Subgroups — C₂²×C₄₁×#343 #23796557

Order 10496.

2494 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₄₁	C₂	1	2	yes	maximal
2	C₂³×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#343	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₄₁	C₄	1	2	yes	—
5	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₄₁	—	2	8	no	—
7	C₂²×M₁₆×C₄₁	—	2	6	no	—
8	C₂²×M₁₆×C₄₁	C₄	1	16	yes	complemented
9	C₂²×M₁₆×C₄₁	C₂²	1	12	yes	—
10	C₂³×C₈×C₄₁	—	2	2	no	—
11	C₂²×C₄⋊C₄×C₄₁	—	2	6	no	—
12	C₂²×C₄⋊C₄×C₄₁	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₄₁	C₄	1	6	yes	—
14	C₂²×C₄⋊C₄×C₄₁	C₂²	1	6	yes	—
15	C₄₁×#343	C₂²	1	16	yes	direct product factor
16	C₂⁵×C₄₁	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₄₁	—	2	6	no	—
18	C₂³×C₄×C₄₁	—	2	2	no	—
19	C₂³×C₄×C₄₁	—	4	8	no	—
20	C₂³×C₄×C₄₁	C₂×C₄	1	12	yes	—
21	C₂³×C₄×C₄₁	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₄₁	D₄	1	4	yes	—
24	C₂³×C₄×C₄₁	D₄	1	6	yes	—
25	C₂³×C₄×C₄₁	D₄	1	2	yes	—
26	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	3	yes	—
28	C₂×M₁₆×C₄₁	—	2	48	no	—
29	C₂×M₁₆×C₄₁	—	2	24	no	—
30	C₂×M₁₆×C₄₁	—	2	12	no	—
31	C₂×M₁₆×C₄₁	C₂×C₄	1	48	yes	complemented
32	C₂×M₁₆×C₄₁	C₂³	1	8	yes	—
33	C₂²×C₈×C₄₁	—	2	16	no	—
34	C₂²×C₈×C₄₁	—	2	12	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	12	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	12	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	12	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	12	no	—
39	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	12	yes	—
40	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
41	C₂⁴×C₄₁	—	2	4	no	—
42	C₂⁴×C₄₁	—	2	4	no	—
43	C₂⁴×C₄₁	C₄²	1	4	yes	—
44	C₂⁴×C₄₁	C₂²⋊C₄	1	3	yes	—
45	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂⁴×C₄₁	C₄⋊C₄	1	4	yes	—
47	C₂⁴×C₄₁	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₄₁	—	2	12	no	—
49	C₂²×C₄×C₄₁	—	2	12	no	—
50	C₂²×C₄×C₄₁	—	4	24	no	—
51	C₂²×C₄×C₄₁	—	4	8	no	—
52	C₂²×C₄×C₄₁	—	4	24	no	—
53	C₂²×C₄×C₄₁	C₄²	1	12	yes	—
54	C₂²×C₄×C₄₁	C₂²⋊C₄	1	6	yes	—
55	C₂²×C₄×C₄₁	C₂²⋊C₄	1	6	yes	—
56	C₂²×C₄×C₄₁	C₂²⋊C₄	1	8	yes	—
57	C₂²×C₄×C₄₁	C₂²⋊C₄	1	24	yes	—
58	C₂²×C₄×C₄₁	C₄⋊C₄	1	12	yes	—
59	C₂²×C₄×C₄₁	C₄⋊C₄	1	24	yes	—
60	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
61	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	12	yes	—
64	C₂²×C₄×C₄₁	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₄₁	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₄₁	C₂×Q₈	1	6	yes	—
68	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₄₁	—	2	48	no	—
70	M₁₆×C₄₁	—	2	8	no	—
71	M₁₆×C₄₁	C₂²×C₄	1	16	yes	complemented
72	C₂×C₈×C₄₁	—	2	48	no	—
73	C₂×C₈×C₄₁	—	2	8	no	—
74	C₄⋊C₄×C₄₁	—	2	24	no	—
75	C₄⋊C₄×C₄₁	—	2	8	no	—
76	C₂³×C₄₁	—	2	16	no	—
77	C₂³×C₄₁	—	2	16	no	—
78	C₂³×C₄₁	—	2	6	no	—
79	C₂³×C₄₁	—	2	6	no	—
80	C₂³×C₄₁	—	2	6	no	—
81	C₂³×C₄₁	—	2	6	no	—
82	C₂³×C₄₁	#95	1	12	yes	—
83	C₂³×C₄₁	#95	1	4	yes	—
84	C₂³×C₄₁	#100	1	4	yes	—
85	C₂³×C₄₁	#101	1	4	yes	—
86	C₂³×C₄₁	C₂×C₄²	1	6	yes	—
87	C₂³×C₄₁	C₂×C₂²⋊C₄	1	3	yes	—
88	C₂³×C₄₁	C₂×C₂²⋊C₄	1	3	yes	—
89	C₂³×C₄₁	C₂×C₄⋊C₄	1	6	yes	—
90	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₁	—	2	8	no	—
92	C₂×C₄×C₄₁	—	4	24	no	—
93	C₂×C₄×C₄₁	—	4	24	no	—
94	C₂×C₄×C₄₁	—	4	8	no	—
95	C₂×C₄×C₄₁	#95	1	24	yes	—
96	C₂×C₄×C₄₁	#95	1	24	yes	—
97	C₂×C₄×C₄₁	C₂×C₄²	1	6	yes	—
98	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	12	yes	—
