Subgroups — C₄₁×#14274 #23790475

Order 10496.

1742 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#391	C₂	1	4	yes	maximal
3	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#376	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1445	C₂	1	8	yes	maximal
6	C₂³×Q₈×C₄₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₄₁	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₄₁	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₄₁	C₄	1	8	yes	complemented
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#391	—	2	16	no	—
15	C₄₁×#382	—	2	4	no	—
16	C₄₁×#376	—	2	4	no	—
17	C₂×#95×C₄₁	C₂²	1	4	yes	—
18	C₂×#95×C₄₁	C₂²	1	16	yes	—
19	C₂×#95×C₄₁	C₂²	1	4	yes	—
20	C₂²×Q₈×C₄₁	—	2	4	no	—
21	C₂²×Q₈×C₄₁	—	2	4	no	—
22	C₂²×Q₈×C₄₁	—	2	8	no	—
23	C₂²×Q₈×C₄₁	—	2	8	no	—
24	C₂²×Q₈×C₄₁	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
29	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₄₁	—	2	32	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
36	C₂×C₄²×C₄₁	—	2	4	no	—
37	C₂×C₄²×C₄₁	—	2	4	no	—
38	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₄₁	D₄	1	8	yes	—
40	C₂×C₄²×C₄₁	D₄	1	4	yes	—
41	C₂×C₄²×C₄₁	D₄	1	4	yes	—
42	#95×C₄₁	—	2	8	no	—
43	#95×C₄₁	—	2	8	no	—
44	#95×C₄₁	—	2	32	no	—
45	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
46	C₂×Q₈×C₄₁	—	2	16	no	—
47	C₂×Q₈×C₄₁	—	2	32	no	—
48	C₂×Q₈×C₄₁	—	4	16	no	—
49	C₂×Q₈×C₄₁	—	4	8	no	—
50	C₂×Q₈×C₄₁	—	4	8	no	—
51	C₂²×C₄×C₄₁	—	2	8	no	—
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	2	8	no	—
54	C₂²×C₄×C₄₁	—	2	4	no	—
55	C₂²×C₄×C₄₁	—	2	4	no	—
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	8	no	—
58	C₂²×C₄×C₄₁	—	2	16	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	8	no	—
61	C₂²×C₄×C₄₁	—	2	16	no	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₄₁	#41	1	4	yes	—
72	C₄⋊C₄×C₄₁	—	2	32	no	—
73	C₄⋊C₄×C₄₁	—	4	8	no	—
74	C₄²×C₄₁	—	2	16	no	—
75	C₄²×C₄₁	C₂²⋊C₄	1	16	yes	—
76	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
77	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
78	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
79	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
80	C₂³×C₄₁	C₂×#41	1	2	yes	—
81	C₂³×C₄₁	2₋¹⁺⁴	1	8	yes	—
82	Q₈×C₄₁	—	4	32	no	—
83	C₂×C₄×C₄₁	—	2	8	no	—
84	C₂×C₄×C₄₁	—	2	8	no	—
85	C₂×C₄×C₄₁	—	2	8	no	—
86	C₂×C₄×C₄₁	—	2	8	no	—
87	C₂×C₄×C₄₁	—	2	16	no	—
88	C₂×C₄×C₄₁	—	2	8	no	—
89	C₂×C₄×C₄₁	—	2	16	no	—
90	C₂×C₄×C₄₁	—	4	8	no	—
91	C₂×C₄×C₄₁	—	4	4	no	—
92	C₂×C₄×C₄₁	—	4	4	no	—
93	C₂×C₄×C₄₁	—	4	8	no	—
94	C₂×C₄×C₄₁	—	4	8	no	—
95	C₂×C₄×C₄₁	—	4	4	no	—
96	C₂×C₄×C₄₁	—	4	8	no	—
97	C₂×C₄×C₄₁	—	4	16	no	—
98	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
99	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂×C₄×C₄₁	C₄×D₄	1	8	yes	—
101	C₂×C₄×C₄₁	#121	1	8	yes	—
102	C₂×C₄×C₄₁	#124	1	4	yes	—
103	C₂×C₄×C₄₁	#127	1	4	yes	—
104	#14274	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₁	C₂×C₄×D₄	1	2	yes	—
107	C₂²×C₄₁	#519	1	4	yes	—
108	C₂²×C₄₁	C₂×#121	1	2	yes	—
109	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
111	C₂²×C₄₁	#536	1	4	yes	—
112	C₂²×C₄₁	#536	1	4	yes	—
113	C₂²×C₄₁	#536	1	8	yes	—
114	C₂²×C₄₁	#541	1	8	yes	—
115	C₄×C₄₁	—	2	8	no	—
116	C₄×C₄₁	—	4	8	no	—
117	C₄×C₄₁	—	4	8	no	—
118	C₄×C₄₁	—	8	4	no	—
119	C₄×C₄₁	#390	1	8	yes	—
120	C₂²×C₄⋊Q₈	C₂×C₄₁	1	1	yes	characteristic
121	C₂×#391	C₂×C₄₁	1	4	yes	—
122	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
123	C₂×#376	C₂×C₄₁	1	1	yes	characteristic
124	#1445	C₂×C₄₁	1	8	yes	—
125	C₂×C₄₁	C₂×#390	1	1	yes	characteristic
126	C₂×C₄₁	#2310	1	2	yes	—
127	C₂×C₄₁	#2327	1	4	yes	—
128	C₂×C₄₁	#2621	1	4	yes	—
129	C₂×C₄₁	#2642	1	2	yes	—
130	C₂×C₄₁	#2658	1	2	yes	—
131	C₂³×Q₈	C₂²×C₄₁	1	2	yes	—
132	C₂×C₄⋊Q₈	—	2	4	no	—
