Subgroups — C₂×C₄₁×#1778 #23796658

Order 10496.

1046 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₄₁	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₄₁	C₂	1	2	yes	maximal
3	C₂²×#97×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1778	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#341	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₄₁	C₂²	1	4	yes	—
7	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#108×C₄₁	C₄	1	16	yes	complemented
11	C₂×#108×C₄₁	C₂²	1	8	yes	—
12	C₄×M₁₆×C₄₁	—	2	4	no	—
13	C₄×M₁₆×C₄₁	C₂²	1	8	yes	—
14	C₂×#97×C₄₁	—	2	2	no	—
15	C₂×#97×C₄₁	C₄	1	4	yes	—
16	C₂×#97×C₄₁	C₄	1	4	yes	—
17	C₂×#97×C₄₁	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₄₁	—	2	2	no	—
19	C₄₁×#341	C₂²	1	8	yes	—
20	C₂³×C₄×C₄₁	Q₈	1	2	yes	—
21	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₁	—	2	4	no	—
23	C₂×M₁₆×C₄₁	—	2	8	no	—
24	C₂×M₁₆×C₄₁	C₂×C₄	1	16	yes	—
25	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
26	C₂²×C₈×C₄₁	—	2	4	no	—
27	C₂²×C₈×C₄₁	—	2	4	no	—
28	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₄₁	D₄	1	2	yes	—
38	C₂×C₄²×C₄₁	D₄	1	2	yes	—
39	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
40	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
41	#108×C₄₁	C₂×C₄	1	32	yes	complemented
42	#97×C₄₁	—	2	8	no	—
43	#97×C₄₁	C₂×C₄	1	16	yes	—
44	C₄×C₈×C₄₁	—	2	8	no	—
45	C₂⁴×C₄₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₄₁	—	4	4	no	—
47	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	#41	1	4	yes	—
57	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
58	M₁₆×C₄₁	—	2	16	no	—
59	M₁₆×C₄₁	—	2	8	no	—
60	M₁₆×C₄₁	C₂²×C₄	1	16	yes	—
61	C₂×C₈×C₄₁	—	2	4	no	—
62	C₂×C₈×C₄₁	—	2	4	no	—
63	C₂×C₈×C₄₁	—	2	16	no	—
64	C₂×C₈×C₄₁	—	2	8	no	—
65	C₂×C₈×C₄₁	—	2	4	no	—
66	C₂×C₈×C₄₁	C₄²	1	8	yes	—
67	C₂×C₈×C₄₁	C₄²	1	8	yes	—
68	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
71	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
72	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
73	C₄²×C₄₁	C₂²×C₄	1	4	yes	—
74	C₄²×C₄₁	C₂×D₄	1	4	yes	—
75	C₂³×C₄₁	—	2	4	no	—
76	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₄₁	C₄×Q₈	1	4	yes	—
79	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₁	—	2	8	no	—
81	C₂×C₄×C₄₁	—	4	4	no	—
82	C₂×C₄×C₄₁	—	4	4	no	—
83	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
86	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₁	#117	1	2	yes	—
89	C₂×C₄×C₄₁	#117	1	2	yes	—
90	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₁	C₄×Q₈	1	4	yes	—
93	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
97	C₈×C₄₁	—	2	4	no	—
98	C₈×C₄₁	—	2	16	no	—
99	C₈×C₄₁	C₂×C₄²	1	8	yes	—
100	C₂×#1778	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₄₁	—	2	2	no	—
102	C₂²×C₄₁	—	2	2	no	—
103	C₂²×C₄₁	—	2	8	no	—
104	C₂²×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
105	C₂²×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
106	C₂²×C₄₁	#430	1	4	yes	—
107	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄₁	C₂×C₄×Q₈	1	2	yes	—
109	C₄×C₄₁	—	2	8	no	—
110	C₄×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₄₁	C₄×C₄⋊C₄	1	2	yes	—
112	C₄×C₄₁	C₂²×C₄²	1	1	yes	characteristic
113	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
114	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
115	C₂²×#108	C₂×C₄₁	1	2	yes	—
116	C₂×C₄×M₁₆	C₂×C₄₁	1	2	yes	—
117	C₂²×#97	C₂×C₄₁	1	1	yes	characteristic
118	#1778	C₂×C₄₁	1	8	yes	direct product factor
119	C₂×#341	C₂×C₄₁	1	2	yes	—
120	C₂×C₄₁	—	2	2	no	—
121	C₂×C₄₁	—	2	2	no	—
122	C₂×C₄₁	#1778	1	4	yes	direct product factor
123	C₂×C₄₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄₁	C₂×#430	1	2	yes	—
125	C₂²×M₁₆	C₂²×C₄₁	1	4	yes	—
126	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
127	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
128	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
129	C₂×#108	C₄×C₄₁	1	16	yes	complemented
130	C₂×#108	C₂²×C₄₁	1	8	yes	—
131	C₄×M₁₆	—	2	4	no	—
132	C₄×M₁₆	C₂²×C₄₁	1	8	yes	—
133	C₂×#97	—	2	2	no	—
134	C₂×#97	C₄×C₄₁	1	4	yes	—
135	C₂×#97	C₄×C₄₁	1	4	yes	—
136	C₂×#97	C₂²×C₄₁	1	4	yes	—
137	C₂×C₄×C₈	—	2	2	no	—
138	#341	C₂²×C₄₁	1	8	yes	—
