Subgroups — C₂²×C₄₁×#358 #23809576

Order 10496.

1214 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#107×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#358	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₄₁	—	2	1	no	—
6	C₂²×Q₁₆×C₄₁	—	2	3	no	—
7	C₂²×Q₁₆×C₄₁	—	2	4	no	—
8	C₂²×Q₁₆×C₄₁	C₄	1	8	yes	complemented
9	C₂²×Q₁₆×C₄₁	C₂²	1	6	yes	—
10	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₄₁	—	2	1	no	—
12	C₂²×C₁₆×C₄₁	—	2	4	no	—
13	C₂²×C₁₆×C₄₁	C₂²	1	6	yes	—
14	C₂×#107×C₄₁	—	2	3	no	—
15	C₂×#107×C₄₁	C₂²	1	6	yes	—
16	C₄₁×#358	C₂²	1	16	yes	direct product factor
17	C₂²×Q₈×C₄₁	—	2	8	no	—
18	C₂²×Q₈×C₄₁	—	2	6	no	—
19	C₂²×Q₈×C₄₁	—	4	3	no	—
20	C₂²×Q₈×C₄₁	—	4	4	no	—
21	C₂³×C₄×C₄₁	—	4	1	no	—
22	C₂³×C₄×C₄₁	—	4	1	no	—
23	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
24	C₂×Q₁₆×C₄₁	—	2	24	no	—
25	C₂×Q₁₆×C₄₁	—	2	6	no	—
26	C₂×Q₁₆×C₄₁	—	2	12	no	—
27	C₂×Q₁₆×C₄₁	C₂×C₄	1	24	yes	complemented
28	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
29	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₄₁	D₄	1	3	yes	—
31	C₂²×C₈×C₄₁	D₄	1	4	yes	—
32	C₂²×C₈×C₄₁	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₄₁	—	2	6	no	—
34	C₂×C₄⋊C₄×C₄₁	—	4	3	no	—
35	C₂×C₁₆×C₄₁	—	2	12	no	—
36	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
37	#107×C₄₁	—	2	6	no	—
38	#107×C₄₁	C₂³	1	4	yes	—
39	C₂⁴×C₄₁	D₈	1	1	yes	characteristic
40	C₂×Q₈×C₄₁	—	2	24	no	—
41	C₂×Q₈×C₄₁	—	2	4	no	—
42	C₂×Q₈×C₄₁	—	4	24	no	—
43	C₂×Q₈×C₄₁	—	4	6	no	—
44	C₂×Q₈×C₄₁	—	4	12	no	—
45	C₂²×C₄×C₄₁	—	4	8	no	—
46	C₂²×C₄×C₄₁	—	4	6	no	—
47	C₂²×C₄×C₄₁	—	4	6	no	—
48	C₂²×C₄×C₄₁	—	8	4	no	—
49	C₂²×C₄×C₄₁	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₁	D₈	1	3	yes	—
51	C₂²×C₄×C₄₁	SD₁₆	1	4	yes	—
52	C₂²×C₄×C₄₁	C₂×D₄	1	3	yes	—
53	Q₁₆×C₄₁	—	2	24	no	—
54	Q₁₆×C₄₁	—	2	4	no	—
55	Q₁₆×C₄₁	C₂²×C₄	1	8	yes	complemented
56	C₂×C₈×C₄₁	C₂²⋊C₄	1	12	yes	—
57	C₂×C₈×C₄₁	C₂²×C₄	1	6	yes	—
58	C₂×C₈×C₄₁	C₂×D₄	1	3	yes	—
59	C₂×C₈×C₄₁	C₂×D₄	1	6	yes	—
60	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₄₁	—	2	4	no	—
62	C₄⋊C₄×C₄₁	—	4	6	no	—
63	C₁₆×C₄₁	—	2	4	no	—
64	C₂³×C₄₁	#102	1	4	yes	—
65	C₂³×C₄₁	SD₃₂	1	4	yes	—
66	C₂³×C₄₁	Q₃₂	1	4	yes	—
67	C₂³×C₄₁	C₂×D₈	1	3	yes	—
68	Q₈×C₄₁	—	2	8	no	—
69	Q₈×C₄₁	—	4	4	no	—
70	Q₈×C₄₁	—	4	24	no	—
71	C₂×C₄×C₄₁	—	4	4	no	—
72	C₂×C₄×C₄₁	—	4	24	no	—
73	C₂×C₄×C₄₁	—	4	4	no	—
74	C₂×C₄×C₄₁	—	8	12	no	—
75	C₂×C₄×C₄₁	#102	1	12	yes	—
76	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	6	yes	—
