Subgroups — C₂×C₄₁×#2164 #23809618

Order 10496.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2164	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄₁×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₁×C₂²⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₄₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆×C₄₁	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₁	C₄	1	1	yes	characteristic
10	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
12	C₄₁×#405	C₄	1	4	yes	—
13	C₄₁×#405	C₂²	1	4	yes	—
14	C₂×#97×C₄₁	C₄	1	1	yes	characteristic
15	C₂×#97×C₄₁	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₄₁×C₄⋊C₁₆	C₂²	1	8	yes	complemented
19	C₄₁×C₂²⋊C₁₆	C₂²	1	8	yes	—
20	C₄₁×#346	C₂²	1	4	yes	complemented
21	C₄×C₁₆×C₄₁	C₂²	1	4	yes	complemented
22	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₄₁	—	2	4	no	—
24	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₄₁	—	2	4	no	—
26	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
32	#117×C₄₁	C₈	1	4	yes	—
33	#117×C₄₁	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₁	C₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₄₁	C₈	1	2	yes	—
36	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₄₁	—	2	4	no	—
39	C₂×C₁₆×C₄₁	—	2	4	no	—
40	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
41	C₂×C₁₆×C₄₁	C₂³	1	4	yes	—
42	#97×C₄₁	C₂×C₄	1	4	yes	—
43	#97×C₄₁	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₁	C₂³	1	2	yes	—
47	C₄×C₈×C₄₁	C₂³	1	2	yes	—
48	C₂⁴×C₄₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	—	2	2	no	—
50	C₂²×C₄×C₄₁	—	2	2	no	—
51	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₄₁	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₄₁	—	2	8	no	—
58	C₂×C₈×C₄₁	—	2	2	no	—
59	C₂×C₈×C₄₁	—	2	2	no	—
60	C₂×C₈×C₄₁	—	2	4	no	—
61	C₂×C₈×C₄₁	—	2	4	no	—
62	C₂×C₈×C₄₁	M₁₆	1	4	yes	—
63	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₄₁	#41	1	4	yes	—
68	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₁	C₂×C₈	1	8	yes	—
70	C₂²⋊C₄×C₄₁	C₂×C₈	1	8	yes	—
71	C₄²×C₄₁	C₂×C₈	1	2	yes	—
72	C₄²×C₄₁	C₂×C₈	1	2	yes	—
73	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
75	C₁₆×C₄₁	—	2	4	no	—
76	C₁₆×C₄₁	—	2	4	no	—
77	C₂³×C₄₁	—	2	2	no	—
78	C₂³×C₄₁	—	2	4	no	—
79	C₂³×C₄₁	C₂²×C₈	1	2	yes	—
80	C₂³×C₄₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₄₁	—	2	2	no	—
82	C₂×C₄×C₄₁	—	2	4	no	—
83	C₂×C₄×C₄₁	—	2	4	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	2	no	—
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	#117	1	2	yes	—
88	C₂×C₄×C₄₁	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₄₁	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
94	C₈×C₄₁	—	2	2	no	—
95	C₈×C₄₁	—	2	2	no	—
96	C₈×C₄₁	#117	1	2	yes	—
97	C₈×C₄₁	C₂×M₁₆	1	4	yes	—
98	C₈×C₄₁	C₂×#41	1	2	yes	—
99	C₂×#2164	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₄₁	—	2	4	no	—
101	C₂²×C₄₁	—	2	2	no	—
102	C₂²×C₄₁	—	4	4	no	—
103	C₂²×C₄₁	#431	1	2	yes	—
104	C₂²×C₄₁	#504	1	4	yes	—
105	C₂²×C₄₁	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₄₁	—	2	4	no	—
107	C₄×C₄₁	—	2	4	no	—
108	C₄×C₄₁	—	2	2	no	—
109	C₄×C₄₁	#431	1	2	yes	—
110	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
111	C₄×C₄₁	C₂²×M₁₆	1	1	yes	characteristic
112	C₂×#405	C₂×C₄₁	1	1	yes	characteristic
113	#2164	C₂×C₄₁	1	8	yes	direct product factor
114	C₂×C₄⋊C₁₆	C₂×C₄₁	1	2	yes	complemented
115	C₂×C₂²⋊C₁₆	C₂×C₄₁	1	2	yes	—
116	C₂×#346	C₂×C₄₁	1	1	yes	characteristic, complemented
117	C₂×C₄×C₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
118	C₂×C₄₁	—	4	2	no	—
119	C₂×C₄₁	#2164	1	4	yes	direct product factor
120	C₂×C₄₁	C₂×#431	1	1	yes	characteristic
121	C₂×C₄₁	C₂×#504	1	2	yes	—
122	C₂²×M₁₆	C₄×C₄₁	1	1	yes	characteristic
123	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
124	C₂×#117	C₄×C₄₁	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
126	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
127	#405	C₄×C₄₁	1	4	yes	—
128	#405	C₂²×C₄₁	1	4	yes	—
