Subgroups — C₄₁×#8184 #23784385

Order 10496.

530 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1476	C₂	1	2	yes	maximal
3	C₄₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈²×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1278	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₄₁	C₄	1	2	yes	—
7	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
8	C₄₁×#431	C₂²	1	2	yes	—
9	C₄₁×#405	—	2	2	no	—
10	C₄₁×#405	C₄	1	2	yes	—
11	C₄×M₁₆×C₄₁	—	2	1	no	—
12	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₄₁	—	2	1	no	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×Q₈⋊C₈	C₄	1	4	yes	—
17	C₄₁×D₄⋊C₈	C₄	1	4	yes	—
18	C₄₁×#322	—	2	1	no	—
19	C₈²×C₄₁	—	2	2	no	—
20	C₈²×C₄₁	—	2	1	no	—
21	C₂×#41×C₄₁	C₂×C₄	1	1	yes	characteristic
22	#131×C₄₁	—	2	4	no	—
23	#131×C₄₁	C₈	1	8	yes	complemented
24	C₂×M₁₆×C₄₁	—	2	2	no	—
25	C₂×M₁₆×C₄₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₄₁	—	2	2	no	—
27	C₂²×C₈×C₄₁	—	2	2	no	—
28	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₄₁	C₂×C₄	1	2	yes	—
30	C₄×D₄×C₄₁	—	2	2	no	—
31	C₄×D₄×C₄₁	C₂×C₄	1	2	yes	—
32	#117×C₄₁	—	2	1	no	—
33	#117×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₁	—	2	1	no	—
35	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₁	C₂×C₄	1	4	yes	—
37	C₂²⋊C₈×C₄₁	—	2	2	no	—
38	#97×C₄₁	—	2	2	no	—
39	#97×C₄₁	—	2	1	no	—
40	C₄×C₈×C₄₁	—	2	2	no	—
41	C₄×C₈×C₄₁	—	2	2	no	—
42	C₄×C₈×C₄₁	—	2	2	no	—
43	C₄×C₈×C₄₁	—	2	1	no	—
44	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₄₁	D₄	1	2	yes	—
46	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
48	#41×C₄₁	—	2	2	no	—
49	#41×C₄₁	C₂×C₈	1	4	yes	—
50	C₂×Q₈×C₄₁	C₄²	1	1	yes	characteristic
51	C₂×D₄×C₄₁	—	2	1	no	—
52	C₂×D₄×C₄₁	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	1	no	—
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	#41	1	2	yes	—
57	M₁₆×C₄₁	—	2	2	no	—
58	M₁₆×C₄₁	—	2	8	no	—
59	M₁₆×C₄₁	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₄₁	—	2	8	no	—
61	C₂×C₈×C₄₁	—	2	2	no	—
62	C₂×C₈×C₄₁	—	2	4	no	—
63	C₂×C₈×C₄₁	—	2	8	no	—
64	C₂×C₈×C₄₁	—	2	4	no	—
65	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₄₁	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₄₁	M₁₆	1	4	yes	—
69	C₄⋊C₄×C₄₁	—	2	2	no	—
70	C₄⋊C₄×C₄₁	C₄²	1	2	yes	—
71	C₂²⋊C₄×C₄₁	—	2	1	no	—
72	C₂²⋊C₄×C₄₁	—	2	2	no	—
73	C₄²×C₄₁	—	2	1	no	—
74	C₄²×C₄₁	—	2	2	no	—
75	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₄₁	—	2	1	no	—
79	C₂³×C₄₁	#117	1	1	yes	characteristic
80	Q₈×C₄₁	—	2	1	no	—
81	Q₈×C₄₁	C₄×C₈	1	2	yes	—
82	D₄×C₄₁	—	2	4	no	—
83	D₄×C₄₁	—	2	1	no	—
84	D₄×C₄₁	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₄₁	—	2	1	no	—
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	—	2	4	no	—
88	C₂×C₄×C₄₁	—	2	2	no	—
89	C₂×C₄×C₄₁	—	4	4	no	—
90	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₁	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₄₁	C₂×M₁₆	1	2	yes	—
96	C₈×C₄₁	—	2	8	no	—
97	C₈×C₄₁	—	2	4	no	—
98	C₈×C₄₁	—	4	4	no	—
99	C₈×C₄₁	C₂²⋊C₈	1	4	yes	—
100	C₈×C₄₁	C₂²⋊C₈	1	4	yes	—
101	#8184	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₄₁	—	2	2	no	—
103	C₂²×C₄₁	—	2	2	no	—
104	C₂²×C₄₁	—	4	2	no	—
105	C₂²×C₄₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₄₁	#431	1	2	yes	—
107	C₄×C₄₁	—	2	2	no	—
108	C₄×C₄₁	—	2	2	no	—
109	C₄×C₄₁	—	4	2	no	—
110	C₄×C₄₁	C₂×C₄×C₈	1	1	yes	characteristic
111	C₄×C₄₁	C₄×M₁₆	1	1	yes	characteristic
112	C₄×C₄₁	C₂×C₂²⋊C₈	1	2	yes	—
113	C₄×C₄₁	#443	1	4	yes	—
114	C₄×#131	C₂×C₄₁	1	1	yes	characteristic
115	#1476	C₂×C₄₁	1	2	yes	—
116	#1450	C₂×C₄₁	1	1	yes	characteristic, complemented
117	C₂×C₈²	C₂×C₄₁	1	1	yes	characteristic, complemented
118	#1278	C₂×C₄₁	1	2	yes	complemented
