Subgroups — C₄×C₄₁×#423 #23796629

Order 10496.

1382 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#423	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₄₁	C₂	1	2	yes	maximal
3	C₂×C₄³×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1767	C₂	1	4	yes	maximal, complemented
5	C₄×C₄⋊C₈×C₄₁	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1728	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₄₁	C₂²	1	4	yes	—
8	C₂²×C₄²×C₄₁	C₄	1	4	yes	—
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₄₁×#423	C₄	1	16	yes	direct product factor
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	8	yes	—
14	C₄×M₁₆×C₄₁	—	2	8	no	—
15	C₂×#97×C₄₁	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
17	C₄³×C₄₁	C₄	1	2	yes	—
18	C₄³×C₄₁	C₄	1	2	yes	—
19	C₄³×C₄₁	C₂²	1	2	yes	—
20	C₄³×C₄₁	C₂²	1	2	yes	—
21	C₄₁×#336	C₂²	1	8	yes	—
22	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₁	—	2	8	no	—
27	C₂×M₁₆×C₄₁	—	2	8	no	—
28	C₂×M₁₆×C₄₁	C₂×C₄	1	16	yes	—
29	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
30	C₂×C₄²×C₄₁	—	2	4	no	—
31	C₂×C₄²×C₄₁	—	2	16	no	—
32	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₁	C₂×C₄	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×C₄²×C₄₁	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₁	D₄	1	4	yes	—
42	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
43	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₁	C₂×C₄	1	32	yes	complemented
48	#97×C₄₁	—	2	8	no	—
49	C₄×C₈×C₄₁	—	2	8	no	—
50	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	—	2	4	no	—
52	C₂²×C₄×C₄₁	—	2	8	no	—
53	C₂²×C₄×C₄₁	—	2	8	no	—
54	C₂²×C₄×C₄₁	—	2	8	no	—
55	C₂²×C₄×C₄₁	C₄²	1	8	yes	—
56	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₁	#41	1	4	yes	—
67	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₄₁	—	2	32	no	—
69	C₂×C₈×C₄₁	—	2	8	no	—
70	C₂×C₈×C₄₁	—	2	8	no	—
71	C₂×C₈×C₄₁	C₂²×C₄	1	16	yes	—
72	C₄²×C₄₁	—	2	4	no	—
73	C₄²×C₄₁	—	2	8	no	—
74	C₄²×C₄₁	—	2	8	no	—
75	C₄²×C₄₁	—	2	8	no	—
76	C₄²×C₄₁	—	2	4	no	—
77	C₄²×C₄₁	C₄²	1	4	yes	—
78	C₄²×C₄₁	C₄²	1	4	yes	—
79	C₄²×C₄₁	C₄⋊C₄	1	8	yes	—
80	C₄²×C₄₁	M₁₆	1	16	yes	complemented
81	C₄²×C₄₁	C₂²×C₄	1	8	yes	—
82	C₄²×C₄₁	C₂×D₄	1	2	yes	—
83	C₄²×C₄₁	C₂×D₄	1	2	yes	—
84	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
85	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
86	C₂³×C₄₁	—	2	4	no	—
87	C₂³×C₄₁	C₂×C₄²	1	2	yes	—
88	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₁	#117	1	2	yes	—
90	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₁	—	2	16	no	—
92	C₂×C₄×C₄₁	—	2	4	no	—
93	C₂×C₄×C₄₁	—	2	4	no	—
94	C₂×C₄×C₄₁	—	2	4	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	2	16	no	—
97	C₂×C₄×C₄₁	—	2	16	no	—
98	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₄₁	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₁	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₁	#117	1	2	yes	—
105	C₂×C₄×C₄₁	C₄×D₄	1	8	yes	—
106	C₂×C₄×C₄₁	C₄×Q₈	1	8	yes	—
107	C₂×C₄×C₄₁	C₂×M₁₆	1	8	yes	—
