Subgroups — C₄₁×#13179 #23789380

Order 10496.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#368	C₂	1	2	yes	maximal
2	C₂×C₄₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1848	C₂	1	1	yes	maximal, characteristic
4	C₄×#108×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1381	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₄₁	C₂²	1	2	yes	—
7	C₂×#108×C₄₁	—	2	1	no	—
8	C₂×#108×C₄₁	C₂²	1	2	yes	—
9	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#423	—	2	1	no	—
12	C₄×M₁₆×C₄₁	—	2	1	no	—
13	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#368	—	2	4	no	—
15	C₄₁×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₄₁×C₄⋊C₁₆	C₄	1	2	yes	—
17	C₄₁×#341	—	2	1	no	—
18	C₂×M₁₆×C₄₁	—	2	1	no	—
19	C₂×M₁₆×C₄₁	—	2	1	no	—
20	C₂×M₁₆×C₄₁	—	2	2	no	—
21	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₁	—	2	2	no	—
25	C₂×C₁₆×C₄₁	—	2	2	no	—
26	C₂×C₁₆×C₄₁	C₂×C₄	1	4	yes	—
27	#108×C₄₁	—	2	2	no	—
28	#108×C₄₁	—	2	1	no	—
29	#108×C₄₁	—	2	1	no	—
30	#108×C₄₁	—	2	2	no	—
31	#108×C₄₁	—	4	2	no	—
32	#108×C₄₁	D₄	1	2	yes	—
33	#108×C₄₁	Q₈	1	2	yes	—
34	C₄⋊C₈×C₄₁	—	4	1	no	—
35	C₄⋊C₈×C₄₁	—	4	1	no	—
36	#97×C₄₁	—	4	1	no	—
37	C₄×C₈×C₄₁	—	4	1	no	—
38	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₄₁	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₄₁	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₄₁	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₄₁	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₄₁	—	2	2	no	—
46	M₁₆×C₄₁	—	2	2	no	—
47	M₁₆×C₄₁	—	4	1	no	—
48	M₁₆×C₄₁	—	4	1	no	—
49	M₁₆×C₄₁	—	4	2	no	—
50	M₁₆×C₄₁	—	4	2	no	—
51	C₂×C₈×C₄₁	—	2	1	no	—
52	C₂×C₈×C₄₁	—	2	1	no	—
53	C₂×C₈×C₄₁	—	4	1	no	—
54	C₂×C₈×C₄₁	—	4	1	no	—
55	C₂×C₈×C₄₁	—	4	2	no	—
56	C₂×C₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₄₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₄₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
63	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
64	C₄²×C₄₁	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₄₁	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₄₁	D₈	1	1	yes	characteristic
67	C₄²×C₄₁	D₈	1	1	yes	characteristic
68	C₁₆×C₄₁	—	2	4	no	—
69	C₂³×C₄₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₄₁	—	2	2	no	—
71	C₂×C₄×C₄₁	—	4	1	no	—
72	C₂×C₄×C₄₁	#107	1	1	yes	characteristic
73	C₂×C₄×C₄₁	#107	1	1	yes	characteristic
74	C₂×C₄×C₄₁	#107	1	1	yes	characteristic
75	C₂×C₄×C₄₁	#107	1	1	yes	characteristic
76	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₁	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
82	C₈×C₄₁	—	2	1	no	—
83	C₈×C₄₁	—	2	1	no	—
84	C₈×C₄₁	—	4	2	no	—
85	C₈×C₄₁	—	4	2	no	—
86	C₈×C₄₁	—	8	2	no	—
87	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
88	C₈×C₄₁	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₄₁	#125	1	1	yes	characteristic
90	C₈×C₄₁	C₄⋊Q₈	1	1	yes	characteristic
91	#13179	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₄₁	—	2	1	no	—
93	C₂²×C₄₁	—	2	1	no	—
94	C₂²×C₄₁	C₂×#107	1	1	yes	characteristic
95	C₂²×C₄₁	#462	1	1	yes	characteristic
96	C₂²×C₄₁	#477	1	1	yes	characteristic
97	C₄×C₄₁	—	2	1	no	—
98	C₄×C₄₁	—	2	1	no	—
99	C₄×C₄₁	—	2	1	no	—
100	C₄×C₄₁	#368	1	2	yes	complemented
101	C₄×C₄₁	#389	1	1	yes	characteristic
102	C₄×C₄₁	#428	1	1	yes	characteristic
103	C₄×C₄₁	#459	1	1	yes	characteristic
104	C₄×C₄₁	#474	1	1	yes	characteristic
105	C₂×#368	C₂×C₄₁	1	2	yes	—
106	C₂×C₄⋊C₁₆	C₂×C₄₁	1	1	yes	characteristic
107	#1848	C₂×C₄₁	1	1	yes	characteristic
