₄₁	1	2	yes	—
133	C₄×C₄⋊C₄	—	2	4	no	—
134	C₄³	C₂²×C₄₁	1	1	yes	characteristic
135	C₄₁	#21691	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
137	C₂×Q₁₆	—	2	8	no	—
138	C₂×Q₁₆	—	2	8	no	—
139	C₂×Q₁₆	C₂×C₄×C₄₁	1	16	yes	—
140	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
141	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
142	C₄⋊Q₈	—	2	4	no	—
143	C₄⋊Q₈	C₂×C₄×C₄₁	1	8	yes	—
144	C₄×Q₈	—	2	16	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
149	C₂×C₄²	—	2	4	no	—
150	C₂×C₄²	D₄×C₄₁	1	4	yes	—
151	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
154	#106	—	2	8	no	—
155	#103	—	2	16	no	—
156	#97	—	2	4	no	—
157	#97	D₄×C₄₁	1	8	yes	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
160	#95	—	2	8	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	2	8	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	C₂²×C₄×C₄₁	1	8	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
168	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
171	Q₁₆	—	2	32	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
175	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
176	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
177	C₂×C₈	#41×C₄₁	1	4	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₄²	—	2	8	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	4	8	no	—
185	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
186	C₄²	C₂×D₄×C₄₁	1	4	yes	—
187	C₄²	#41×C₄₁	1	2	yes	—
188	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	4	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
200	C₂×C₄	#127×C₄₁	1	4	yes	—
201	C₂×C₄	#137×C₄₁	1	8	yes	—
202	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
204	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
205	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
206	C₈	—	2	4	no	—
207	C₈	C₄×D₄×C₄₁	1	8	yes	—
208	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
209	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
210	C₂²	C₄₁×#532	1	1	yes	characteristic
211	C₂²	C₂×#137×C₄₁	1	4	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	4	8	no	—
214	C₄	—	4	2	no	—
215	C₄	—	8	4	no	—
216	C₄	C₄₁×#441	1	8	yes	complemented
217	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
218	C₄	C₄₁×#532	1	2	yes	—
219	C₂	C₄×#127×C₄₁	1	1	yes	characteristic
220	C₂	C₂×C₄₁×#441	1	2	yes	—
221	C₂	C₄₁×#3152	1	2	yes	—
222	C₂	C₄₁×#3185	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄₁×#494	C₄	semidirect
96. #21691	C₄₁	direct (N × H)
105. C₄×C₄₁	#441	semidirect
122. #494	C₄×C₄₁	semidirect
135. C₄₁	#21691	direct (N × H)
216. C₄	C₄₁×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#494	2
C₂×C₄₁×#441	2
C₄×C₄⋊Q₈×C₄₁	2
C₄₁×#1971	2
C₄₁×#1956	2
C₄₁×#1844	2
C₄×#97×C₄₁	2
#21691	41

Subgroups — C41×#21691 #23797892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21691 #23797892