Subgroups — #20248306

Order 6016.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094000	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	#15689045	C₂	1	4	yes	maximal, complemented
4	#15689025	C₂	1	2	yes	maximal
5	#15689021	C₂	1	2	yes	maximal, complemented
6	#15688993	C₂	1	1	yes	maximal, characteristic, complemented
7	#15688980	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
11	#13094000	—	2	8	no	—
12	C₄×D₁₈₈	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
19	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
21	C₄₇⋊#95	C₂²	1	4	yes	—
22	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₄₇	—	2	4	no	—
24	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₂³×D₄₇	C₂³	1	2	yes	—
29	C₂×Q₈×C₄₇	—	2	2	no	—
30	C₂×Q₈×C₄₇	—	2	2	no	—
31	C₂×Q₈×C₄₇	D₄	1	4	yes	complemented
32	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₄₇	C₂³	1	2	yes	—
38	C₂²×Dic₄₇	C₂³	1	4	yes	—
39	C₂×D₁₈₈	—	2	2	no	—
40	C₂×D₁₈₈	—	2	2	no	—
41	C₂×D₁₈₈	Q₈	1	4	yes	complemented
42	C₂×C₄×D₄₇	—	2	4	no	—
43	C₂×C₄×D₄₇	—	2	4	no	—
44	C₂×C₄×D₄₇	—	2	4	no	—
45	C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₄²×C₄₇	—	2	4	no	—
48	C₄²×C₄₇	D₄	1	4	yes	—
49	(C₂×D₄₇)⋊C₄	—	2	4	no	—
50	(C₂×D₄₇)⋊C₄	—	2	8	no	—
51	(C₂×D₄₇)⋊C₄	—	4	4	no	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	C₄⋊Dic₄₇	—	2	4	no	—
54	C₄⋊Dic₄₇	—	2	4	no	—
55	C₄⋊Dic₄₇	—	4	2	no	—
56	Dic₄₇⋊C₄	—	2	8	no	—
57	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
58	C₂²×D₄₇	—	2	2	no	—
59	C₂²×D₄₇	—	2	2	no	—
60	C₂²×D₄₇	—	2	2	no	—
61	C₂²×D₄₇	—	2	2	no	—
62	C₂²×D₄₇	—	2	2	no	—
63	C₂²×D₄₇	—	2	2	no	—
64	C₂²×D₄₇	C₂×Q₈	1	4	yes	—
65	Q₈×C₄₇	—	2	8	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₄₇	#41	1	2	yes	—
80	C₂×C₄×C₄₇	#41	1	2	yes	—
81	C₂×Dic₄₇	—	2	2	no	—
82	C₂×Dic₄₇	—	2	4	no	—
83	C₂×Dic₄₇	—	2	2	no	—
84	C₂×Dic₄₇	—	2	4	no	—
85	C₂×Dic₄₇	—	2	2	no	—
86	C₂×Dic₄₇	—	2	4	no	—
87	D₁₈₈	—	2	8	no	—
88	C₄×D₄₇	—	4	8	no	—
89	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
93	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
94	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂×D₄₇	—	2	4	no	—
97	C₂×D₄₇	—	2	4	no	—
98	C₂×D₄₇	—	2	4	no	—
99	C₂×D₄₇	—	4	2	no	—
100	C₂×D₄₇	—	4	2	no	—
101	C₂×D₄₇	—	4	2	no	—
102	C₂×D₄₇	—	4	2	no	—
103	C₄×C₄₇	—	2	4	no	—
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	4	2	no	—
106	C₄×C₄₇	#121	1	4	yes	—
107	C₄×C₄₇	C₂²⋊Q₈	1	4	yes	complemented
108	Dic₄₇	—	4	2	no	—
109	Dic₄₇	—	4	4	no	—
110	#2538	—	47	1	no	maximal, Sylow
111	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
112	C₂×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄₇	#537	1	1	yes	characteristic
114	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
115	C₂×C₄₇	#550	1	1	yes	characteristic
116	C₂×C₄₇	#554	1	1	yes	characteristic
117	C₂×C₄₇	#556	1	1	yes	characteristic
118	D₄₇	—	4	4	no	—
119	C₂×C₂²⋊Q₈	—	47	2	no	—
120	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
121	C₂×C₄×D₄	—	47	1	no	—
122	#396	—	47	4	no	—
123	#390	—	47	2	no	—
124	#389	—	47	2	no	—
125	#384	—	47	1	no	—
126	#380	—	47	2	no	—
127	C₄₇	#2538	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
129	C₂²×D₄	—	47	1	no	—
130	C₂³×C₄	—	47	2	no	—
131	C₂²⋊Q₈	—	94	8	no	—
