Subgroups — C₂×C₄₁×#2176 #23809650

Order 10496.

670 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₄₁	C₂	1	2	yes	maximal
3	C₄₁×#2176	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₁×#367	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₁×#358	C₂	1	2	yes	maximal
6	C₂×C₄₁×#346	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₄₁	—	2	2	no	—
9	C₂×Q₃₂×C₄₁	C₄	1	8	yes	complemented
10	C₂×Q₃₂×C₄₁	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₄×Q₁₆×C₄₁	—	2	4	no	—
14	C₄×Q₁₆×C₄₁	C₂²	1	8	yes	—
15	C₂×#107×C₄₁	C₂²	1	2	yes	—
16	C₂×#103×C₄₁	—	2	2	no	—
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×#367	C₂²	1	4	yes	—
19	C₄₁×#358	C₂²	1	8	yes	—
20	C₄₁×#346	C₂²	1	4	yes	—
21	C₂²×Q₈×C₄₁	—	2	2	no	—
22	C₂×Q₁₆×C₄₁	—	2	2	no	—
23	C₂×Q₁₆×C₄₁	—	2	4	no	—
24	C₂×Q₁₆×C₄₁	C₂×C₄	1	8	yes	—
25	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₄₁	—	2	8	no	—
29	C₄×Q₈×C₄₁	—	4	4	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₁	—	4	2	no	—
32	C₂×C₄²×C₄₁	—	4	2	no	—
33	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
34	Q₃₂×C₄₁	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆×C₄₁	—	2	2	no	—
36	C₂×C₁₆×C₄₁	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
38	C₂×C₁₆×C₄₁	C₂³	1	2	yes	—
39	#107×C₄₁	—	2	2	no	—
40	#107×C₄₁	C₂³	1	4	yes	—
41	#103×C₄₁	—	2	8	no	—
42	C₄×C₈×C₄₁	D₄	1	2	yes	—
43	C₄×C₈×C₄₁	C₂³	1	2	yes	—
44	C₂×Q₈×C₄₁	—	2	8	no	—
45	C₂×Q₈×C₄₁	—	2	4	no	—
46	C₂×Q₈×C₄₁	—	4	2	no	—
47	C₂×Q₈×C₄₁	—	4	4	no	—
48	C₂²×C₄×C₄₁	—	4	2	no	—
49	C₂²×C₄×C₄₁	—	4	2	no	—
50	C₂²×C₄×C₄₁	D₈	1	2	yes	—
51	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
52	Q₁₆×C₄₁	—	2	8	no	—
53	Q₁₆×C₄₁	—	2	4	no	—
54	Q₁₆×C₄₁	C₂²×C₄	1	8	yes	—
55	C₂×C₈×C₄₁	—	2	2	no	—
56	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₄₁	#41	1	2	yes	—
60	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₄₁	—	2	4	no	—
62	C₄⋊C₄×C₄₁	—	4	2	no	—
63	C₄⋊C₄×C₄₁	—	4	8	no	—
64	C₄²×C₄₁	—	4	8	no	—
65	C₄²×C₄₁	D₈	1	2	yes	—
66	C₄²×C₄₁	C₂×D₄	1	2	yes	—
67	C₁₆×C₄₁	—	2	2	no	—
68	C₁₆×C₄₁	C₂²×C₄	1	4	yes	—
69	C₂³×C₄₁	C₂×D₈	1	1	yes	characteristic
70	Q₈×C₄₁	—	2	8	no	—
71	Q₈×C₄₁	—	4	4	no	—
72	Q₈×C₄₁	—	4	8	no	—
73	C₂×C₄×C₄₁	—	2	4	no	—
74	C₂×C₄×C₄₁	—	4	4	no	—
75	C₂×C₄×C₄₁	—	4	8	no	—
76	C₂×C₄×C₄₁	—	4	4	no	—
77	C₂×C₄×C₄₁	—	8	4	no	—
78	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₄₁	C₂×D₈	1	4	yes	—
80	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₄₁	#135	1	2	yes	—
82	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₄₁	—	2	2	no	—
84	C₈×C₄₁	C₄×D₄	1	2	yes	—
85	C₈×C₄₁	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₄₁	C₂×#41	1	1	yes	characteristic
87	C₂×#2176	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₄₁	C₄×D₈	1	2	yes	—
89	C₂²×C₄₁	#510	1	4	yes	—
90	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
91	C₄×C₄₁	—	2	4	no	—
92	C₄×C₄₁	—	4	8	no	—
93	C₄×C₄₁	—	8	4	no	—
94	C₄×C₄₁	C₄×D₈	1	2	yes	—
95	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
96	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
97	C₂²×Q₃₂	C₂×C₄₁	1	1	yes	characteristic
98	C₂×C₄×Q₁₆	C₂×C₄₁	1	2	yes	—
99	#2176	C₂×C₄₁	1	8	yes	direct product factor
100	C₂×#367	C₂×C₄₁	1	1	yes	characteristic
101	C₂×#358	C₂×C₄₁	1	2	yes	—
102	C₂×#346	C₂×C₄₁	1	1	yes	characteristic
103	C₂×C₄₁	#2176	1	4	yes	direct product factor
104	C₂×C₄₁	C₂×C₄×D₈	1	1	yes	characteristic
105	C₂×C₄₁	C₂×#510	1	2	yes	—
