Subgroups — C₄₁×#2344 #19106124

Order 5248.

604 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₄₁	C₂	1	2	yes	maximal
2	C₂²×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#389	C₂	1	2	yes	maximal, complemented
4	C₄₁×#387	C₂	1	4	yes	maximal, complemented
5	C₄₁×#381	C₂	1	2	yes	maximal
6	C₄₁×#380	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄×C₄₁	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄×C₄₁	C₂²	1	2	yes	—
10	C₂³×C₄×C₄₁	C₂²	1	1	yes	characteristic
11	#117×C₄₁	—	2	8	no	—
12	C₂×C₄⋊C₄×C₄₁	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₄₁	C₂²	1	4	yes	—
22	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
23	#95×C₄₁	C₂²	1	2	yes	—
24	#95×C₄₁	C₂²	1	4	yes	—
25	#95×C₄₁	C₂²	1	2	yes	—
26	#95×C₄₁	C₂²	1	4	yes	—
27	C₂⁴×C₄₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₁	—	2	4	no	—
29	C₂²×C₄×C₄₁	—	2	4	no	—
30	C₂²×C₄×C₄₁	—	2	4	no	—
31	C₂²×C₄×C₄₁	C₂×C₄	1	8	yes	—
32	C₂²×C₄×C₄₁	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₄₁	—	2	8	no	—
41	C₄⋊C₄×C₄₁	—	2	4	no	—
42	C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
43	C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
44	C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
45	C₂²⋊C₄×C₄₁	—	2	4	no	—
46	C₂²⋊C₄×C₄₁	—	2	2	no	—
47	C₂²⋊C₄×C₄₁	D₄	1	4	yes	—
48	C₄²×C₄₁	—	2	8	no	—
49	C₄²×C₄₁	—	2	4	no	—
50	C₂³×C₄₁	—	2	2	no	—
51	C₂³×C₄₁	—	2	2	no	—
52	C₂³×C₄₁	C₂²×C₄	1	2	yes	—
53	C₂³×C₄₁	C₂×D₄	1	2	yes	—
54	C₂³×C₄₁	#41	1	2	yes	—
55	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₄₁	—	2	16	no	—
57	C₂×C₄×C₄₁	—	2	2	no	—
58	C₂×C₄×C₄₁	—	2	2	no	—
59	C₂×C₄×C₄₁	—	2	2	no	—
60	C₂×C₄×C₄₁	—	2	2	no	—
61	C₂×C₄×C₄₁	—	2	2	no	—
62	C₂×C₄×C₄₁	—	2	2	no	—
63	C₂×C₄×C₄₁	—	2	2	no	—
64	C₂×C₄×C₄₁	—	2	8	no	—
65	C₂×C₄×C₄₁	—	2	4	no	—
66	C₂×C₄×C₄₁	—	2	2	no	—
67	C₂×C₄×C₄₁	—	2	2	no	—
68	C₂×C₄×C₄₁	—	2	4	no	—
69	C₂×C₄×C₄₁	—	2	4	no	—
70	C₂×C₄×C₄₁	—	2	2	no	—
71	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
73	C₂×C₄×C₄₁	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₄₁	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₄₁	#41	1	4	yes	—
77	C₂²×C₄₁	—	2	2	no	—
78	C₂²×C₄₁	—	2	2	no	—
79	C₂²×C₄₁	—	2	4	no	—
80	C₂²×C₄₁	—	2	4	no	—
81	C₂²×C₄₁	C₄×D₄	1	4	yes	—
82	C₂²×C₄₁	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₄₁	C₂×#41	1	2	yes	—
85	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
86	C₂²×C₄₁	2₋¹⁺⁴	1	2	yes	—
87	C₄×C₄₁	—	2	2	no	—
88	C₄×C₄₁	—	2	4	no	—
89	C₄×C₄₁	—	2	4	no	—
90	C₄×C₄₁	—	2	4	no	—
91	C₄×C₄₁	—	4	4	no	—
92	C₄×C₄₁	—	4	2	no	—
93	C₄×C₄₁	#117	1	4	yes	complemented
94	#2344	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₄₁	—	2	4	no	—
96	C₂×C₄₁	C₂×#117	1	1	yes	characteristic
97	C₂×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
98	C₂×C₄₁	#519	1	1	yes	characteristic
99	C₂×C₄₁	#547	1	2	yes	—
100	C₂×C₄₁	#552	1	2	yes	—
101	C₂×#117	C₂×C₄₁	1	2	yes	—
102	C₂²×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
103	#389	C₂×C₄₁	1	2	yes	complemented
104	#387	C₂×C₄₁	1	4	yes	complemented
105	#381	C₂×C₄₁	1	2	yes	—
106	#380	C₂×C₄₁	1	1	yes	characteristic
