Subgroups — #20251957

Order 6016.

484 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#15689151	C₂	1	2	yes	maximal, complemented
4	#15689148	C₂	1	4	yes	maximal, complemented
5	#15689145	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₄₇	C₂	1	2	yes	maximal, complemented
8	C₄₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625350	C₂²	1	1	yes	characteristic
10	#13094032	—	2	8	no	—
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	1	yes	characteristic
15	C₄×Dic₉₄	C₄	1	1	yes	characteristic
16	C₄×D₁₈₈	C₄	1	1	yes	characteristic
17	C₄×D₁₈₈	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₄₇	C₄	1	2	yes	—
19	C₄×C₂²⋊D₄₇	C₄	1	2	yes	—
20	#13093983	C₂²	1	1	yes	characteristic
21	#13093981	C₂²	1	2	yes	—
22	C₄²×D₄₇	C₂²	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
26	Dic₄₇⋊C₈	C₂²	1	4	yes	—
27	Dic₄₇⋊C₈	C₂²	1	4	yes	—
28	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
36	#10625350	D₄	1	4	yes	—
37	#10625350	Q₈	1	4	yes	—
38	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₄₇	C₂³	1	2	yes	—
42	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
43	M₁₆×C₄₇	—	2	4	no	—
44	C₂×C₈×C₄₇	C₂³	1	4	yes	—
45	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
50	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
51	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
54	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
57	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
58	C₄×Dic₄₇	C₂³	1	2	yes	—
59	C₂×C₄₇⋊C₈	—	2	4	no	—
60	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
61	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
62	#10625324	—	2	8	no	—
63	C₈×D₄₇	—	2	8	no	—
64	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	4	1	no	—
68	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₄₇	C₄⋊C₄	1	8	yes	—
77	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
79	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
80	D₁₈₈	C₄⋊C₄	1	4	yes	—
81	C₄×D₄₇	—	2	4	no	—
82	C₄×D₄₇	—	4	2	no	—
83	C₄×D₄₇	C₂×D₄	1	4	yes	—
84	C₄×D₄₇	C₂×Q₈	1	4	yes	—
85	Dic₉₄	C₄⋊C₄	1	4	yes	—
86	C₈×C₄₇	—	2	4	no	—
87	C₄₇⋊C₈	—	2	2	no	—
88	C₄₇⋊C₈	—	2	2	no	—
89	C₂²×C₄₇	—	2	1	no	—
90	C₂²×C₄₇	—	2	1	no	—
91	C₂²×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₄₇	—	2	2	no	—
94	C₂×D₄₇	—	2	2	no	—
95	C₂×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	2	1	no	—
99	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₄×C₄₇	#131	1	2	yes	complemented
101	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	Dic₄₇	—	2	2	no	—
104	Dic₄₇	—	2	2	no	—
105	Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	#2906	—	47	1	no	maximal, Sylow
107	C₂×C₄₇	—	2	2	no	—
108	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
110	C₂×C₄₇	#568	1	1	yes	characteristic
111	D₄₇	—	2	4	no	—
112	C₂×#131	—	47	2	no	—
113	C₄×#41	—	47	1	no	—
114	#424	—	47	2	no	—
115	#424	—	47	4	no	—
116	#423	—	47	2	no	—
117	#423	D₄₇	1	1	yes	characteristic, complemented
118	C₂×C₄⋊C₈	—	47	1	no	—
119	C₂×C₄⋊C₈	—	47	2	no	—
