Subgroups — C₄₁×#21903 #23798104

Order 10496.

894 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2297	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1923	C₂	1	8	yes	maximal, complemented
3	C₄₁×#1798	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1780	C₂	1	2	yes	maximal
5	C₄×C₂²⋊C₈×C₄₁	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
7	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₄₁	C₂²	1	4	yes	—
14	C₄₁×#391	C₄	1	2	yes	—
15	C₄₁×#390	C₄	1	2	yes	—
16	C₄₁×#386	C₄	1	4	yes	—
17	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
18	C₄₁×#336	C₂²	1	4	yes	—
19	C₄₁×#336	C₂²	1	4	yes	—
20	C₂×#41×C₄₁	—	2	4	no	—
21	C₂×#41×C₄₁	—	4	2	no	—
22	C₂×#41×C₄₁	—	4	2	no	—
23	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₄₁	C₂³	1	8	yes	—
30	#117×C₄₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
33	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₁	—	4	2	no	—
35	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
37	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
38	C₄⋊C₈×C₄₁	—	4	4	no	—
39	C₂²⋊C₈×C₄₁	—	2	8	no	—
40	C₂²⋊C₈×C₄₁	—	4	4	no	—
41	C₂²⋊C₈×C₄₁	—	4	4	no	—
42	C₄×C₈×C₄₁	—	4	4	no	—
43	#95×C₄₁	C₂×C₄	1	8	yes	—
44	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
45	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
46	#41×C₄₁	—	4	16	no	—
47	C₂×Q₈×C₄₁	—	2	2	no	—
48	C₂×Q₈×C₄₁	—	4	1	no	—
49	C₂×Q₈×C₄₁	—	4	1	no	—
50	C₂×D₄×C₄₁	—	2	4	no	—
51	C₂×D₄×C₄₁	—	2	2	no	—
52	C₂×D₄×C₄₁	—	4	1	no	—
53	C₂×D₄×C₄₁	—	4	1	no	—
54	C₂×D₄×C₄₁	—	4	2	no	—
55	C₂×D₄×C₄₁	—	4	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	—	2	2	no	—
58	C₂²×C₄×C₄₁	—	2	8	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	4	2	no	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₁	#41	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₄₁	—	2	8	no	—
70	C₂×C₈×C₄₁	—	2	8	no	—
71	C₂×C₈×C₄₁	—	2	8	no	—
72	C₄⋊C₄×C₄₁	—	4	2	no	—
73	C₂²⋊C₄×C₄₁	—	2	16	no	—
74	C₂²⋊C₄×C₄₁	—	4	2	no	—
75	C₄²×C₄₁	—	2	4	no	—
76	C₄²×C₄₁	—	2	4	no	—
77	C₄²×C₄₁	—	4	2	no	—
78	C₂³×C₄₁	—	2	1	no	—
79	C₂³×C₄₁	—	2	1	no	—
80	C₂³×C₄₁	—	2	4	no	—
81	C₂³×C₄₁	—	2	2	no	—
82	C₂³×C₄₁	—	2	2	no	—
83	C₂³×C₄₁	—	2	2	no	—
84	C₂³×C₄₁	—	2	8	no	—
85	C₂³×C₄₁	#117	1	2	yes	—
86	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
87	Q₈×C₄₁	—	4	4	no	—
88	D₄×C₄₁	—	4	4	no	—
89	D₄×C₄₁	—	4	8	no	—
90	C₂×C₄×C₄₁	—	2	4	no	—
91	C₂×C₄×C₄₁	—	2	1	no	—
92	C₂×C₄×C₄₁	—	2	1	no	—
93	C₂×C₄×C₄₁	—	2	4	no	—
94	C₂×C₄×C₄₁	—	2	8	no	—
95	C₂×C₄×C₄₁	—	2	2	no	—
96	C₂×C₄×C₄₁	—	2	4	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	2	no	—
99	C₂×C₄×C₄₁	—	2	8	no	—
100	C₂×C₄×C₄₁	—	4	8	no	—
101	C₂×C₄×C₄₁	—	4	8	no	—
102	C₂×C₄×C₄₁	#117	1	2	yes	—
103	C₂×C₄×C₄₁	#131	1	8	yes	—
104	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
105	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
106	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
107	C₈×C₄₁	—	4	8	no	—
108	#21903	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₄₁	—	2	2	no	—
110	C₂²×C₄₁	—	2	4	no	—
111	C₂²×C₄₁	—	4	2	no	—
112	C₂²×C₄₁	—	4	2	no	—
113	C₂²×C₄₁	—	4	2	no	—
114	C₂²×C₄₁	—	4	4	no	—
115	C₂²×C₄₁	—	4	4	no	—
116	C₂²×C₄₁	—	4	4	no	—
117	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
118	C₂²×C₄₁	#518	1	1	yes	characteristic
119	C₂²×C₄₁	#518	1	1	yes	characteristic
120	C₂²×C₄₁	C₂×#131	1	4	yes	—
121	C₄×C₄₁	—	2	2	no	—
122	C₄×C₄₁	—	2	8	no	—
123	C₄×C₄₁	—	4	2	no	—
124	C₄×C₄₁	—	4	4	no	—
125	C₄×C₄₁	#538	1	4	yes	—
126	#2297	C₂×C₄₁	1	1	yes	characteristic
127	#1923	C₂×C₄₁	1	8	yes	complemented
128	#1798	C₂×C₄₁	1	2	yes	complemented
129	#1780	C₂×C₄₁	1	2	yes	—
130	C₄×C₂²⋊C₈	C₂×C₄₁	1	2	yes	complemented
131	C₂×C₄₁	—	4	2	no	—
132	C₂×C₄₁	—	8	2	no	—
133	C₂×C₄₁	#2312	1	1	yes	characteristic
134	C₂×C₄₁	#2922	1	2	yes	—