132	C₄×Q₈	—	2	4	no	—
133	C₄×D₄	—	94	4	no	—
134	C₂×C₄⋊C₄	—	47	1	no	—
135	C₂×C₄⋊C₄	—	47	4	no	—
136	C₂×C₄⋊C₄	—	47	2	no	—
137	C₂×C₄⋊C₄	—	47	2	no	—
138	C₂×C₄⋊C₄	—	47	2	no	—
139	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
141	C₂×C₂²⋊C₄	—	47	2	no	—
142	C₂×C₂²⋊C₄	—	47	4	no	—
143	C₂×C₂²⋊C₄	—	47	4	no	—
144	C₂×C₄²	C₂×D₄₇	1	2	yes	—
145	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
146	#95	—	47	4	no	—
147	C₂⁴	—	47	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	D₁₈₈	1	4	yes	complemented
151	C₂×D₄	—	47	4	no	—
152	C₂×D₄	—	94	2	no	—
153	C₂×D₄	—	94	2	no	—
154	C₂²×C₄	—	47	2	no	—
155	C₂²×C₄	—	47	4	no	—
156	C₂²×C₄	—	94	4	no	—
157	C₂²×C₄	—	94	4	no	—
158	C₂²×C₄	—	94	4	no	—
159	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
161	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	94	8	no	—
167	C₄⋊C₄	—	94	4	no	—
168	C₄⋊C₄	—	94	4	no	—
169	C₄⋊C₄	—	94	2	no	—
170	C₄⋊C₄	—	188	2	no	—
171	C₂²⋊C₄	—	94	8	no	—
172	C₂²⋊C₄	—	94	4	no	—
173	C₂²⋊C₄	—	188	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₂²⋊D₄₇	1	4	yes	—
176	C₂³	—	47	4	no	—
177	C₂³	—	94	2	no	—
178	C₂³	—	94	2	no	—
179	C₂³	—	94	2	no	—
180	C₂³	—	94	2	no	—
181	C₂³	—	94	2	no	—
182	C₂³	—	94	2	no	—
183	C₂³	C₂³×D₄₇	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	D₄	—	94	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	94	2	no	—
194	C₂×C₄	—	94	4	no	—
195	C₂×C₄	—	94	2	no	—
196	C₂×C₄	—	94	2	no	—
197	C₂×C₄	—	94	4	no	—
198	C₂×C₄	—	94	4	no	—
199	C₂×C₄	—	188	8	no	—
200	C₂×C₄	C₂×D₁₈₈	1	4	yes	—
201	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
202	C₂×C₄	#10625350	1	2	yes	—
203	C₂×C₄	Q₈×D₄₇	1	2	yes	—
204	C₂×C₄	#10625354	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
208	C₂²	—	94	4	no	—
209	C₂²	—	94	4	no	—
210	C₂²	—	94	4	no	—
211	C₂²	—	188	2	no	—
212	C₂²	—	188	2	no	—
213	C₂²	—	188	2	no	—
214	C₂²	—	188	2	no	—
215	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
216	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
217	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
218	C₂²	C₂×#10625350	1	1	yes	characteristic
219	C₂²	C₂×#10625354	1	1	yes	characteristic
220	C₂²	#13094070	1	1	yes	characteristic
221	C₂²	#13094071	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	188	4	no	—
226	C₄	—	188	2	no	—
227	C₄	#13093998	1	4	yes	—
228	C₄	#13094000	1	4	yes	complemented
229	C₂	—	188	4	no	—
230	C₂	C₂×#13093998	1	1	yes	characteristic
231	C₂	C₂×#13094000	1	1	yes	characteristic
232	C₂	#15689701	1	1	yes	characteristic
233	C₂	Q₈×D₁₈₈	1	1	yes	characteristic
234	C₂	#15689795	1	1	yes	characteristic
235	C₂	#15689833	1	1	yes	characteristic
236	C₂	#15689846	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×C₄×Q₈×C₄₇	C₂
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#2538
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13094000	C₂	semidirect
3. #15689045	C₂	semidirect
5. #15689021	C₂	semidirect
6. #15688993	C₂	semidirect
7. #15688980	C₂	semidirect
8. C₂×C₄×Q₈×C₄₇	C₂	semidirect
31. C₂×Q₈×C₄₇	D₄	semidirect
41. C₂×D₁₈₈	Q₈	semidirect
107. C₄×C₄₇	C₂²⋊Q₈	semidirect
120. C₂×C₄×Q₈	D₄₇	semidirect
127. C₄₇	#2538	semidirect
150. C₂×Q₈	D₁₈₈	semidirect
228. C₄	#13094000	semidirect