Subgroups — C₄₁×#7854 #23784055

Order 10496.

714 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	2	yes	maximal
2	C₄×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1274	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₄₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
6	C₄₁×#431	—	2	2	no	—
7	C₄₁×#431	—	2	2	no	—
8	C₄₁×#431	C₄	1	4	yes	complemented
9	C₄₁×#431	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₄₁	—	2	2	no	—
12	C₂×C₄×C₈×C₄₁	C₄	1	4	yes	complemented
13	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
15	C₄₁×#376	C₂²	1	2	yes	—
16	C₄³×C₄₁	C₄	1	2	yes	—
17	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₁	D₄	1	2	yes	—
19	C₂²×C₈×C₄₁	—	2	4	no	—
20	#117×C₄₁	—	2	4	no	—
21	#117×C₄₁	C₈	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
23	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
24	C₂×C₄²×C₄₁	—	2	4	no	—
25	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₁	D₄	1	2	yes	—
30	C₂×C₄²×C₄₁	D₄	1	2	yes	—
31	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₁	—	2	4	no	—
34	C₄⋊C₈×C₄₁	—	2	4	no	—
35	C₂²⋊C₈×C₄₁	—	2	4	no	—
36	C₂²⋊C₈×C₄₁	—	2	4	no	—
37	C₄×C₈×C₄₁	—	2	4	no	—
38	C₄×C₈×C₄₁	C₂×C₄	1	8	yes	—
39	#95×C₄₁	—	2	4	no	—
40	C₂⁴×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₁	—	2	4	no	—
42	C₂²×C₄×C₄₁	—	4	4	no	—
43	C₂²×C₄×C₄₁	C₄²	1	2	yes	—
44	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₁	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₄₁	M₁₆	1	4	yes	—
53	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₄₁	—	2	8	no	—
56	C₂×C₈×C₄₁	—	4	8	no	—
57	C₄⋊C₄×C₄₁	—	2	8	no	—
58	C₂²⋊C₄×C₄₁	—	2	8	no	—
59	C₄²×C₄₁	—	2	4	no	—
60	C₄²×C₄₁	—	4	8	no	—
61	C₄²×C₄₁	C₄²	1	1	yes	characteristic
62	C₄²×C₄₁	C₄²	1	1	yes	characteristic
63	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₄₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
66	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₄₁	C₂×C₈	1	8	yes	complemented
69	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
72	C₂³×C₄₁	—	4	2	no	—
73	C₂³×C₄₁	#95	1	2	yes	—
74	C₂³×C₄₁	C₂²⋊C₈	1	4	yes	—
75	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₁	—	2	4	no	—
77	C₂×C₄×C₄₁	—	2	4	no	—
78	C₂×C₄×C₄₁	—	2	4	no	—
79	C₂×C₄×C₄₁	—	2	8	no	—
80	C₂×C₄×C₄₁	—	4	2	no	—
81	C₂×C₄×C₄₁	—	4	2	no	—
82	C₂×C₄×C₄₁	—	4	2	no	—
83	C₂×C₄×C₄₁	—	4	8	no	—
84	C₂×C₄×C₄₁	#95	1	2	yes	—
85	C₂×C₄×C₄₁	#95	1	2	yes	—
86	C₂×C₄×C₄₁	#95	1	2	yes	—
87	C₂×C₄×C₄₁	C₄×C₈	1	4	yes	—
88	C₂×C₄×C₄₁	#97	1	4	yes	—
89	C₂×C₄×C₄₁	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₄₁	C₄⋊C₈	1	8	yes	—
91	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
97	C₈×C₄₁	—	4	8	no	—
98	#7854	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₄₁	—	2	2	no	—
100	C₂²×C₄₁	—	2	2	no	—
101	C₂²×C₄₁	—	2	2	no	—
102	C₂²×C₄₁	—	4	4	no	—
103	C₂²×C₄₁	#336	1	4	yes	—
104	C₂²×C₄₁	C₂×#95	1	1	yes	characteristic
105	C₂²×C₄₁	C₂×C₂²⋊C₈	1	2	yes	—
106	C₄×C₄₁	—	2	2	no	—
107	C₄×C₄₁	—	2	2	no	—
108	C₄×C₄₁	—	2	2	no	—
109	C₄×C₄₁	—	4	8	no	—
110	C₄×C₄₁	#336	1	4	yes	—
111	C₄×C₄₁	H(3,ℤ/4)	1	4	yes	—
112	C₄×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
113	C₄×C₄₁	C₂×#97	1	1	yes	characteristic
114	C₄×C₄₁	C₂×C₄⋊C₈	1	2	yes	—
115	C₂×#431	C₂×C₄₁	1	2	yes	—
116	C₄×#117	C₂×C₄₁	1	1	yes	characteristic
117	#1274	C₂×C₄₁	1	4	yes	complemented
118	C₂×C₄₁	—	2	2	no	—
119	C₂×C₄₁	—	2	4	no	—
120	C₂×C₄₁	C₂×#336	1	1	yes	characteristic
121	C₂×C₄₁	C₂×H(3,ℤ/4)	1	2	yes	—
122	C₂×#117	C₄×C₄₁	1	2	yes	—
123	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
124	#431	—	2	2	no	—
125	#431	—	2	2	no	—
126	#431	C₄×C₄₁	1	4	yes	complemented
127	#431	C₂²×C₄₁	1	4	yes	—