77	C₂×C₄×C₄₁	C₂×D₈	1	3	yes	—
78	C₂×C₄×C₄₁	C₂×SD₁₆	1	6	yes	—
79	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
80	C₈×C₄₁	C₂×C₂²⋊C₄	1	6	yes	—
81	C₈×C₄₁	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₄₁	C₂²×D₄	1	1	yes	characteristic
83	C₂²×#358	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₄₁	#358	1	16	yes	direct product factor
85	C₂²×C₄₁	C₂×#102	1	6	yes	—
86	C₂²×C₄₁	C₂×SD₃₂	1	6	yes	—
87	C₂²×C₄₁	C₂×Q₃₂	1	6	yes	—
88	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₄₁	—	4	8	no	—
90	C₄×C₄₁	—	8	4	no	—
91	C₄×C₄₁	C₂×#102	1	6	yes	—
92	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
94	C₂³×Q₁₆	C₂×C₄₁	1	1	yes	characteristic
95	C₂³×C₁₆	C₂×C₄₁	1	1	yes	characteristic
96	C₂²×#107	C₂×C₄₁	1	1	yes	characteristic
97	C₂×#358	C₂×C₄₁	1	12	yes	direct product factor
98	C₂×C₄₁	C₂×#358	1	12	yes	direct product factor
99	C₂×C₄₁	C₂²×#102	1	1	yes	characteristic
100	C₂×C₄₁	C₂²×SD₃₂	1	1	yes	characteristic
101	C₂×C₄₁	C₂²×Q₃₂	1	1	yes	characteristic
102	C₂³×Q₈	—	2	1	no	—
103	C₂²×Q₁₆	—	2	3	no	—
104	C₂²×Q₁₆	—	2	4	no	—
105	C₂²×Q₁₆	C₄×C₄₁	1	8	yes	complemented
106	C₂²×Q₁₆	C₂²×C₄₁	1	6	yes	—
107	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	—	2	1	no	—
109	C₂²×C₁₆	—	2	4	no	—
110	C₂²×C₁₆	C₂²×C₄₁	1	6	yes	—
111	C₂×#107	—	2	3	no	—
112	C₂×#107	C₂²×C₄₁	1	6	yes	—
113	#358	C₂²×C₄₁	1	16	yes	direct product factor
114	C₄₁	C₂²×#358	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×Q₈	—	2	8	no	—
116	C₂²×Q₈	—	2	6	no	—
117	C₂²×Q₈	—	4	3	no	—
118	C₂²×Q₈	—	4	4	no	—
119	C₂³×C₄	—	4	1	no	—
120	C₂³×C₄	—	4	1	no	—
121	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
122	C₂×Q₁₆	—	2	24	no	—
123	C₂×Q₁₆	—	2	6	no	—
124	C₂×Q₁₆	—	2	12	no	—
125	C₂×Q₁₆	C₂×C₄×C₄₁	1	24	yes	complemented
126	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
127	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
128	C₂²×C₈	D₄×C₄₁	1	3	yes	—
129	C₂²×C₈	D₄×C₄₁	1	4	yes	—
130	C₂²×C₈	C₂³×C₄₁	1	3	yes	—
131	C₂×C₄⋊C₄	—	2	6	no	—
132	C₂×C₄⋊C₄	—	4	3	no	—
133	C₂×C₁₆	—	2	12	no	—
134	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
135	#107	—	2	6	no	—
136	#107	C₂³×C₄₁	1	4	yes	—
137	C₂⁴	D₈×C₄₁	1	1	yes	characteristic
138	C₂×Q₈	—	2	24	no	—
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	—	4	24	no	—
141	C₂×Q₈	—	4	6	no	—
142	C₂×Q₈	—	4	12	no	—
143	C₂²×C₄	—	4	8	no	—
144	C₂²×C₄	—	4	6	no	—
145	C₂²×C₄	—	4	6	no	—
146	C₂²×C₄	—	8	4	no	—
147	C₂²×C₄	C₂²⋊C₄×C₄₁	1	4	yes	—
148	C₂²×C₄	D₈×C₄₁	1	3	yes	—