99	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	12	yes	—
100	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	12	yes	—
102	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	12	yes	—
103	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	6	yes	—
104	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₄₁	—	2	16	no	—
109	C₂²×#343	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₄₁	—	2	3	no	—
111	C₂²×C₄₁	—	2	1	no	—
112	C₂²×C₄₁	—	2	3	no	—
113	C₂²×C₄₁	—	2	1	no	—
114	C₂²×C₄₁	—	2	24	no	—
115	C₂²×C₄₁	—	2	12	no	—
116	C₂²×C₄₁	—	2	12	no	—
117	C₂²×C₄₁	#343	1	16	yes	direct product factor
118	C₂²×C₄₁	C₂×#95	1	6	yes	—
119	C₂²×C₄₁	C₂×#95	1	6	yes	—
120	C₂²×C₄₁	C₂×#100	1	6	yes	—
121	C₂²×C₄₁	C₂×#101	1	6	yes	—
122	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
123	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₄₁	—	4	8	no	—
126	C₄×C₄₁	C₂×#95	1	12	yes	—
127	C₄×C₄₁	C₂²×C₂²⋊C₄	1	2	yes	—
128	C₄×C₄₁	C₂²×C₄⋊C₄	1	2	yes	—
129	C₂³×M₁₆	C₂×C₄₁	1	2	yes	—
130	C₂³×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
131	C₂×#343	C₂×C₄₁	1	12	yes	direct product factor
132	C₂×C₄₁	—	2	1	no	—
133	C₂×C₄₁	—	2	6	no	—
134	C₂×C₄₁	—	2	1	no	—
135	C₂×C₄₁	C₂×#343	1	12	yes	direct product factor
136	C₂×C₄₁	C₂²×#95	1	1	yes	characteristic
137	C₂×C₄₁	C₂²×#100	1	1	yes	characteristic
138	C₂×C₄₁	C₂²×#101	1	1	yes	characteristic
139	C₂⁴×C₄	C₄×C₄₁	1	2	yes	—
140	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
141	C₂²×M₁₆	—	2	8	no	—
142	C₂²×M₁₆	—	2	6	no	—
143	C₂²×M₁₆	C₄×C₄₁	1	16	yes	complemented
144	C₂²×M₁₆	C₂²×C₄₁	1	12	yes	—
145	C₂³×C₈	—	2	2	no	—
146	C₂²×C₄⋊C₄	—	2	2	no	—
147	C₂²×C₄⋊C₄	—	2	6	no	—
148	C₂²×C₄⋊C₄	C₄×C₄₁	1	6	yes	—
149	C₂²×C₄⋊C₄	C₂²×C₄₁	1	6	yes	—
150	#343	C₂²×C₄₁	1	16	yes	direct product factor
151	C₄₁	C₂²×#343	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁵	C₂×C₄×C₄₁	1	1	yes	characteristic
153	C₂³×C₄	—	2	6	no	—
154	C₂³×C₄	—	2	2	no	—
155	C₂³×C₄	—	4	8	no	—
156	C₂³×C₄	C₂×C₄×C₄₁	1	12	yes	—
157	C₂³×C₄	C₂×C₄×C₄₁	1	3	yes	—
158	C₂³×C₄	C₂×C₄×C₄₁	1	8	yes	—
159	C₂³×C₄	D₄×C₄₁	1	4	yes	—
160	C₂³×C₄	D₄×C₄₁	1	2	yes	—
161	C₂³×C₄	D₄×C₄₁	1	6	yes	—
162	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
163	C₂³×C₄	C₂³×C₄₁	1	3	yes	—
164	C₂×M₁₆	—	2	48	no	—
165	C₂×M₁₆	—	2	24	no	—
166	C₂×M₁₆	—	2	12	no	—
167	C₂×M₁₆	C₂×C₄×C₄₁	1	48	yes	complemented
168	C₂×M₁₆	C₂³×C₄₁	1	8	yes	—
169	C₂²×C₈	—	2	16	no	—
170	C₂²×C₈	—	2	12	no	—
171	C₂×C₄⋊C₄	—	2	12	no	—
172	C₂×C₄⋊C₄	—	2	12	no	—
173	C₂×C₄⋊C₄	—	2	12	no	—
174	C₂×C₄⋊C₄	—	2	12	no	—
175	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	12	yes	—
176	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
177	C₂⁴	—	2	4	no	—
178	C₂⁴	—	2	4	no	—
179	C₂⁴	C₄²×C₄₁	1	4	yes	—
180	C₂⁴	C₂²⋊C₄×C₄₁	1	3	yes	—
181	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
182	C₂⁴	C₄⋊C₄×C₄₁	1	4	yes	—
183	C₂⁴	C₂²×C₄×C₄₁	1	3	yes	—
184	C₂²×C₄	—	2	12	no	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	4	24	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	—	4	24	no	—
189	C₂²×C₄	C₄²×C₄₁	1	12	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₄₁	1	6	yes	—
191	C₂²×C₄	C₂²⋊C₄×C₄₁	1	6	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₄₁	1	8	yes	—
193	C₂²×C₄	C₂²⋊C₄×C₄₁	1	24	yes	—
194	C₂²×C₄	C₄⋊C₄×C₄₁	1	12	yes	—
195	C₂²×C₄	C₄⋊C₄×C₄₁	1	24	yes	—
196	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
197	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
