Subgroups — C₄₁×#33232 #23809433

Order 10496.

894 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#491	C₂	1	4	yes	maximal
2	C₄₁×#2868	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2859	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1960	C₂	1	4	yes	maximal
5	C₄₁×#1956	C₂	1	4	yes	maximal
6	C₄×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#125×C₄₁	C₂²	1	2	yes	—
11	C₄₁×#491	—	2	16	no	—
12	C₂×#103×C₄₁	C₂²	1	16	yes	—
13	C₂×#97×C₄₁	C₂²	1	2	yes	—
14	C₂×#97×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
16	C₄₁×#389	—	2	4	no	—
17	C₄₁×#384	—	2	4	no	—
18	C₄₁×#384	C₂²	1	2	yes	—
19	C₄₁×#382	C₂²	1	2	yes	—
20	C₄³×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
22	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
23	C₄⋊Q₈×C₄₁	—	2	8	no	—
24	C₄⋊Q₈×C₄₁	—	2	8	no	—
25	C₄⋊Q₈×C₄₁	—	2	8	no	—
26	#125×C₄₁	—	2	8	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
33	C₂×C₄²×C₄₁	D₄	1	4	yes	—
34	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
37	#103×C₄₁	—	2	32	no	—
38	#97×C₄₁	—	2	4	no	—
39	#97×C₄₁	D₄	1	8	yes	—
40	C₄×C₈×C₄₁	—	2	4	no	—
41	#95×C₄₁	—	2	4	no	—
42	C₂×Q₈×C₄₁	—	2	8	no	—
43	C₂×Q₈×C₄₁	—	2	8	no	—
44	C₂×Q₈×C₄₁	—	2	8	no	—
45	C₂²×C₄×C₄₁	—	2	4	no	—
46	C₂²×C₄×C₄₁	—	2	4	no	—
47	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₄₁	—	2	4	no	—
52	C₂×C₈×C₄₁	—	2	4	no	—
53	C₂×C₈×C₄₁	C₂×D₄	1	8	yes	—
54	C₄⋊C₄×C₄₁	—	2	8	no	—
55	C₄⋊C₄×C₄₁	—	4	8	no	—
56	C₄⋊C₄×C₄₁	—	4	8	no	—
57	C₄⋊C₄×C₄₁	—	4	8	no	—
58	C₄⋊C₄×C₄₁	—	4	8	no	—
59	C₄⋊C₄×C₄₁	—	4	8	no	—
60	C₄²×C₄₁	—	2	8	no	—
61	C₄²×C₄₁	C₂×D₄	1	4	yes	—
62	C₄²×C₄₁	#41	1	4	yes	—
63	C₄²×C₄₁	#41	1	4	yes	—
64	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
65	Q₈×C₄₁	—	4	16	no	—
66	C₂×C₄×C₄₁	—	2	4	no	—
67	C₂×C₄×C₄₁	—	2	4	no	—
68	C₂×C₄×C₄₁	—	4	4	no	—
69	C₂×C₄×C₄₁	—	4	4	no	—
70	C₂×C₄×C₄₁	—	4	4	no	—
71	C₂×C₄×C₄₁	—	4	4	no	—
72	C₂×C₄×C₄₁	—	4	8	no	—
73	C₂×C₄×C₄₁	#124	1	4	yes	—
74	C₂×C₄×C₄₁	#124	1	4	yes	—
75	C₂×C₄×C₄₁	#127	1	4	yes	—
76	C₂×C₄×C₄₁	#137	1	8	yes	—
77	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
79	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
80	C₈×C₄₁	—	2	8	no	—
81	#33232	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
83	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
84	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
85	C₂²×C₄₁	C₂×#137	1	4	yes	—
86	C₄×C₄₁	—	2	4	no	—
87	C₄×C₄₁	—	2	4	no	—
88	C₄×C₄₁	—	8	4	no	—
89	C₄×C₄₁	—	8	4	no	—
90	C₄×C₄₁	#491	1	8	yes	complemented
91	C₄×C₄₁	#532	1	4	yes	—
92	C₂×#491	C₂×C₄₁	1	4	yes	—
93	#2868	C₂×C₄₁	1	1	yes	characteristic
94	#2859	C₂×C₄₁	1	1	yes	characteristic
95	#1960	C₂×C₄₁	1	4	yes	—
96	#1956	C₂×C₄₁	1	4	yes	—
97	C₄×#97	C₂×C₄₁	1	1	yes	characteristic
98	C₂×C₄₁	#2858	1	1	yes	characteristic
99	C₂×C₄₁	C₂×#491	1	2	yes	—
100	C₂×C₄₁	#3139	1	2	yes	—
