Subgroups — C₄₁×#2548 #19106328

Order 5248.

676 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#127×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#124×C₄₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#394	C₂	1	4	yes	maximal, complemented
5	C₄₁×#390	C₂	1	4	yes	maximal, complemented
6	C₄₁×#390	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
10	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×D₄×C₄₁	C₂²	1	2	yes	—
12	#127×C₄₁	—	2	4	no	—
13	#124×C₄₁	—	2	8	no	—
14	C₄×Q₈×C₄₁	—	2	4	no	—
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	2	yes	—
18	C₂×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₄×C₄₁	C₂²	1	8	yes	—
20	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₁	C₂²	1	1	yes	characteristic
23	#95×C₄₁	C₂²	1	2	yes	—
24	C₂⁴×C₄₁	C₂³	1	4	yes	—
25	C₂×Q₈×C₄₁	—	2	2	no	—
26	C₂×Q₈×C₄₁	—	2	2	no	—
27	C₂×Q₈×C₄₁	D₄	1	4	yes	complemented
28	C₂×D₄×C₄₁	—	2	4	no	—
29	C₂×D₄×C₄₁	—	2	4	no	—
30	C₂×D₄×C₄₁	—	2	4	no	—
31	C₂×D₄×C₄₁	—	2	4	no	—
32	C₂×D₄×C₄₁	—	2	4	no	—
33	C₂×D₄×C₄₁	—	2	8	no	—
34	C₂×D₄×C₄₁	—	4	4	no	—
35	C₂²×C₄×C₄₁	C₂³	1	4	yes	—
36	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₄₁	—	2	4	no	—
42	C₄⋊C₄×C₄₁	—	2	2	no	—
43	C₄⋊C₄×C₄₁	—	2	4	no	—
44	C₂²⋊C₄×C₄₁	—	2	16	no	—
45	C₂²⋊C₄×C₄₁	—	4	8	no	—
46	C₄²×C₄₁	—	2	4	no	—
47	C₄²×C₄₁	—	2	4	no	—
48	C₄²×C₄₁	D₄	1	4	yes	—
49	C₂³×C₄₁	—	2	4	no	—
50	C₂³×C₄₁	—	2	4	no	—
51	C₂³×C₄₁	—	2	4	no	—
52	C₂³×C₄₁	—	2	8	no	—
53	C₂³×C₄₁	—	2	8	no	—
54	C₂³×C₄₁	C₂⁴	1	1	yes	characteristic
55	Q₈×C₄₁	—	2	8	no	—
56	D₄×C₄₁	—	4	8	no	—
57	D₄×C₄₁	—	4	8	no	—
58	D₄×C₄₁	—	4	8	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	4	no	—
63	C₂×C₄×C₄₁	—	2	2	no	—
64	C₂×C₄×C₄₁	—	2	4	no	—
65	C₂×C₄×C₄₁	—	2	4	no	—
66	C₂×C₄×C₄₁	—	2	4	no	—
67	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₄₁	C₂×D₄	1	4	yes	—
69	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₄₁	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₄₁	#41	1	2	yes	—
73	C₂×C₄×C₄₁	#41	1	4	yes	—
74	C₂²×C₄₁	—	4	4	no	—
75	C₂²×C₄₁	—	4	4	no	—
76	C₂²×C₄₁	—	4	4	no	—
77	C₂²×C₄₁	—	4	8	no	—
78	C₂²×C₄₁	—	4	8	no	—
79	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₄₁	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₄₁	C₂×#41	1	1	yes	characteristic
82	C₂²×C₄₁	C₂×#41	1	2	yes	—
83	C₂²×C₄₁	H(5,𝔽₂)	1	2	yes	—
84	C₄×C₄₁	—	2	4	no	—
85	C₄×C₄₁	—	2	2	no	—
86	C₄×C₄₁	—	4	2	no	—
87	C₄×C₄₁	—	4	4	no	—
88	C₄×C₄₁	#121	1	4	yes	—
89	C₄×C₄₁	#124	1	4	yes	complemented
90	#2548	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₄₁	—	8	4	no	—
92	C₂×C₄₁	C₂×#121	1	1	yes	characteristic
93	C₂×C₄₁	C₂×#124	1	1	yes	characteristic
94	C₂×C₄₁	#535	1	1	yes	characteristic
95	C₂×C₄₁	#550	1	2	yes	—
96	C₂×C₄₁	#559	1	2	yes	—
97	C₂×#127	C₂×C₄₁	1	1	yes	characteristic
98	C₂×#124	C₂×C₄₁	1	2	yes	complemented
99	C₂×C₄×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
100	#394	C₂×C₄₁	1	4	yes	complemented
101	#390	C₂×C₄₁	1	4	yes	complemented
102	#390	C₂×C₄₁	1	2	yes	complemented
103	C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
104	C₄₁	#2548	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²×Q₈	C₂²×C₄₁	1	1	yes	characteristic