108	C₂×C₄×C₄₁	C₂×M₁₆	1	8	yes	—
109	C₂×C₄×C₄₁	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₄₁	C₂³×C₄	1	2	yes	—
111	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
114	C₈×C₄₁	—	2	16	no	—
115	C₄×#423	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₄₁	—	2	4	no	—
117	C₂²×C₄₁	—	2	4	no	—
118	C₂²×C₄₁	—	2	4	no	—
119	C₂²×C₄₁	C₄×C₄⋊C₄	1	4	yes	—
120	C₂²×C₄₁	C₂²×C₄²	1	1	yes	characteristic
121	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
123	C₂²×C₄₁	C₂²×M₁₆	1	4	yes	—
124	C₄×C₄₁	—	2	4	no	—
125	C₄×C₄₁	—	2	4	no	—
126	C₄×C₄₁	—	2	8	no	—
127	C₄×C₄₁	C₄×C₄⋊C₄	1	4	yes	—
128	C₄×C₄₁	C₄×M₁₆	1	8	yes	complemented
129	C₄×C₄₁	#423	1	8	yes	direct product factor
130	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
131	C₄×C₄₁	C₂×C₄×Q₈	1	2	yes	—
132	C₂×#423	C₂×C₄₁	1	2	yes	—
133	C₂×C₄×M₁₆	C₂×C₄₁	1	2	yes	—
134	C₂×C₄³	C₂×C₄₁	1	1	yes	characteristic
135	#1767	C₂×C₄₁	1	4	yes	complemented
136	C₄×C₄⋊C₈	C₂×C₄₁	1	4	yes	complemented
137	#1728	C₂×C₄₁	1	2	yes	—
138	C₂×C₄₁	—	2	4	no	—
139	C₂×C₄₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄₁	C₂×C₄×M₁₆	1	2	yes	—
141	C₂×C₄₁	C₂×#423	1	2	yes	—
142	C₂×C₄₁	#2920	1	2	yes	—
143	C₂²×M₁₆	C₂²×C₄₁	1	4	yes	—
144	C₂²×C₄²	C₄×C₄₁	1	4	yes	—
145	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
147	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
148	#423	C₄×C₄₁	1	16	yes	direct product factor
149	C₂×C₄⋊C₈	C₂²×C₄₁	1	8	yes	—
150	C₄×M₁₆	—	2	8	no	—
151	C₂×#97	C₂²×C₄₁	1	4	yes	—
152	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
153	C₄³	C₄×C₄₁	1	2	yes	—
154	C₄³	C₄×C₄₁	1	2	yes	—
155	C₄³	C₂²×C₄₁	1	2	yes	—
156	C₄³	C₂²×C₄₁	1	2	yes	—
157	#336	C₂²×C₄₁	1	8	yes	—
158	C₄₁	C₄×#423	1	1	yes	characteristic, direct product factor, Sylow
159	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
160	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
161	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
162	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
163	C₂×M₁₆	—	2	8	no	—
164	C₂×M₁₆	—	2	8	no	—
165	C₂×M₁₆	C₂×C₄×C₄₁	1	16	yes	—
166	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
167	C₂×C₄²	—	2	4	no	—
168	C₂×C₄²	—	2	16	no	—
169	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
171	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
173	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₄₁	1	8	yes	—
176	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
178	C₂×C₄²	D₄×C₄₁	1	4	yes	—
179	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
180	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
181	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
182	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
184	C₄⋊C₈	C₂×C₄×C₄₁	1	32	yes	complemented
185	#97	—	2	8	no	—
186	C₄×C₈	—	2	8	no	—
187	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	C₄²×C₄₁	1	8	yes	—
193	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
194	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