119	C₂×C₄₁	—	2	2	no	—
120	C₂×C₄₁	—	4	2	no	—
121	C₂×C₄₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₄₁	#1764	1	2	yes	—
123	C₂×#131	C₄×C₄₁	1	2	yes	—
124	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
125	#431	C₂²×C₄₁	1	2	yes	—
126	#405	—	2	2	no	—
127	#405	C₄×C₄₁	1	2	yes	—
128	C₄×M₁₆	—	2	1	no	—
129	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
133	Q₈⋊C₈	C₄×C₄₁	1	4	yes	—
134	D₄⋊C₈	C₄×C₄₁	1	4	yes	—
135	#322	—	2	1	no	—
136	C₈²	—	2	2	no	—
137	C₈²	—	2	1	no	—
138	C₄₁	#8184	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#41	C₂×C₄×C₄₁	1	1	yes	characteristic
140	#131	—	2	4	no	—
141	#131	C₈×C₄₁	1	8	yes	complemented
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	C₂×C₄×C₄₁	1	2	yes	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
147	C₄×Q₈	C₂×C₄×C₄₁	1	2	yes	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	C₂×C₄×C₄₁	1	2	yes	—
150	#117	—	2	1	no	—
151	#117	C₂×C₄×C₄₁	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
154	C₄⋊C₈	C₂×C₄×C₄₁	1	4	yes	—
155	C₂²⋊C₈	—	2	2	no	—
156	#97	—	2	2	no	—
157	#97	—	2	1	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
163	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
164	C₄×C₈	D₄×C₄₁	1	2	yes	—
165	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
166	#41	—	2	2	no	—
167	#41	C₂×C₈×C₄₁	1	4	yes	—
168	C₂×Q₈	C₄²×C₄₁	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₄²×C₄₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
174	C₂²×C₄	#41×C₄₁	1	2	yes	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	C₂×C₈×C₄₁	1	4	yes	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
184	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
185	C₂×C₈	C₂×C₈×C₄₁	1	4	yes	—
186	C₂×C₈	M₁₆×C₄₁	1	4	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	C₄²×C₄₁	1	2	yes	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
194	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	C₂³	#117×C₄₁	1	1	yes	characteristic
198	Q₈	—	2	1	no	—
199	Q₈	C₄×C₈×C₄₁	1	2	yes	—
200	D₄	—	2	4	no	—
201	D₄	—	2	1	no	—
202	D₄	C₄×C₈×C₄₁	1	2	yes	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
209	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
210	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
211	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
212	C₂×C₄	C₂²×C₈×C₄₁	1	2	yes	—
213	C₂×C₄	C₂×M₁₆×C₄₁	1	2	yes	—
214	C₈	—	2	8	no	—
215	C₈	—	2	4	no	—
216	C₈	—	4	4	no	—
217	C₈	C₂²⋊C₈×C₄₁	1	4	yes	—
218	C₈	C₂²⋊C₈×C₄₁	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₄×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
223	C₂²	C₄₁×#431	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₄×C₈×C₄₁	1	1	yes	characteristic
228	C₄	C₄×M₁₆×C₄₁	1	1	yes	characteristic
229	C₄	C₂×C₂²⋊C₈×C₄₁	1	2	yes	—
230	C₄	C₄₁×#443	1	4	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	C₄×C₂²⋊C₈×C₄₁	1	1	yes	characteristic
234	C₂	C₄₁×#1764	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₄₁	D₄
commutator	C₄	C₂×C₄×C₈×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂²×C₄₁	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₄₁	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₁×#1450	C₂	semidirect
4. C₂×C₈²×C₄₁	C₂	semidirect
5. C₄₁×#1278	C₂	semidirect
23. #131×C₄₁	C₈	semidirect
101. #8184	C₄₁	direct (N × H)
116. #1450	C₂×C₄₁	semidirect
117. C₂×C₈²	C₂×C₄₁	semidirect
118. #1278	C₂×C₄₁	semidirect
138. C₄₁	#8184	direct (N × H)
141. #131	C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#131×C₄₁	2
C₄₁×#1476	2
C₄₁×#1450	2
C₂×C₈²×C₄₁	2
C₄₁×#1278	2
#8184	41

Subgroups — C41×#8184 #23784385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8184 #23784385