106	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	—
107	C₂×C₄×Q₈	—	2	2	no	—
108	C₂×Q₃₂	C₄×C₄₁	1	8	yes	complemented
109	C₂×Q₃₂	C₂²×C₄₁	1	4	yes	—
110	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
111	C₂²×C₁₆	C₂²×C₄₁	1	1	yes	characteristic
112	C₄×Q₁₆	—	2	4	no	—
113	C₄×Q₁₆	C₂²×C₄₁	1	8	yes	—
114	C₂×#107	C₂²×C₄₁	1	2	yes	—
115	C₂×#103	—	2	2	no	—
116	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
117	#367	C₂²×C₄₁	1	4	yes	—
118	#358	C₂²×C₄₁	1	8	yes	—
119	#346	C₂²×C₄₁	1	4	yes	—
120	C₄₁	C₂×#2176	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×Q₈	—	2	2	no	—
122	C₂×Q₁₆	—	2	2	no	—
123	C₂×Q₁₆	—	2	4	no	—
124	C₂×Q₁₆	C₂×C₄×C₄₁	1	8	yes	—
125	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
126	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
128	C₄×Q₈	—	2	8	no	—
129	C₄×Q₈	—	4	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	2	no	—
132	C₂×C₄²	—	4	2	no	—
133	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
134	Q₃₂	C₂×C₄×C₄₁	1	16	yes	complemented
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₂×C₄×C₄₁	1	4	yes	—
137	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
138	C₂×C₁₆	C₂³×C₄₁	1	2	yes	—
139	#107	—	2	2	no	—
140	#107	C₂³×C₄₁	1	4	yes	—
141	#103	—	2	8	no	—
142	C₄×C₈	D₄×C₄₁	1	2	yes	—
143	C₄×C₈	C₂³×C₄₁	1	2	yes	—
144	C₂×Q₈	—	2	8	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	D₈×C₄₁	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
152	Q₁₆	—	2	8	no	—
153	Q₁₆	—	2	4	no	—
154	Q₁₆	C₂²×C₄×C₄₁	1	8	yes	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
157	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
158	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
159	C₂×C₈	#41×C₄₁	1	2	yes	—
160	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	8	no	—
164	C₄²	—	4	8	no	—
165	C₄²	D₈×C₄₁	1	2	yes	—
166	C₄²	C₂×D₄×C₄₁	1	2	yes	—
167	C₁₆	—	2	2	no	—
168	C₁₆	C₂²×C₄×C₄₁	1	4	yes	—
169	C₂³	C₂×D₈×C₄₁	1	1	yes	characteristic
170	Q₈	—	2	8	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	4	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
179	C₂×C₄	C₂×D₈×C₄₁	1	4	yes	—
180	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
181	C₂×C₄	#135×C₄₁	1	2	yes	—
182	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	C₄×D₄×C₄₁	1	2	yes	—
185	C₈	C₂³×C₄×C₄₁	1	1	yes	characteristic
186	C₈	C₂×#41×C₄₁	1	1	yes	characteristic
187	C₂²	C₄×D₈×C₄₁	1	2	yes	—
188	C₂²	C₄₁×#510	1	4	yes	—
189	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	4	8	no	—
192	C₄	—	8	4	no	—
193	C₄	C₄×D₈×C₄₁	1	2	yes	—
194	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
195	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
196	C₂	C₄₁×#2176	1	4	yes	direct product factor
197	C₂	C₂×C₄×D₈×C₄₁	1	1	yes	characteristic
198	C₂	C₂×C₄₁×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#2176	C₂	direct (N × H)
9. C₂×Q₃₂×C₄₁	C₄	semidirect
34. Q₃₂×C₄₁	C₂×C₄	semidirect
87. C₂×#2176	C₄₁	direct (N × H)
99. #2176	C₂×C₄₁	direct (N × H)
103. C₂×C₄₁	#2176	direct (N × H)
108. C₂×Q₃₂	C₄×C₄₁	semidirect
120. C₄₁	C₂×#2176	direct (N × H)
134. Q₃₂	C₂×C₄×C₄₁	semidirect
196. C₂	C₄₁×#2176	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₄₁	2
C₂×C₄×Q₁₆×C₄₁	2
C₄₁×#2176	2
C₂×C₄₁×#367	2
C₂×C₄₁×#358	2
C₂×C₄₁×#346	2
C₂×#2176	41

Subgroups — C2×C41×#2176 #23809650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2176 #23809650