120	C₄₇	#2906	1	1	yes	characteristic, complemented, Sylow
121	C₂×#41	—	47	1	no	—
122	#131	—	94	8	no	—
123	C₂×M₁₆	—	47	4	no	—
124	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
125	C₂²×C₈	—	47	2	no	—
126	C₂²×C₈	—	47	4	no	—
127	C₄×Q₈	—	47	1	no	—
128	C₄×Q₈	—	47	1	no	—
129	C₄×D₄	—	47	2	no	—
130	C₄×D₄	—	47	2	no	—
131	C₄×D₄	—	47	1	no	—
132	C₄×D₄	—	47	1	no	—
133	#117	—	47	1	no	—
134	#117	—	47	2	no	—
135	C₂×C₄²	—	47	2	no	—
136	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
137	C₄⋊C₈	—	47	4	no	—
138	C₄⋊C₈	—	47	4	no	—
139	C₄⋊C₈	—	47	1	no	—
140	C₄⋊C₈	—	47	1	no	—
141	C₄⋊C₈	—	47	2	no	—
142	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
143	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
144	#41	—	47	4	no	—
145	#41	—	47	4	no	—
146	C₂×Q₈	—	47	1	no	—
147	C₂×D₄	—	47	2	no	—
148	C₂×D₄	—	47	1	no	—
149	C₂²×C₄	—	47	2	no	—
150	C₂²×C₄	—	47	2	no	—
151	C₂²×C₄	—	47	2	no	—
152	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
155	M₁₆	—	2	4	no	—
156	M₁₆	—	94	8	no	—
157	C₂×C₈	—	47	2	no	—
158	C₂×C₈	—	47	2	no	—
159	C₂×C₈	—	94	8	no	—
160	C₂×C₈	—	94	4	no	—
161	C₂×C₈	C₂²×D₄₇	1	4	yes	—
162	C₄⋊C₄	—	47	2	no	—
163	C₄⋊C₄	—	47	2	no	—
164	C₄⋊C₄	—	47	1	no	—
165	C₄⋊C₄	—	47	1	no	—
166	C₂²⋊C₄	—	47	2	no	—
167	C₂²⋊C₄	—	47	2	no	—
168	C₂²⋊C₄	—	47	2	no	—
169	C₄²	—	47	2	no	—
170	C₄²	—	47	2	no	—
171	C₄²	—	47	2	no	—
172	C₄²	C₄×D₄₇	1	1	yes	characteristic
173	C₄²	C₄×D₄₇	1	1	yes	characteristic
174	C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₂³	—	47	2	no	—
177	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
178	Q₈	—	47	4	no	—
179	D₄	—	47	8	no	—
180	D₄	—	47	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	47	2	no	—
184	C₂×C₄	—	47	4	no	—
185	C₂×C₄	—	47	4	no	—
186	C₂×C₄	—	47	2	no	—
187	C₂×C₄	—	47	2	no	—
188	C₂×C₄	—	94	4	no	—
189	C₂×C₄	—	188	2	no	—
190	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
195	C₂×C₄	D₄×D₄₇	1	2	yes	—
196	C₂×C₄	Q₈×D₄₇	1	2	yes	—
197	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	—	94	2	no	—
200	C₈	—	94	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	47	4	no	—
204	C₂²	—	94	2	no	—
205	C₂²	—	94	2	no	—
206	C₂²	C₄⋊C₄×D₄₇	1	2	yes	—
207	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	47	4	no	—
212	C₄	—	94	2	no	—
213	C₄	—	94	2	no	—
214	C₄	C₄⋊C₄×D₄₇	1	2	yes	—
215	C₄	#13094032	1	2	yes	complemented
216	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	94	4	no	—
220	C₂	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
221	C₂	C₂×#13094032	1	1	yes	characteristic
222	C₂	#15689889	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#423	C₂
hypercenter	#423	D₄₇
hypocenter	C₄₇	#2906
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
3. #15689151	C₂	semidirect
4. #15689148	C₂	semidirect
5. #15689145	C₂	semidirect
6. C₂×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
7. C₄⋊C₈×D₄₇	C₂	semidirect
8. C₄₇×#423	C₂	semidirect
100. C₄×C₄₇	#131	semidirect
117. #423	D₄₇	semidirect
120. C₄₇	#2906	semidirect
215. C₄	#13094032	semidirect