Subgroups — C₂×C₄₁×#2397 #23810745

Order 10496.

5150 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₄₁	C₂	1	6	yes	maximal, complemented
2	C₄₁×#2397	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄₁×#399	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₁×#392	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₄₁	C₂²	1	6	yes	—
7	C₂⁴×C₄×C₄₁	C₂²	1	4	yes	—
8	C₂×C₄×D₄×C₄₁	—	2	48	no	—
9	C₂×C₄×D₄×C₄₁	C₂²	1	48	yes	complemented
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	6	yes	—
11	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	12	yes	—
12	C₂²×C₄²×C₄₁	C₂²	1	3	yes	—
13	C₄₁×#399	C₂²	1	32	yes	complemented
14	C₄₁×#392	C₂²	1	32	yes	complemented
15	C₄₁×#376	C₂²	1	8	yes	complemented
16	C₂×#95×C₄₁	C₂²	1	4	yes	—
17	C₂⁵×C₄₁	C₂³	1	4	yes	—
18	C₂²×D₄×C₄₁	—	2	48	no	—
19	C₂²×D₄×C₄₁	—	2	24	no	—
20	C₂²×D₄×C₄₁	C₂³	1	24	yes	—
21	C₂³×C₄×C₄₁	—	2	48	no	—
22	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₄₁	C₂³	1	16	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	6	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₄₁	—	2	192	no	—
27	C₂×C₄⋊C₄×C₄₁	—	2	24	no	—
28	C₂×C₄⋊C₄×C₄₁	C₂³	1	24	yes	—
29	C₂×C₂²⋊C₄×C₄₁	—	2	48	no	—
30	C₂×C₂²⋊C₄×C₄₁	C₂³	1	48	yes	—
31	C₂×C₄²×C₄₁	D₄	1	24	yes	complemented
32	C₂×C₄²×C₄₁	C₂³	1	12	yes	—
33	#95×C₄₁	C₂³	1	16	yes	—
34	C₂⁴×C₄₁	—	2	24	no	—
35	C₂⁴×C₄₁	—	2	24	no	—
36	C₂⁴×C₄₁	#41	1	16	yes	—
37	C₂⁴×C₄₁	C₂⁴	1	8	yes	—
38	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
39	C₂×D₄×C₄₁	—	2	192	no	—
40	C₂×D₄×C₄₁	—	2	96	no	—
41	C₂×D₄×C₄₁	—	2	48	no	—
42	C₂²×C₄×C₄₁	—	2	24	no	—
43	C₂²×C₄×C₄₁	—	2	24	no	—
44	C₂²×C₄×C₄₁	—	2	96	no	—
45	C₂²×C₄×C₄₁	—	4	64	no	—
46	C₂²×C₄×C₄₁	C₂×D₄	1	24	yes	—
47	C₂²×C₄×C₄₁	C₂×D₄	1	12	yes	—
48	C₂²×C₄×C₄₁	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₄₁	C₂⁴	1	12	yes	—
50	C₂²×C₄×C₄₁	C₂⁴	1	2	yes	—
51	C₄⋊C₄×C₄₁	—	2	48	no	—
52	C₂²⋊C₄×C₄₁	—	2	96	no	—
53	C₄²×C₄₁	C₂×D₄	1	48	yes	complemented
54	C₂³×C₄₁	—	2	96	no	—
55	C₂³×C₄₁	—	2	24	no	—
56	C₂³×C₄₁	—	2	24	no	—
57	C₂³×C₄₁	—	4	48	no	—
58	C₂³×C₄₁	—	4	16	no	—
59	C₂³×C₄₁	C₂²×D₄	1	6	yes	—
60	C₂³×C₄₁	C₂×#41	1	16	yes	—
61	C₂³×C₄₁	C₂×#41	1	8	yes	—
62	C₂³×C₄₁	C₂⁵	1	1	yes	characteristic
63	D₄×C₄₁	—	2	192	no	—
64	C₂×C₄×C₄₁	—	2	48	no	—
65	C₂×C₄×C₄₁	—	2	24	no	—