139	C₄₁	C₂×#1778	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³×C₄	Q₈×C₄₁	1	2	yes	—
141	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
142	C₂×M₁₆	—	2	4	no	—
143	C₂×M₁₆	—	2	8	no	—
144	C₂×M₁₆	C₂×C₄×C₄₁	1	16	yes	—
145	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
149	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
152	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	—
153	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
154	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
157	C₂×C₄²	D₄×C₄₁	1	2	yes	—
158	C₂×C₄²	D₄×C₄₁	1	2	yes	—
159	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
160	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
161	#108	C₂×C₄×C₄₁	1	32	yes	complemented
162	#97	—	2	8	no	—
163	#97	C₂×C₄×C₄₁	1	16	yes	—
164	C₄×C₈	—	2	8	no	—
165	C₂⁴	C₂×Q₈×C₄₁	1	1	yes	characteristic
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
168	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
169	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
172	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
174	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
175	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
176	C₂²×C₄	#41×C₄₁	1	4	yes	—
177	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
178	M₁₆	—	2	16	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	C₂²×C₄×C₄₁	1	16	yes	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	16	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₄²×C₄₁	1	8	yes	—
187	C₂×C₈	C₄²×C₄₁	1	8	yes	—
188	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
189	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
190	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
191	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
192	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
193	C₄²	C₂²×C₄×C₄₁	1	4	yes	—
194	C₄²	C₂×D₄×C₄₁	1	4	yes	—
195	C₂³	—	2	4	no	—
196	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
197	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
198	C₂³	C₄×Q₈×C₄₁	1	4	yes	—
199	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
204	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
207	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
208	C₂×C₄	#117×C₄₁	1	2	yes	—
209	C₂×C₄	#117×C₄₁	1	2	yes	—
210	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
211	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
212	C₂×C₄	C₄×Q₈×C₄₁	1	4	yes	—
213	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
215	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
216	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
217	C₈	—	2	4	no	—
218	C₈	—	2	16	no	—
219	C₈	C₂×C₄²×C₄₁	1	8	yes	—
220	C₂²	—	2	8	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
224	C₂²	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
225	C₂²	C₄₁×#430	1	4	yes	—
226	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
227	C₂²	C₂×C₄×Q₈×C₄₁	1	2	yes	—
228	C₄	—	2	8	no	—
229	C₄	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
230	C₄	C₄×C₄⋊C₄×C₄₁	1	2	yes	—
231	C₄	C₂²×C₄²×C₄₁	1	1	yes	characteristic
232	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
233	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
234	C₂	—	2	2	no	—
235	C₂	—	2	2	no	—
236	C₂	C₄₁×#1778	1	4	yes	direct product factor
237	C₂	C₂×C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
238	C₂	C₂×C₄₁×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₄	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₁	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1778	C₂	direct (N × H)
10. C₂×#108×C₄₁	C₄	semidirect
41. #108×C₄₁	C₂×C₄	semidirect
100. C₂×#1778	C₄₁	direct (N × H)
118. #1778	C₂×C₄₁	direct (N × H)
122. C₂×C₄₁	#1778	direct (N × H)
129. C₂×#108	C₄×C₄₁	semidirect
139. C₄₁	C₂×#1778	direct (N × H)
161. #108	C₂×C₄×C₄₁	semidirect
236. C₂	C₄₁×#1778	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×#108×C₄₁	2
C₂×C₄×M₁₆×C₄₁	2
C₂²×#97×C₄₁	2
C₄₁×#1778	2
C₂×C₄₁×#341	2
C₂×#1778	41

Subgroups — C2×C41×#1778 #23796658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1778 #23796658