135	C₂×C₄₁	#2977	1	2	yes	—
136	C₂×C₄₁	#2979	1	2	yes	—
137	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
138	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
139	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
140	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
141	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
142	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
143	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
144	C₂×C₄×C₈	C₂²×C₄₁	1	4	yes	—
145	#391	C₄×C₄₁	1	2	yes	—
146	#390	C₄×C₄₁	1	2	yes	—
147	#386	C₄×C₄₁	1	4	yes	—
148	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
149	#336	C₂²×C₄₁	1	4	yes	—
150	#336	C₂²×C₄₁	1	4	yes	—
151	C₄₁	#21903	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×#41	—	2	4	no	—
153	C₂×#41	—	4	2	no	—
154	C₂×#41	—	4	2	no	—
155	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
156	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
157	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
158	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
159	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
160	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₄₁	1	8	yes	—
162	#117	—	4	2	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
165	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
166	C₂×C₄²	—	4	2	no	—
167	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
168	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
169	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
170	C₄⋊C₈	—	4	4	no	—
171	C₂²⋊C₈	—	2	8	no	—
172	C₂²⋊C₈	—	4	4	no	—
173	C₂²⋊C₈	—	4	4	no	—
174	C₄×C₈	—	4	4	no	—
175	#95	C₂×C₄×C₄₁	1	8	yes	—
176	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
177	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
178	#41	—	4	16	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	1	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
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	4	yes	—
199	C₂²×C₄	#41×C₄₁	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	—	2	8	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	2	16	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	—	4	2	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	4	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	8	no	—
217	C₂³	#117×C₄₁	1	2	yes	—
218	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	8	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	#117×C₄₁	1	2	yes	—
235	C₂×C₄	#131×C₄₁	1	8	yes	—
236	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
237	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
238	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
239	C₈	—	4	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
249	C₂²	C₄₁×#518	1	1	yes	characteristic
250	C₂²	C₄₁×#518	1	1	yes	characteristic
251	C₂²	C₂×#131×C₄₁	1	4	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	8	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	4	no	—
256	C₄	C₄₁×#538	1	4	yes	—
257	C₂	—	4	2	no	—
258	C₂	—	8	2	no	—
259	C₂	C₄₁×#2312	1	1	yes	characteristic
260	C₂	C₄₁×#2922	1	2	yes	—
261	C₂	C₄₁×#2977	1	2	yes	—
262	C₂	C₄₁×#2979	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂²×#41×C₄₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1923	C₂	semidirect
3. C₄₁×#1798	C₂	semidirect
5. C₄×C₂²⋊C₈×C₄₁	C₂	semidirect
108. #21903	C₄₁	direct (N × H)
127. #1923	C₂×C₄₁	semidirect
128. #1798	C₂×C₄₁	semidirect
130. C₄×C₂²⋊C₈	C₂×C₄₁	semidirect
151. C₄₁	#21903	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#2297	2
C₄₁×#1923	2
C₄₁×#1798	2
C₄₁×#1780	2
C₄×C₂²⋊C₈×C₄₁	2
#21903	41

Subgroups — C41×#21903 #23798104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21903 #23798104