128	C₂×C₄⋊C₈	—	2	2	no	—
129	C₂×C₂²⋊C₈	—	2	2	no	—
130	C₂×C₄×C₈	C₄×C₄₁	1	4	yes	complemented
131	C₄×C₄⋊C₄	—	2	2	no	—
132	C₄×C₂²⋊C₄	—	2	2	no	—
133	#376	C₂²×C₄₁	1	2	yes	—
134	C₄³	C₄×C₄₁	1	2	yes	—
135	C₄₁	#7854	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
137	C₂³×C₄	D₄×C₄₁	1	2	yes	—
138	C₂²×C₈	—	2	4	no	—
139	#117	—	2	4	no	—
140	#117	C₈×C₄₁	1	8	yes	complemented
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₂²⋊C₄	—	2	2	no	—
143	C₂×C₄²	—	2	4	no	—
144	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
145	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
147	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
148	C₂×C₄²	D₄×C₄₁	1	2	yes	—
149	C₂×C₄²	D₄×C₄₁	1	2	yes	—
150	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
151	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
152	C₄⋊C₈	—	2	4	no	—
153	C₄⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₄×C₈	C₂×C₄×C₄₁	1	8	yes	—
158	#95	—	2	4	no	—
159	C₂⁴	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₄²×C₄₁	1	2	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
168	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₄₁	1	4	yes	—
171	C₂²×C₄	M₁₆×C₄₁	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	4	8	no	—
176	C₄⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	4	8	no	—
180	C₄²	C₄²×C₄₁	1	1	yes	characteristic
181	C₄²	C₄²×C₄₁	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₄₁	1	2	yes	—
184	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
185	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
186	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
187	C₄²	C₂×C₈×C₄₁	1	8	yes	complemented
188	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
189	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
190	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
191	C₂³	—	4	2	no	—
192	C₂³	#95×C₄₁	1	2	yes	—
193	C₂³	C₂²⋊C₈×C₄₁	1	4	yes	—
194	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	#95×C₄₁	1	2	yes	—
204	C₂×C₄	#95×C₄₁	1	2	yes	—
205	C₂×C₄	#95×C₄₁	1	2	yes	—
206	C₂×C₄	C₄×C₈×C₄₁	1	4	yes	—
207	C₂×C₄	#97×C₄₁	1	4	yes	—
208	C₂×C₄	C₂²⋊C₈×C₄₁	1	4	yes	—
209	C₂×C₄	C₄⋊C₈×C₄₁	1	8	yes	—
210	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
212	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
215	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
216	C₈	—	4	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₄₁×#336	1	4	yes	—
222	C₂²	C₂×#95×C₄₁	1	1	yes	characteristic
223	C₂²	C₂×C₂²⋊C₈×C₄₁	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	8	no	—
228	C₄	C₄₁×#336	1	4	yes	—
229	C₄	C₄₁×H(3,ℤ/4)	1	4	yes	—
230	C₄	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
231	C₄	C₂×#97×C₄₁	1	1	yes	characteristic
232	C₄	C₂×C₄⋊C₈×C₄₁	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂×C₄₁×#336	1	1	yes	characteristic
236	C₂	C₂×C₄₁×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₄²×C₄₁	C₄²
commutator	C₄	C₂×C₄×C₈×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×#95
prime radical	C₂⁴×C₄₁	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1274	C₂	semidirect
8. C₄₁×#431	C₄	semidirect
12. C₂×C₄×C₈×C₄₁	C₄	semidirect
21. #117×C₄₁	C₈	semidirect
68. C₄²×C₄₁	C₂×C₈	semidirect
98. #7854	C₄₁	direct (N × H)
117. #1274	C₂×C₄₁	semidirect
126. #431	C₄×C₄₁	semidirect
130. C₂×C₄×C₈	C₄×C₄₁	semidirect
135. C₄₁	#7854	direct (N × H)
140. #117	C₈×C₄₁	semidirect
187. C₄²	C₂×C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#431	2
C₄×#117×C₄₁	2
C₄₁×#1274	2
#7854	41

Subgroups — C41×#7854 #23784055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#7854 #23784055