66	C₂×C₄×C₄₁	—	2	24	no	—
67	C₂×C₄×C₄₁	—	4	16	no	—
68	C₂×C₄×C₄₁	—	4	64	no	—
69	C₂×C₄×C₄₁	#127	1	16	yes	—
70	C₂×C₄×C₄₁	C₂²×D₄	1	24	yes	—
71	C₂×C₄×C₄₁	C₂²×D₄	1	12	yes	—
72	C₂×#2397	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂²×C₄₁	—	2	48	no	—
74	C₂²×C₄₁	—	4	64	no	—
75	C₂²×C₄₁	—	4	24	no	—
76	C₂²×C₄₁	—	4	8	no	—
77	C₂²×C₄₁	#525	1	24	yes	—
78	C₂²×C₄₁	C₂×#127	1	4	yes	—
79	C₂²×C₄₁	C₂³×D₄	1	3	yes	—
80	C₂²×C₄₁	C₂²×#41	1	4	yes	—
81	C₄×C₄₁	—	2	24	no	—
82	C₄×C₄₁	—	4	8	no	—
83	C₄×C₄₁	C₂×#127	1	8	yes	—
84	C₂²×C₄×D₄	C₂×C₄₁	1	6	yes	complemented
85	#2397	C₂×C₄₁	1	16	yes	direct product factor
86	C₂×#399	C₂×C₄₁	1	4	yes	complemented
87	C₂×#392	C₂×C₄₁	1	4	yes	complemented
88	C₂×#376	C₂×C₄₁	1	1	yes	characteristic, complemented
89	C₂×C₄₁	—	4	16	no	—
90	C₂×C₄₁	#2397	1	8	yes	direct product factor
91	C₂×C₄₁	C₂×#525	1	6	yes	—
92	C₂×C₄₁	C₂²×#127	1	1	yes	characteristic
93	C₂³×D₄	C₂²×C₄₁	1	6	yes	—
94	C₂⁴×C₄	C₂²×C₄₁	1	4	yes	—
95	C₂×C₄×D₄	—	2	48	no	—
96	C₂×C₄×D₄	C₂²×C₄₁	1	48	yes	complemented
97	C₂²×C₄⋊C₄	C₂²×C₄₁	1	6	yes	—
98	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	12	yes	—
99	C₂²×C₄²	C₂²×C₄₁	1	3	yes	—
100	#399	C₂²×C₄₁	1	32	yes	complemented
101	#392	C₂²×C₄₁	1	32	yes	complemented
102	#376	C₂²×C₄₁	1	8	yes	complemented
103	C₂×#95	C₂²×C₄₁	1	4	yes	—
104	C₄₁	C₂×#2397	1	1	yes	characteristic, direct product factor, Sylow
105	C₂⁵	C₂³×C₄₁	1	4	yes	—
106	C₂²×D₄	—	2	48	no	—
107	C₂²×D₄	—	2	24	no	—
108	C₂²×D₄	C₂³×C₄₁	1	24	yes	—
109	C₂³×C₄	—	2	48	no	—
110	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
111	C₂³×C₄	C₂³×C₄₁	1	16	yes	—
112	C₂³×C₄	C₂³×C₄₁	1	6	yes	—
113	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
114	C₄×D₄	—	2	192	no	—
115	C₂×C₄⋊C₄	—	2	24	no	—
116	C₂×C₄⋊C₄	C₂³×C₄₁	1	24	yes	—
117	C₂×C₂²⋊C₄	—	2	48	no	—
118	C₂×C₂²⋊C₄	C₂³×C₄₁	1	48	yes	—
119	C₂×C₄²	D₄×C₄₁	1	24	yes	complemented
120	C₂×C₄²	C₂³×C₄₁	1	12	yes	—
121	#95	C₂³×C₄₁	1	16	yes	—
122	C₂⁴	—	2	24	no	—
123	C₂⁴	—	2	24	no	—
124	C₂⁴	#41×C₄₁	1	16	yes	—
125	C₂⁴	C₂⁴×C₄₁	1	8	yes	—
126	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
127	C₂×D₄	—	2	192	no	—
128	C₂×D₄	—	2	96	no	—
129	C₂×D₄	—	2	48	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	—	2	24	no	—
132	C₂²×C₄	—	2	96	no	—
133	C₂²×C₄	—	4	64	no	—