129	C₂×#97	C₄×C₄₁	1	1	yes	characteristic
130	C₂×#97	C₄×C₄₁	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂²×C₄₁	1	8	yes	complemented
134	C₂²⋊C₁₆	C₂²×C₄₁	1	8	yes	—
135	#346	C₂²×C₄₁	1	4	yes	complemented
136	C₄×C₁₆	C₂²×C₄₁	1	4	yes	complemented
137	C₄₁	C₂×#2164	1	1	yes	characteristic, direct product factor, Sylow
138	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
143	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
145	C₂²×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
147	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
148	#117	C₈×C₄₁	1	4	yes	—
149	#117	C₂×C₄×C₄₁	1	4	yes	—
150	C₂×C₄⋊C₄	C₈×C₄₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₈×C₄₁	1	2	yes	—
152	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
153	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	2	4	no	—
156	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
157	C₂×C₁₆	C₂³×C₄₁	1	4	yes	—
158	#97	C₂×C₄×C₄₁	1	4	yes	—
159	#97	C₂×C₄×C₄₁	1	4	yes	—
160	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
161	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
162	C₄×C₈	C₂³×C₄₁	1	2	yes	—
163	C₄×C₈	C₂³×C₄₁	1	2	yes	—
164	C₂⁴	C₂×C₈×C₄₁	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
168	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
169	C₂²×C₄	C₂×C₈×C₄₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	M₁₆×C₄₁	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
180	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
181	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
183	C₂×C₈	#41×C₄₁	1	4	yes	—
184	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
185	C₄⋊C₄	C₂×C₈×C₄₁	1	8	yes	—
186	C₂²⋊C₄	C₂×C₈×C₄₁	1	8	yes	—
187	C₄²	C₂×C₈×C₄₁	1	2	yes	—
188	C₄²	C₂×C₈×C₄₁	1	2	yes	—
189	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
190	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₁₆	—	2	4	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂²×C₈×C₄₁	1	2	yes	—
196	C₂³	C₂²×C₈×C₄₁	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	#117×C₄₁	1	2	yes	—
204	C₂×C₄	C₂²×C₈×C₄₁	1	4	yes	—
205	C₂×C₄	C₂²×C₈×C₄₁	1	4	yes	—
206	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
207	C₂×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
208	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
209	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	#117×C₄₁	1	2	yes	—
213	C₈	C₂×M₁₆×C₄₁	1	4	yes	—
214	C₈	C₂×#41×C₄₁	1	2	yes	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	C₄₁×#431	1	2	yes	—
219	C₂²	C₄₁×#504	1	4	yes	—
220	C₂²	C₂³×C₈×C₄₁	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	C₄₁×#431	1	2	yes	—
225	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
226	C₄	C₂²×M₁₆×C₄₁	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	C₄₁×#2164	1	4	yes	direct product factor
229	C₂	C₂×C₄₁×#431	1	1	yes	characteristic
230	C₂	C₂×C₄₁×#504	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₄₁	C₂³
commutator	C₂²	C₂³×C₈×C₄₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂²×C₈
prime radical	C₂⁴×C₄₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2164	C₂	direct (N × H)
3. C₂×C₄₁×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₄₁×#346	C₂	semidirect
6. C₂×C₄×C₁₆×C₄₁	C₂	semidirect
18. C₄₁×C₄⋊C₁₆	C₂²	semidirect
20. C₄₁×#346	C₂²	semidirect
21. C₄×C₁₆×C₄₁	C₂²	semidirect
99. C₂×#2164	C₄₁	direct (N × H)
113. #2164	C₂×C₄₁	direct (N × H)
114. C₂×C₄⋊C₁₆	C₂×C₄₁	semidirect
116. C₂×#346	C₂×C₄₁	semidirect
117. C₂×C₄×C₁₆	C₂×C₄₁	semidirect
119. C₂×C₄₁	#2164	direct (N × H)
133. C₄⋊C₁₆	C₂²×C₄₁	semidirect
135. #346	C₂²×C₄₁	semidirect
136. C₄×C₁₆	C₂²×C₄₁	semidirect
137. C₄₁	C₂×#2164	direct (N × H)
228. C₂	C₄₁×#2164	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#405	2
C₄₁×#2164	2
C₂×C₄₁×C₄⋊C₁₆	2
C₂×C₄₁×C₂²⋊C₁₆	2
C₂×C₄₁×#346	2
C₂×C₄×C₁₆×C₄₁	2
C₂×#2164	41

Subgroups — C2×C41×#2164 #23809618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2164 #23809618