101	C₂×C₄₁	#3152	1	2	yes	—
102	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
104	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
105	C₂×#125	C₂²×C₄₁	1	2	yes	—
106	#491	—	2	16	no	—
107	C₂×#103	C₂²×C₄₁	1	16	yes	—
108	C₂×#97	C₂²×C₄₁	1	2	yes	—
109	C₂×#97	C₂²×C₄₁	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
111	#389	—	2	4	no	—
112	#384	—	2	4	no	—
113	#384	C₂²×C₄₁	1	2	yes	—
114	#382	C₂²×C₄₁	1	2	yes	—
115	C₄³	C₂²×C₄₁	1	1	yes	characteristic
116	C₄₁	#33232	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
118	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
119	C₄⋊Q₈	—	2	8	no	—
120	C₄⋊Q₈	—	2	8	no	—
121	C₄⋊Q₈	—	2	8	no	—
122	#125	—	2	8	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	4	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
129	C₂×C₄²	D₄×C₄₁	1	4	yes	—
130	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
131	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
133	#103	—	2	32	no	—
134	#97	—	2	4	no	—
135	#97	D₄×C₄₁	1	8	yes	—
136	C₄×C₈	—	2	4	no	—
137	#95	—	2	4	no	—
138	C₂×Q₈	—	2	8	no	—
139	C₂×Q₈	—	2	8	no	—
140	C₂×Q₈	—	2	8	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
146	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂×D₄×C₄₁	1	8	yes	—
150	C₄⋊C₄	—	2	8	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	8	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄²	—	2	8	no	—
157	C₄²	C₂×D₄×C₄₁	1	4	yes	—
158	C₄²	#41×C₄₁	1	4	yes	—
159	C₄²	#41×C₄₁	1	4	yes	—
160	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
161	Q₈	—	4	16	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	#124×C₄₁	1	4	yes	—
170	C₂×C₄	#124×C₄₁	1	4	yes	—
171	C₂×C₄	#127×C₄₁	1	4	yes	—
172	C₂×C₄	#137×C₄₁	1	8	yes	—
173	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
174	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
175	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
176	C₈	—	2	8	no	—
177	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
178	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
179	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
180	C₂²	C₂×#137×C₄₁	1	4	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	8	4	no	—
185	C₄	C₄₁×#491	1	8	yes	complemented
186	C₄	C₄₁×#532	1	4	yes	—
187	C₂	C₄₁×#2858	1	1	yes	characteristic
188	C₂	C₂×C₄₁×#491	1	2	yes	—
189	C₂	C₄₁×#3139	1	2	yes	—
190	C₂	C₄₁×#3152	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×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
81. #33232	C₄₁	direct (N × H)
90. C₄×C₄₁	#491	semidirect
116. C₄₁	#33232	direct (N × H)
185. C₄	C₄₁×#491	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#491	2
C₄₁×#2868	2
C₄₁×#2859	2
C₄₁×#1960	2
C₄₁×#1956	2
C₄×#97×C₄₁	2
#33232	41

Subgroups — C41×#33232 #23809433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33232 #23809433