196	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
202	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
203	C₂²×C₄	#41×C₄₁	1	4	yes	—
204	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
205	M₁₆	—	2	32	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	2	8	no	—
208	C₂×C₈	C₂²×C₄×C₄₁	1	16	yes	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	8	no	—
211	C₄²	—	2	8	no	—
212	C₄²	—	2	8	no	—
213	C₄²	—	2	4	no	—
214	C₄²	C₄²×C₄₁	1	4	yes	—
215	C₄²	C₄²×C₄₁	1	4	yes	—
216	C₄²	C₄⋊C₄×C₄₁	1	8	yes	—
217	C₄²	M₁₆×C₄₁	1	16	yes	complemented
218	C₄²	C₂²×C₄×C₄₁	1	8	yes	—
219	C₄²	C₂×D₄×C₄₁	1	2	yes	—
220	C₄²	C₂×D₄×C₄₁	1	2	yes	—
221	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
222	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
223	C₂³	—	2	4	no	—
224	C₂³	C₂×C₄²×C₄₁	1	2	yes	—
225	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
226	C₂³	#117×C₄₁	1	2	yes	—
227	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
236	C₂×C₄	C₂×C₄²×C₄₁	1	2	yes	—
237	C₂×C₄	C₂×C₄²×C₄₁	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
241	C₂×C₄	#117×C₄₁	1	2	yes	—
242	C₂×C₄	C₄×D₄×C₄₁	1	8	yes	—
243	C₂×C₄	C₄×Q₈×C₄₁	1	8	yes	—
244	C₂×C₄	C₂×M₁₆×C₄₁	1	8	yes	—
245	C₂×C₄	C₂×M₁₆×C₄₁	1	8	yes	—
246	C₂×C₄	C₂×M₁₆×C₄₁	1	8	yes	—
247	C₂×C₄	C₂³×C₄×C₄₁	1	2	yes	—
248	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
249	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
250	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
251	C₈	—	2	16	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	C₄×C₄⋊C₄×C₄₁	1	4	yes	—
256	C₂²	C₂²×C₄²×C₄₁	1	1	yes	characteristic
257	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
258	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
259	C₂²	C₂²×M₁₆×C₄₁	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	C₄×C₄⋊C₄×C₄₁	1	4	yes	—
264	C₄	C₄×M₁₆×C₄₁	1	8	yes	complemented
265	C₄	C₄₁×#423	1	8	yes	direct product factor
266	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
267	C₄	C₂×C₄×Q₈×C₄₁	1	2	yes	—
268	C₂	—	2	4	no	—
269	C₂	C₂×C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
270	C₂	C₂×C₄×M₁₆×C₄₁	1	2	yes	—
271	C₂	C₂×C₄₁×#423	1	2	yes	—
272	C₂	C₄₁×#2920	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1767	C₂	semidirect
5. C₄×C₄⋊C₈×C₄₁	C₂	semidirect
12. C₄₁×#423	C₄	direct (N × H)
47. C₄⋊C₈×C₄₁	C₂×C₄	semidirect
80. C₄²×C₄₁	M₁₆	semidirect
115. C₄×#423	C₄₁	direct (N × H)
128. C₄×C₄₁	C₄×M₁₆	semidirect
129. C₄×C₄₁	#423	direct (N × H)
135. #1767	C₂×C₄₁	semidirect
136. C₄×C₄⋊C₈	C₂×C₄₁	semidirect
148. #423	C₄×C₄₁	direct (N × H)
158. C₄₁	C₄×#423	direct (N × H)
184. C₄⋊C₈	C₂×C₄×C₄₁	semidirect
217. C₄²	M₁₆×C₄₁	semidirect
264. C₄	C₄×M₁₆×C₄₁	semidirect
265. C₄	C₄₁×#423	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×#423	2
C₂×C₄×M₁₆×C₄₁	2
C₂×C₄³×C₄₁	2
C₄₁×#1767	2
C₄×C₄⋊C₈×C₄₁	2
C₄₁×#1728	2
C₄×#423	41

Subgroups — C4×C41×#423 #23796629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₁×#423 #23796629