198	C₂²×C₄	C₂²×C₄×C₄₁	1	3	yes	—
199	C₂²×C₄	C₂²×C₄×C₄₁	1	12	yes	—
200	C₂²×C₄	C₂×D₄×C₄₁	1	6	yes	—
201	C₂²×C₄	C₂×D₄×C₄₁	1	6	yes	—
202	C₂²×C₄	C₂×D₄×C₄₁	1	6	yes	—
203	C₂²×C₄	C₂×Q₈×C₄₁	1	6	yes	—
204	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
205	M₁₆	—	2	48	no	—
206	M₁₆	—	2	8	no	—
207	M₁₆	C₂²×C₄×C₄₁	1	16	yes	complemented
208	C₂×C₈	—	2	48	no	—
209	C₂×C₈	—	2	8	no	—
210	C₄⋊C₄	—	2	24	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₂³	—	2	16	no	—
213	C₂³	—	2	16	no	—
214	C₂³	—	2	6	no	—
215	C₂³	—	2	6	no	—
216	C₂³	—	2	6	no	—
217	C₂³	—	2	6	no	—
218	C₂³	#95×C₄₁	1	12	yes	—
219	C₂³	#95×C₄₁	1	4	yes	—
220	C₂³	#100×C₄₁	1	4	yes	—
221	C₂³	#101×C₄₁	1	4	yes	—
222	C₂³	C₂×C₄²×C₄₁	1	6	yes	—
223	C₂³	C₂×C₂²⋊C₄×C₄₁	1	3	yes	—
224	C₂³	C₂×C₂²⋊C₄×C₄₁	1	3	yes	—
225	C₂³	C₂×C₄⋊C₄×C₄₁	1	6	yes	—
226	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	4	24	no	—
229	C₂×C₄	—	4	24	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	#95×C₄₁	1	24	yes	—
232	C₂×C₄	#95×C₄₁	1	24	yes	—
233	C₂×C₄	C₂×C₄²×C₄₁	1	6	yes	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	12	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	12	yes	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	6	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	12	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	12	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	6	yes	—
240	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
241	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
242	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
243	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
244	C₈	—	2	16	no	—
245	C₂²	—	2	3	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	3	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	24	no	—
250	C₂²	—	2	12	no	—
251	C₂²	—	2	12	no	—
252	C₂²	C₄₁×#343	1	16	yes	direct product factor
253	C₂²	C₂×#95×C₄₁	1	6	yes	—
254	C₂²	C₂×#95×C₄₁	1	6	yes	—
255	C₂²	C₂×#100×C₄₁	1	6	yes	—
256	C₂²	C₂×#101×C₄₁	1	6	yes	—
257	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
258	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
259	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
260	C₄	—	4	8	no	—
261	C₄	C₂×#95×C₄₁	1	12	yes	—
262	C₄	C₂²×C₂²⋊C₄×C₄₁	1	2	yes	—
263	C₄	C₂²×C₄⋊C₄×C₄₁	1	2	yes	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	2	6	no	—
267	C₂	C₂×C₄₁×#343	1	12	yes	direct product factor
268	C₂	C₂²×#95×C₄₁	1	1	yes	characteristic
269	C₂	C₂²×#100×C₄₁	1	1	yes	characteristic
270	C₂	C₂²×#101×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁵×C₄₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂⁴×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄₁×#343	C₂	direct (N × H)
8. C₂²×M₁₆×C₄₁	C₄	semidirect
15. C₄₁×#343	C₂²	direct (N × H)
31. C₂×M₁₆×C₄₁	C₂×C₄	semidirect
71. M₁₆×C₄₁	C₂²×C₄	semidirect
109. C₂²×#343	C₄₁	direct (N × H)
117. C₂²×C₄₁	#343	direct (N × H)
131. C₂×#343	C₂×C₄₁	direct (N × H)
135. C₂×C₄₁	C₂×#343	direct (N × H)
143. C₂²×M₁₆	C₄×C₄₁	semidirect
150. #343	C₂²×C₄₁	direct (N × H)
151. C₄₁	C₂²×#343	direct (N × H)
167. C₂×M₁₆	C₂×C₄×C₄₁	semidirect
207. M₁₆	C₂²×C₄×C₄₁	semidirect
252. C₂²	C₄₁×#343	direct (N × H)
267. C₂	C₂×C₄₁×#343	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×M₁₆×C₄₁	2
C₂³×C₄⋊C₄×C₄₁	2
C₂×C₄₁×#343	2
C₂²×#343	41

Subgroups — C22×C41×#343 #23796557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#343 #23796557