108	C₄×#108	C₂×C₄₁	1	1	yes	characteristic
109	#1381	C₂×C₄₁	1	2	yes	—
110	C₂×C₄₁	—	2	1	no	—
111	C₂×C₄₁	—	2	1	no	—
112	C₂×C₄₁	#1947	1	1	yes	characteristic
113	C₂×C₄₁	C₂×#368	1	1	yes	characteristic
114	C₂×C₄₁	#2161	1	1	yes	characteristic
115	C₂²×C₁₆	C₂²×C₄₁	1	2	yes	—
116	C₂×#108	—	2	1	no	—
117	C₂×#108	C₂²×C₄₁	1	2	yes	—
118	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
119	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
120	#423	—	2	1	no	—
121	C₄×M₁₆	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
123	#368	—	2	4	no	—
124	C₄⋊C₁₆	C₄×C₄₁	1	2	yes	—
125	C₄⋊C₁₆	C₄×C₄₁	1	2	yes	—
126	#341	—	2	1	no	—
127	C₄₁	#13179	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
133	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₂×C₄×C₄₁	1	4	yes	—
137	#108	—	2	2	no	—
138	#108	—	2	1	no	—
139	#108	—	2	1	no	—
140	#108	—	2	2	no	—
141	#108	—	4	2	no	—
142	#108	D₄×C₄₁	1	2	yes	—
143	#108	Q₈×C₄₁	1	2	yes	—
144	C₄⋊C₈	—	4	1	no	—
145	C₄⋊C₈	—	4	1	no	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
149	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
150	C₄×C₈	Q₈×C₄₁	1	1	yes	characteristic
151	C₄×C₈	Q₈×C₄₁	1	1	yes	characteristic
152	C₂²×C₄	Q₁₆×C₄₁	1	1	yes	characteristic
153	C₂²×C₄	Q₁₆×C₄₁	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	4	1	no	—
158	M₁₆	—	4	1	no	—
159	M₁₆	—	4	2	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
167	C₂×C₈	C₄⋊C₄×C₄₁	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
171	C₂×C₈	C₂×Q₈×C₄₁	1	1	yes	characteristic
172	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
173	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
174	C₄²	C₄⋊C₄×C₄₁	1	1	yes	characteristic
175	C₄²	C₄⋊C₄×C₄₁	1	1	yes	characteristic
176	C₄²	D₈×C₄₁	1	1	yes	characteristic
177	C₄²	D₈×C₄₁	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₂³	C₂×Q₁₆×C₄₁	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#107×C₄₁	1	1	yes	characteristic
183	C₂×C₄	#107×C₄₁	1	1	yes	characteristic
184	C₂×C₄	#107×C₄₁	1	1	yes	characteristic
185	C₂×C₄	#107×C₄₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
187	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
190	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
191	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	8	2	no	—
197	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
198	C₈	C₄×Q₈×C₄₁	1	1	yes	characteristic
199	C₈	#125×C₄₁	1	1	yes	characteristic
200	C₈	C₄⋊Q₈×C₄₁	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#107×C₄₁	1	1	yes	characteristic
204	C₂²	C₄₁×#462	1	1	yes	characteristic
205	C₂²	C₄₁×#477	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	C₄₁×#368	1	2	yes	complemented
210	C₄	C₄₁×#389	1	1	yes	characteristic
211	C₄	C₄₁×#428	1	1	yes	characteristic
212	C₄	C₄₁×#459	1	1	yes	characteristic
213	C₄	C₄₁×#474	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	C₄₁×#1947	1	1	yes	characteristic
217	C₂	C₂×C₄₁×#368	1	1	yes	characteristic
218	C₂	C₄₁×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
socle	C₂²×C₄₁	C₂×#107
prime radical	C₂³×C₄₁	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ C₂×C₄×C₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
91. #13179	C₄₁	direct (N × H)
100. C₄×C₄₁	#368	semidirect
127. C₄₁	#13179	direct (N × H)
209. C₄	C₄₁×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#368	2
C₂×C₄₁×C₄⋊C₁₆	2
C₄₁×#1848	2
C₄×#108×C₄₁	2
C₄₁×#1381	2
#13179	41

Subgroups — C41×#13179 #23789380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13179 #23789380