134	C₂²×C₄	C₂×D₄×C₄₁	1	24	yes	—
135	C₂²×C₄	C₂×D₄×C₄₁	1	12	yes	—
136	C₂²×C₄	C₂⁴×C₄₁	1	8	yes	—
137	C₂²×C₄	C₂⁴×C₄₁	1	12	yes	—
138	C₂²×C₄	C₂⁴×C₄₁	1	2	yes	—
139	C₄⋊C₄	—	2	48	no	—
140	C₂²⋊C₄	—	2	96	no	—
141	C₄²	C₂×D₄×C₄₁	1	48	yes	complemented
142	C₂³	—	2	96	no	—
143	C₂³	—	2	24	no	—
144	C₂³	—	2	24	no	—
145	C₂³	—	4	48	no	—
146	C₂³	—	4	16	no	—
147	C₂³	C₂²×D₄×C₄₁	1	6	yes	—
148	C₂³	C₂×#41×C₄₁	1	16	yes	—
149	C₂³	C₂×#41×C₄₁	1	8	yes	—
150	C₂³	C₂⁵×C₄₁	1	1	yes	characteristic
151	D₄	—	2	192	no	—
152	C₂×C₄	—	2	48	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	—	4	64	no	—
157	C₂×C₄	#127×C₄₁	1	16	yes	—
158	C₂×C₄	C₂²×D₄×C₄₁	1	24	yes	—
159	C₂×C₄	C₂²×D₄×C₄₁	1	12	yes	—
160	C₂²	—	2	48	no	—
161	C₂²	—	4	64	no	—
162	C₂²	—	4	24	no	—
163	C₂²	—	4	8	no	—
164	C₂²	C₄₁×#525	1	24	yes	—
165	C₂²	C₂×#127×C₄₁	1	4	yes	—
166	C₂²	C₂³×D₄×C₄₁	1	3	yes	—
167	C₂²	C₂²×#41×C₄₁	1	4	yes	—
168	C₄	—	2	24	no	—
169	C₄	—	4	8	no	—
170	C₄	C₂×#127×C₄₁	1	8	yes	—
171	C₂	—	4	16	no	—
172	C₂	C₄₁×#2397	1	8	yes	direct product factor
173	C₂	C₂×C₄₁×#525	1	6	yes	—
174	C₂	C₂²×#127×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₄₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₄₁
socle	C₂⁴×C₄₁	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×D₄×C₄₁	C₂	semidirect
2. C₄₁×#2397	C₂	direct (N × H)
3. C₂×C₄₁×#399	C₂	semidirect
4. C₂×C₄₁×#392	C₂	semidirect
5. C₂×C₄₁×#376	C₂	semidirect
9. C₂×C₄×D₄×C₄₁	C₂²	semidirect
13. C₄₁×#399	C₂²	semidirect
14. C₄₁×#392	C₂²	semidirect
15. C₄₁×#376	C₂²	semidirect
31. C₂×C₄²×C₄₁	D₄	semidirect
53. C₄²×C₄₁	C₂×D₄	semidirect
72. C₂×#2397	C₄₁	direct (N × H)
84. C₂²×C₄×D₄	C₂×C₄₁	semidirect
85. #2397	C₂×C₄₁	direct (N × H)
86. C₂×#399	C₂×C₄₁	semidirect
87. C₂×#392	C₂×C₄₁	semidirect
88. C₂×#376	C₂×C₄₁	semidirect
90. C₂×C₄₁	#2397	direct (N × H)
96. C₂×C₄×D₄	C₂²×C₄₁	semidirect
100. #399	C₂²×C₄₁	semidirect
101. #392	C₂²×C₄₁	semidirect
102. #376	C₂²×C₄₁	semidirect
104. C₄₁	C₂×#2397	direct (N × H)
119. C₂×C₄²	D₄×C₄₁	semidirect
141. C₄²	C₂×D₄×C₄₁	semidirect
172. C₂	C₄₁×#2397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₄₁	2
C₄₁×#2397	2
C₂×C₄₁×#399	2
C₂×C₄₁×#392	2
C₂×C₄₁×#376	2
C₂×#2397	41

Subgroups — C2×C41×#2397 #23810745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2397 #23810745