106	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
107	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
108	C₂²×D₄	C₂²×C₄₁	1	2	yes	—
109	#127	—	2	4	no	—
110	#124	—	2	8	no	—
111	C₄×Q₈	—	2	4	no	—
112	C₂×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₄₁	1	8	yes	—
117	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
118	C₂×C₄²	C₂²×C₄₁	1	2	yes	—
119	C₂×C₄²	C₂²×C₄₁	1	1	yes	characteristic
120	#95	C₂²×C₄₁	1	2	yes	—
121	C₂⁴	C₂³×C₄₁	1	4	yes	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	D₄×C₄₁	1	4	yes	complemented
125	C₂×D₄	—	2	4	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	2	4	no	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	2	8	no	—
131	C₂×D₄	—	4	4	no	—
132	C₂²×C₄	C₂³×C₄₁	1	4	yes	—
133	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
134	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₄₁	1	2	yes	—
136	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₄₁	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	2	16	no	—
142	C₂²⋊C₄	—	4	8	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	4	no	—
145	C₄²	D₄×C₄₁	1	4	yes	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	4	no	—
148	C₂³	—	2	4	no	—
149	C₂³	—	2	8	no	—
150	C₂³	—	2	8	no	—
151	C₂³	C₂⁴×C₄₁	1	1	yes	characteristic
152	Q₈	—	2	8	no	—
153	D₄	—	4	8	no	—
154	D₄	—	4	8	no	—
155	D₄	—	4	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
165	C₂×C₄	C₂×D₄×C₄₁	1	4	yes	—
166	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
167	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
168	C₂×C₄	C₂×D₄×C₄₁	1	2	yes	—
169	C₂×C₄	#41×C₄₁	1	2	yes	—
170	C₂×C₄	#41×C₄₁	1	4	yes	—
171	C₂²	—	4	4	no	—
172	C₂²	—	4	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	8	no	—
175	C₂²	—	4	8	no	—
176	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
177	C₂²	C₂²×D₄×C₄₁	1	1	yes	characteristic
178	C₂²	C₂×#41×C₄₁	1	1	yes	characteristic
179	C₂²	C₂×#41×C₄₁	1	2	yes	—
180	C₂²	H(5,𝔽₂)×C₄₁	1	2	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	4	no	—
185	C₄	#121×C₄₁	1	4	yes	—
186	C₄	#124×C₄₁	1	4	yes	complemented
187	C₂	—	8	4	no	—
188	C₂	C₂×#121×C₄₁	1	1	yes	characteristic
189	C₂	C₂×#124×C₄₁	1	1	yes	characteristic
190	C₂	C₄₁×#535	1	1	yes	characteristic
191	C₂	C₄₁×#550	1	2	yes	—
192	C₂	C₄₁×#559	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₄₁
prime radical	C₂×#127×C₄₁	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#124×C₄₁	C₂	semidirect
3. C₂×C₄×Q₈×C₄₁	C₂	semidirect
4. C₄₁×#394	C₂	semidirect
5. C₄₁×#390	C₂	semidirect
6. C₄₁×#390	C₂	semidirect
7. C₄×C₄⋊C₄×C₄₁	C₂	semidirect
27. C₂×Q₈×C₄₁	D₄	semidirect
89. C₄×C₄₁	#124	semidirect
90. #2548	C₄₁	direct (N × H)
98. C₂×#124	C₂×C₄₁	semidirect
99. C₂×C₄×Q₈	C₂×C₄₁	semidirect
100. #394	C₂×C₄₁	semidirect
101. #390	C₂×C₄₁	semidirect
102. #390	C₂×C₄₁	semidirect
103. C₄×C₄⋊C₄	C₂×C₄₁	semidirect
104. C₄₁	#2548	direct (N × H)
124. C₂×Q₈	D₄×C₄₁	semidirect
186. C₄	#124×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#127×C₄₁	2
C₂×#124×C₄₁	2
C₂×C₄×Q₈×C₄₁	2
C₄₁×#394	2
C₄₁×#390	2
C₄×C₄⋊C₄×C₄₁	2
#2548	41

Subgroups — C41×#2548 #19106328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#2548 #19106328