133	C₂×C₄⋊Q₈	—	2	4	no	—
134	C₂×C₄⋊Q₈	C₄×C₄₁	1	8	yes	complemented
135	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
136	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
137	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
138	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
139	#391	—	2	16	no	—
140	#382	—	2	4	no	—
141	#376	—	2	4	no	—
142	C₂×#95	C₂²×C₄₁	1	4	yes	—
143	C₂×#95	C₂²×C₄₁	1	4	yes	—
144	C₂×#95	C₂²×C₄₁	1	16	yes	—
145	C₄₁	#14274	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×Q₈	—	2	4	no	—
147	C₂²×Q₈	—	2	4	no	—
148	C₂²×Q₈	—	2	8	no	—
149	C₂²×Q₈	—	2	8	no	—
150	C₂²×Q₈	C₂×C₄×C₄₁	1	8	yes	—
151	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
152	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
153	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
154	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
155	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	32	no	—
157	C₂×C₄⋊C₄	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	8	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	8	no	—
161	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
162	C₂×C₄²	—	2	4	no	—
163	C₂×C₄²	—	2	4	no	—
164	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
165	C₂×C₄²	D₄×C₄₁	1	8	yes	—
166	C₂×C₄²	D₄×C₄₁	1	4	yes	—
167	C₂×C₄²	D₄×C₄₁	1	4	yes	—
168	#95	—	2	8	no	—
169	#95	—	2	8	no	—
170	#95	—	2	32	no	—
171	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
172	C₂×Q₈	—	2	16	no	—
173	C₂×Q₈	—	2	32	no	—
174	C₂×Q₈	—	4	16	no	—
175	C₂×Q₈	—	4	8	no	—
176	C₂×Q₈	—	4	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	16	no	—
188	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
189	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
192	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
193	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
194	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
197	C₂²×C₄	#41×C₄₁	1	4	yes	—
198	C₄⋊C₄	—	2	32	no	—
199	C₄⋊C₄	—	4	8	no	—
200	C₄²	—	2	16	no	—
201	C₄²	C₂²⋊C₄×C₄₁	1	16	yes	—
202	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
203	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
204	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
205	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
206	C₂³	C₂×#41×C₄₁	1	2	yes	—
207	C₂³	2₋¹⁺⁴×C₄₁	1	8	yes	—
208	Q₈	—	4	32	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	16	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	8	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
226	C₂×C₄	C₄×D₄×C₄₁	1	8	yes	—
227	C₂×C₄	#121×C₄₁	1	8	yes	—
228	C₂×C₄	#124×C₄₁	1	4	yes	—
229	C₂×C₄	#127×C₄₁	1	4	yes	—
230	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
231	C₂²	C₂×C₄×D₄×C₄₁	1	2	yes	—
232	C₂²	C₄₁×#519	1	4	yes	—
233	C₂²	C₂×#121×C₄₁	1	2	yes	—
234	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
235	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
236	C₂²	C₄₁×#536	1	4	yes	—
237	C₂²	C₄₁×#536	1	4	yes	—
238	C₂²	C₄₁×#536	1	8	yes	—
239	C₂²	C₄₁×#541	1	8	yes	—
240	C₄	—	2	8	no	—
241	C₄	—	4	8	no	—
242	C₄	—	4	8	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄₁×#390	1	8	yes	—
245	C₂	C₂×C₄₁×#390	1	1	yes	characteristic
246	C₂	C₄₁×#2310	1	2	yes	—
247	C₂	C₄₁×#2327	1	4	yes	—
248	C₂	C₄₁×#2621	1	4	yes	—
249	C₂	C₄₁×#2642	1	2	yes	—
250	C₂	C₄₁×#2658	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Q₈×C₄₁	C₄	semidirect
104. #14274	C₄₁	direct (N × H)
134. C₂×C₄⋊Q₈	C₄×C₄₁	semidirect
145. C₄₁	#14274	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×C₄⋊Q₈×C₄₁	2
C₂×C₄₁×#391	2
C₂×C₄₁×#382	2
C₂×C₄₁×#376	2
C₄₁×#1445	2
#14274	41

Subgroups — C41×#14274 #23790475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14274 #23790475