107	C₄×C₄⋊C₄	C₂×C₄₁	1	2	yes	complemented
108	C₄×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic
109	C₄₁	#2344	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³×C₄	C₂²×C₄₁	1	2	yes	—
111	C₂³×C₄	C₂²×C₄₁	1	1	yes	characteristic
112	#117	—	2	8	no	—
113	C₂×C₄⋊C₄	C₄×C₄₁	1	8	yes	complemented
114	C₂×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
115	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₄₁	1	4	yes	—
123	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
124	#95	C₂²×C₄₁	1	2	yes	—
125	#95	C₂²×C₄₁	1	4	yes	—
126	#95	C₂²×C₄₁	1	2	yes	—
127	#95	C₂²×C₄₁	1	4	yes	—
128	C₂⁴	C₂³×C₄₁	1	1	yes	characteristic
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂×C₄×C₄₁	1	8	yes	—
133	C₂²×C₄	C₂×C₄×C₄₁	1	4	yes	—
134	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
135	C₂²×C₄	C₂³×C₄₁	1	4	yes	—
136	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
138	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
139	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
140	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
144	C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
145	C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	D₄×C₄₁	1	4	yes	—
149	C₄²	—	2	8	no	—
150	C₄²	—	2	4	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	C₂²×C₄×C₄₁	1	2	yes	—
154	C₂³	C₂×D₄×C₄₁	1	2	yes	—
155	C₂³	#41×C₄₁	1	2	yes	—
156	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
157	C₂×C₄	—	2	16	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
173	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
174	C₂×C₄	C₂²×C₄×C₄₁	1	4	yes	—
175	C₂×C₄	C₂²×C₄×C₄₁	1	2	yes	—
176	C₂×C₄	C₂×D₄×C₄₁	1	4	yes	—
177	C₂×C₄	#41×C₄₁	1	4	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	4	no	—
182	C₂²	C₄×D₄×C₄₁	1	4	yes	—
183	C₂²	C₂³×C₄×C₄₁	1	1	yes	characteristic
184	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
185	C₂²	C₂×#41×C₄₁	1	2	yes	—
186	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
187	C₂²	2₋¹⁺⁴×C₄₁	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	4	4	no	—
193	C₄	—	4	2	no	—
194	C₄	#117×C₄₁	1	4	yes	complemented
195	C₂	—	2	4	no	—
196	C₂	C₂×#117×C₄₁	1	1	yes	characteristic
197	C₂	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
198	C₂	C₄₁×#519	1	1	yes	characteristic
199	C₂	C₄₁×#547	1	2	yes	—
200	C₂	C₄₁×#552	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₁×#389	C₂	semidirect
4. C₄₁×#387	C₂	semidirect
7. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
12. C₂×C₄⋊C₄×C₄₁	C₄	semidirect
93. C₄×C₄₁	#117	semidirect
94. #2344	C₄₁	direct (N × H)
103. #389	C₂×C₄₁	semidirect
104. #387	C₂×C₄₁	semidirect
107. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
109. C₄₁	#2344	direct (N × H)
113. C₂×C₄⋊C₄	C₄×C₄₁	semidirect
194. C₄	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#117×C₄₁	2
C₂²×C₄⋊C₄×C₄₁	2
C₄₁×#389	2
C₄₁×#387	2
C₄₁×#381	2
C₄₁×#380	2
C₄×C₄⋊C₄×C₄₁	2
C₄×C₂²⋊C₄×C₄₁	2
#2344	41

Subgroups — C41×#2344 #19106124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2344 #19106124