149	C₂²×C₄	SD₁₆×C₄₁	1	4	yes	—
150	C₂²×C₄	C₂×D₄×C₄₁	1	3	yes	—
151	Q₁₆	—	2	24	no	—
152	Q₁₆	—	2	4	no	—
153	Q₁₆	C₂²×C₄×C₄₁	1	8	yes	complemented
154	C₂×C₈	C₂²⋊C₄×C₄₁	1	12	yes	—
155	C₂×C₈	C₂²×C₄×C₄₁	1	6	yes	—
156	C₂×C₈	C₂×D₄×C₄₁	1	3	yes	—
157	C₂×C₈	C₂×D₄×C₄₁	1	6	yes	—
158	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	6	no	—
161	C₁₆	—	2	4	no	—
162	C₂³	#102×C₄₁	1	4	yes	—
163	C₂³	SD₃₂×C₄₁	1	4	yes	—
164	C₂³	Q₃₂×C₄₁	1	4	yes	—
165	C₂³	C₂×D₈×C₄₁	1	3	yes	—
166	Q₈	—	2	8	no	—
167	Q₈	—	4	4	no	—
168	Q₈	—	4	24	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	24	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	12	no	—
173	C₂×C₄	#102×C₄₁	1	12	yes	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	6	yes	—
175	C₂×C₄	C₂×D₈×C₄₁	1	3	yes	—
176	C₂×C₄	C₂×SD₁₆×C₄₁	1	6	yes	—
177	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
178	C₈	C₂×C₂²⋊C₄×C₄₁	1	6	yes	—
179	C₈	C₂³×C₄×C₄₁	1	1	yes	characteristic
180	C₈	C₂²×D₄×C₄₁	1	1	yes	characteristic
181	C₂²	C₄₁×#358	1	16	yes	direct product factor
182	C₂²	C₂×#102×C₄₁	1	6	yes	—
183	C₂²	C₂×SD₃₂×C₄₁	1	6	yes	—
184	C₂²	C₂×Q₃₂×C₄₁	1	6	yes	—
185	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
186	C₄	—	4	8	no	—
187	C₄	—	8	4	no	—
188	C₄	C₂×#102×C₄₁	1	6	yes	—
189	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
190	C₄	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
191	C₂	C₂×C₄₁×#358	1	12	yes	direct product factor
192	C₂	C₂²×#102×C₄₁	1	1	yes	characteristic
193	C₂	C₂²×SD₃₂×C₄₁	1	1	yes	characteristic
194	C₂	C₂²×Q₃₂×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	D₈
commutator	C₈	C₂³×C₄×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ C₂³×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₄₁×#358	C₂	direct (N × H)
8. C₂²×Q₁₆×C₄₁	C₄	semidirect
16. C₄₁×#358	C₂²	direct (N × H)
27. C₂×Q₁₆×C₄₁	C₂×C₄	semidirect
55. Q₁₆×C₄₁	C₂²×C₄	semidirect
83. C₂²×#358	C₄₁	direct (N × H)
84. C₂²×C₄₁	#358	direct (N × H)
97. C₂×#358	C₂×C₄₁	direct (N × H)
98. C₂×C₄₁	C₂×#358	direct (N × H)
105. C₂²×Q₁₆	C₄×C₄₁	semidirect
113. #358	C₂²×C₄₁	direct (N × H)
114. C₄₁	C₂²×#358	direct (N × H)
125. C₂×Q₁₆	C₂×C₄×C₄₁	semidirect
153. Q₁₆	C₂²×C₄×C₄₁	semidirect
181. C₂²	C₄₁×#358	direct (N × H)
191. C₂	C₂×C₄₁×#358	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₄₁	2
C₂³×C₁₆×C₄₁	2
C₂²×#107×C₄₁	2
C₂×C₄₁×#358	2
C₂²×#358	41

Subgroups — C22×C41×#358 #23809576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#358 #23809576