Subgroups — #20251817

Order 6016.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094032	C₂	1	2	yes	maximal
2	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
3	#15689081	C₂	1	2	yes	maximal, complemented
4	#15689077	C₂	1	4	yes	maximal, complemented
5	#15689075	C₂	1	2	yes	maximal, complemented
6	C₂×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×D₄₇	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625350	C₂²	1	1	yes	characteristic
10	#13094032	C₄	1	16	yes	complemented
11	C₂×#10625324	C₂²	1	4	yes	—
12	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₈×D₄₇	C₂²	1	4	yes	—
14	C₄×Dic₉₄	C₄	1	2	yes	—
15	C₄×D₁₈₈	C₄	1	2	yes	—
16	C₄×C₂²⋊D₄₇	C₄	1	4	yes	—
17	#13093983	C₂²	1	1	yes	characteristic
18	#13093981	C₂²	1	2	yes	—
19	C₄²×D₄₇	C₂²	1	2	yes	—
20	#13093935	C₂²	1	1	yes	characteristic
21	#13093935	C₂²	1	1	yes	characteristic
22	#13093935	C₂²	1	2	yes	—
23	#13093934	C₂²	1	4	yes	—
24	C₈×Dic₄₇	C₂²	1	4	yes	—
25	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	#97×C₄₇	C₂²	1	2	yes	—
28	C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
32	#10625350	C₂×C₄	1	8	yes	—
33	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₄₇	C₂³	1	2	yes	—
36	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
37	M₁₆×C₄₇	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₄₇	C₂³	1	4	yes	—
39	C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
46	C₄×Dic₄₇	C₂×C₄	1	4	yes	—
47	C₄×Dic₄₇	C₂³	1	2	yes	—
48	C₂×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
49	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
50	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
51	#10625324	C₂×C₄	1	16	yes	complemented
52	C₈×D₄₇	C₂×C₄	1	16	yes	complemented
53	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂²⋊D₄₇	C₄²	1	8	yes	—
62	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
63	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
64	D₁₈₈	C₄²	1	4	yes	—
65	C₄×D₄₇	—	4	4	no	—
66	C₄×D₄₇	C₂²×C₄	1	8	yes	—
67	Dic₉₄	C₄²	1	4	yes	—
68	C₈×C₄₇	C₂²×C₄	1	8	yes	—
69	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
70	C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	C₂×C₄²	1	2	yes	—
73	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₄₇	—	2	4	no	—
75	C₂×D₄₇	C₂×C₄²	1	4	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	C₂×C₄²	1	2	yes	—
79	C₄×C₄₇	C₂³×C₄	1	2	yes	—
80	Dic₄₇	—	2	4	no	—
81	Dic₄₇	C₂×C₄²	1	4	yes	—
82	#2875	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	—	2	2	no	—
84	C₂×C₄₇	C₂²×C₄²	1	1	yes	characteristic
85	C₂×C₄₇	#568	1	2	yes	—
86	D₄₇	—	2	4	no	—
87	C₂×#131	—	47	2	no	—
88	C₄×#41	—	47	1	no	—
89	#405	—	47	2	no	—
90	#405	—	47	4	no	—
91	C₄×M₁₆	—	47	2	no	—
92	C₄×M₁₆	D₄₇	1	1	yes	characteristic, complemented
93	C₂×#97	—	47	1	no	—
94	C₂×#97	—	47	2	no	—
95	C₄₇	#2875	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	47	1	no	—
97	#131	—	47	16	no	—
98	C₂×M₁₆	—	47	4	no	—
99	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
100	C₂²×C₈	—	47	2	no	—
101	C₂²×C₈	—	47	4	no	—
102	C₄×Q₈	—	47	2	no	—
103	C₄×D₄	—	47	4	no	—
104	C₄×D₄	—	47	2	no	—
105	#117	—	47	1	no	—
106	#117	—	47	2	no	—
107	C₂×C₄²	—	47	2	no	—
108	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
109	#97	—	47	4	no	—
110	#97	—	47	1	no	—
111	#97	—	47	1	no	—
112	#97	—	47	2	no	—
113	#97	C₂×D₄₇	1	2	yes	—
114	C₄×C₈	—	47	4	no	—
115	C₄×C₈	C₂×D₄₇	1	2	yes	—
116	#41	—	47	8	no	—
117	C₂×Q₈	—	47	1	no	—
118	C₂×D₄	—	47	2	no	—
119	C₂×D₄	—	47	1	no	—
120	C₂²×C₄	—	47	2	no	—
121	C₂²×C₄	—	47	4	no	—
122	C₂²×C₄	C₄×D₄₇	1	2	yes	—
123	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	M₁₆	—	47	16	no	—
125	M₁₆	C₄×D₄₇	1	8	yes	complemented
126	C₂×C₈	—	47	8	no	—
127	C₂×C₈	—	47	2	no	—
128	C₂×C₈	—	47	2	no	—
129	C₂×C₈	—	47	16	no	—
130	C₂×C₈	C₂²×D₄₇	1	4	yes	—
131	C₄⋊C₄	—	47	4	no	—
132	C₄⋊C₄	—	47	2	no	—
133	C₂²⋊C₄	—	47	2	no	—
134	C₂²⋊C₄	—	47	4	no	—
135	C₄²	—	47	4	no	—
136	C₄²	—	47	2	no	—
137	C₄²	C₄×D₄₇	1	2	yes	—
138	C₄²	C₂²×D₄₇	1	1	yes	characteristic
139	C₄²	C₂²×D₄₇	1	1	yes	characteristic
140	C₂³	—	47	2	no	—
141	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
142	Q₈	—	47	4	no	—
143	D₄	—	47	8	no	—
144	D₄	—	47	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	47	2	no	—
147	C₂×C₄	—	47	8	no	—
148	C₂×C₄	—	47	4	no	—
149	C₂×C₄	—	188	4	no	—
150	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
154	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
155	C₈	—	47	4	no	—
156	C₈	—	47	4	no	—
157	C₈	C₂×C₄×D₄₇	1	8	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	47	4	no	—
160	C₂²	—	94	4	no	—
161	C₂²	C₄²×D₄₇	1	2	yes	—
162	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	47	4	no	—
166	C₄	—	94	4	no	—
167	C₄	C₄²×D₄₇	1	2	yes	—
168	C₄	C₂²×C₄×D₄₇	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	94	4	no	—
171	C₂	C₂×C₄²×D₄₇	1	1	yes	characteristic
172	C₂	#15689889	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂²×C₄²
Fitting	C₄×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×C₄
hypercenter	C₄×M₁₆	D₄₇
hypocenter	C₄₇	#2875
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #15689081	C₂	semidirect
4. #15689077	C₂	semidirect
5. #15689075	C₂	semidirect
6. C₂×#13093935	C₂	semidirect
7. #97×D₄₇	C₂	semidirect
8. C₄×M₁₆×C₄₇	C₂	semidirect
10. #13094032	C₄	semidirect
37. M₁₆×C₄₇	C₂×C₄	semidirect
48. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
51. #10625324	C₂×C₄	semidirect
52. C₈×D₄₇	C₂×C₄	semidirect
92. C₄×M₁₆	D₄₇	semidirect
95. C₄₇	#2875	semidirect
125. M₁₆	C₄×D₄₇	semidirect