Subgroups — #20248419

Order 6016.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094015	C₂	1	2	yes	maximal
2	#15689055	C₂	1	4	yes	maximal
3	#15689022	C₂	1	2	yes	maximal
4	#15689021	C₂	1	2	yes	maximal
5	#15688993	C₂	1	2	yes	maximal
6	#15688993	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#13094015	—	2	8	no	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
19	C₄₇⋊#95	C₂²	1	2	yes	—
20	C₄₇⋊#95	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	#95×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	C₂³	1	2	yes	—
32	C₂²×Dic₄₇	C₂³	1	2	yes	—
33	C₂²×Dic₄₇	C₂³	1	4	yes	—
34	C₄⋊C₄×C₄₇	D₄	1	4	yes	—
35	C₄⋊C₄×C₄₇	Q₈	1	4	yes	—
36	C₄²×C₄₇	—	2	4	no	—
37	C₄²×C₄₇	Q₈	1	4	yes	—
38	C₄⋊Dic₄₇	—	2	8	no	—
39	C₄⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	2	2	no	—
41	C₄⋊Dic₄₇	—	2	2	no	—
42	C₄⋊Dic₄₇	—	2	4	no	—
43	C₄⋊Dic₄₇	—	4	4	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	Dic₄₇⋊C₄	—	2	8	no	—
46	C₄×Dic₄₇	—	2	4	no	—
47	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
58	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	#41	1	2	yes	—
62	C₂×C₄×C₄₇	#41	1	2	yes	—
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	—	2	2	no	—
65	C₂×Dic₄₇	—	2	4	no	—
66	C₂×Dic₄₇	—	2	2	no	—
67	C₂×Dic₄₇	—	2	2	no	—
68	C₂×Dic₄₇	—	2	4	no	—
69	C₂×Dic₄₇	—	2	2	no	—
70	C₂×Dic₄₇	—	2	4	no	—
71	C₂×Dic₄₇	—	2	2	no	—
72	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
76	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
77	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	2	4	no	—
81	C₄×C₄₇	—	2	4	no	—
82	C₄×C₄₇	#125	1	4	yes	complemented
83	C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
84	Dic₄₇	—	4	2	no	—
85	Dic₄₇	—	4	2	no	—
86	Dic₄₇	—	4	4	no	—
87	#2552	—	47	1	no	maximal, Sylow
88	C₂×C₄₇	C₂×#125	1	1	yes	characteristic
89	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₄₇	#544	1	1	yes	characteristic
91	C₂×C₄₇	#547	1	1	yes	characteristic
92	C₂×C₄₇	#550	1	1	yes	characteristic
93	C₂×C₄₇	#554	1	1	yes	characteristic
94	C₂×C₄₇	#557	1	1	yes	characteristic
95	C₂×#125	—	47	2	no	—
96	#398	—	47	4	no	—
97	#389	—	47	2	no	—
98	#389	—	47	2	no	—
99	#384	—	47	2	no	—
100	#384	—	47	1	no	—
101	C₄×C₄⋊C₄	—	47	1	no	—
102	C₄×C₄⋊C₄	D₄₇	1	1	yes	characteristic
103	C₄₇	#2552	1	1	yes	characteristic, complemented, Sylow
104	#125	—	94	8	no	—
105	C₂×C₄⋊C₄	—	47	4	no	—
106	C₂×C₄⋊C₄	—	47	2	no	—
107	C₂×C₄⋊C₄	—	47	4	no	—
108	C₂×C₄⋊C₄	—	47	1	no	—
109	C₂×C₄⋊C₄	—	47	1	no	—
110	C₂×C₄⋊C₄	—	47	2	no	—
111	C₂×C₄⋊C₄	—	47	4	no	—
112	C₂×C₄⋊C₄	—	47	2	no	—
113	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
115	C₂×C₄²	—	47	2	no	—
116	C₂×C₄²	C₂×D₄₇	1	2	yes	—
117	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
118	#95	—	47	2	no	—
119	#95	—	47	4	no	—
120	#95	C₂×D₄₇	1	2	yes	—
121	C₂²×C₄	—	47	2	no	—
122	C₂²×C₄	—	47	2	no	—
123	C₂²×C₄	—	47	4	no	—
124	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
127	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
129	C₄⋊C₄	—	94	8	no	—
130	C₄⋊C₄	—	94	8	no	—
131	C₄⋊C₄	—	94	4	no	—
132	C₄⋊C₄	—	94	2	no	—
133	C₄⋊C₄	—	94	2	no	—
134	C₄⋊C₄	—	94	4	no	—
135	C₄⋊C₄	—	188	4	no	—
136	C₄⋊C₄	—	188	2	no	—
137	C₄⋊C₄	Dic₉₄	1	4	yes	—
138	C₄⋊C₄	D₁₈₈	1	4	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	94	4	no	—
141	C₄²	Dic₉₄	1	4	yes	—
142	C₂³	C₂³×D₄₇	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	94	2	no	—
150	C₂×C₄	—	94	2	no	—
151	C₂×C₄	—	94	4	no	—
152	C₂×C₄	—	94	2	no	—
153	C₂×C₄	—	94	2	no	—
154	C₂×C₄	—	94	4	no	—
155	C₂×C₄	—	94	2	no	—
156	C₂×C₄	—	94	4	no	—
157	C₂×C₄	—	94	2	no	—
158	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
159	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
160	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
161	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
162	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
163	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
164	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
165	C₂×C₄	#10625352	1	2	yes	—
166	C₂×C₄	#10625354	1	2	yes	—
167	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
168	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
169	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
170	C₂²	C₂×#10625352	1	1	yes	characteristic
171	C₂²	C₂×#10625354	1	1	yes	characteristic
172	C₂²	#13094070	1	1	yes	characteristic
173	C₂²	#13094071	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	2	4	no	—
177	C₄	—	188	2	no	—
178	C₄	—	188	2	no	—
179	C₄	—	188	4	no	—
180	C₄	#13094015	1	4	yes	complemented
181	C₄	#13094024	1	4	yes	—
182	C₂	C₂×#13094015	1	1	yes	characteristic
183	C₂	C₂×#13094024	1	1	yes	characteristic
184	C₂	#15689751	1	1	yes	characteristic
185	C₂	#15689777	1	1	yes	characteristic
186	C₂	#15689795	1	1	yes	characteristic
187	C₂	#15689830	1	1	yes	characteristic
188	C₂	#15689852	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₇	C₂
hypercenter	C₄×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2552
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
82. C₄×C₄₇	#125	semidirect
103. C₄₇	#2552	semidirect
180. C₄	#13094015	semidirect