Subgroups — #20252540

Order 6016.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689660	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689202	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689200	C₂	1	2	yes	maximal, complemented
5	#15689196	C₂	1	2	yes	maximal, complemented
6	#15689188	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689185	C₂	1	2	yes	maximal, complemented
8	#15689174	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689148	C₂	1	2	yes	maximal, complemented
10	#15689145	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625324	C₂²	1	2	yes	—
13	C₂×#10625324	C₂²	1	2	yes	—
14	C₂×C₈×D₄₇	C₂²	1	2	yes	—
15	#13094019	C₄	1	2	yes	—
16	#13094015	C₄	1	1	yes	characteristic
17	#13094013	C₄	1	1	yes	characteristic
18	#13094009	C₄	1	2	yes	—
19	#13094000	C₄	1	1	yes	characteristic
20	#13093999	C₄	1	1	yes	characteristic
21	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
22	C₄×D₁₈₈	C₂²	1	2	yes	—
23	#13093981	C₂²	1	2	yes	—
24	C₄²×D₄₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
27	Dic₄₇⋊C₈	C₂²	1	2	yes	—
28	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
29	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
34	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
36	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
39	#97×C₄₇	C₂²	1	2	yes	—
40	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₄₇	C₂³	1	2	yes	—
48	C₂×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄²×C₄₇	C₂³	1	2	yes	—
55	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
62	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
70	#10625324	—	4	2	no	—
71	#10625324	—	4	2	no	—
72	C₈×D₄₇	—	4	2	no	—
73	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×D₄₇	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₄₇	—	2	2	no	—
76	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
85	D₁₈₈	—	2	2	no	—
86	D₁₈₈	—	4	2	no	—
87	C₄×D₄₇	—	2	2	no	—
88	C₄×D₄₇	—	2	2	no	—
89	C₄×D₄₇	—	2	1	no	—
90	C₄×D₄₇	—	2	1	no	—
91	C₄×D₄₇	—	2	2	no	—
92	C₄×D₄₇	—	4	1	no	—
93	C₈×C₄₇	—	2	2	no	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	—	2	1	no	—
96	C₄₇⋊C₈	—	2	1	no	—
97	C₄₇⋊C₈	—	2	2	no	—
98	C₄₇⋊C₈	—	2	1	no	—
99	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₄₇	—	2	2	no	—
101	C₂×D₄₇	—	2	1	no	—
102	C₂×D₄₇	—	2	1	no	—
103	C₂×D₄₇	—	2	2	no	—
104	C₂×D₄₇	—	2	1	no	—
105	C₂×D₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	#131	1	2	yes	—
110	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
111	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
112	Dic₄₇	—	2	2	no	—
113	Dic₄₇	—	2	1	no	—
114	Dic₄₇	—	2	1	no	—
115	Dic₄₇	—	2	1	no	—
116	#2993	—	47	1	no	maximal, Sylow
117	C₂×C₄₇	#520	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
119	C₂×C₄₇	#568	1	1	yes	characteristic
120	D₄₇	—	4	1	no	—
121	D₄₇	—	4	2	no	—
122	#529	—	47	1	no	—
123	#446	D₄₇	1	1	yes	characteristic, complemented
124	C₈×Q₈	—	47	1	no	—
125	#436	—	47	2	no	—
126	#436	—	47	1	no	—
127	#435	—	47	2	no	—
128	C₈×D₄	—	47	1	no	—
129	#433	—	47	2	no	—
130	#431	—	47	1	no	—
131	#424	—	47	2	no	—
132	#423	—	47	1	no	—
133	C₄₇	#2993	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	47	2	no	—
135	C₂×M₁₆	—	47	2	no	—
136	C₂²×C₈	—	47	2	no	—
137	#126	—	47	2	no	—
138	#125	—	47	1	no	—
139	#124	—	47	1	no	—
140	#123	—	47	2	no	—
141	C₂²⋊Q₈	—	47	1	no	—
142	#121	—	47	1	no	—
143	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
144	C₄×D₄	—	47	1	no	—
145	C₄×D₄	—	47	2	no	—
146	#117	—	47	2	no	—
147	C₂×C₄²	—	47	1	no	—
148	C₄⋊C₈	—	47	2	no	—
149	C₄⋊C₈	—	47	1	no	—
150	C₄⋊C₈	—	47	1	no	—
151	C₄⋊C₈	—	47	1	no	—
152	C₄⋊C₈	—	47	2	no	—
153	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
155	C₂²⋊C₈	—	47	1	no	—
156	C₂²⋊C₈	—	47	1	no	—
157	C₂²⋊C₈	—	47	2	no	—
158	C₂²⋊C₈	—	47	2	no	—
159	#97	C₂×D₄₇	1	2	yes	—
160	C₄×C₈	—	47	1	no	—
161	C₄×C₈	—	47	2	no	—
162	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
163	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
164	C₂×D₄	—	47	1	no	—
165	C₂×D₄	—	47	2	no	—
166	C₂²×C₄	—	47	1	no	—
167	C₂²×C₄	—	47	1	no	—
168	C₂²×C₄	—	47	2	no	—
169	C₂²×C₄	—	47	1	no	—
170	M₁₆	—	188	2	no	—
171	M₁₆	—	188	2	no	—
172	C₂×C₈	—	47	1	no	—
173	C₂×C₈	—	47	1	no	—
174	C₂×C₈	—	47	2	no	—
175	C₂×C₈	—	188	2	no	—
176	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₄₇	1	2	yes	—
179	C₄⋊C₄	—	47	2	no	—
180	C₄⋊C₄	—	47	2	no	—
181	C₄⋊C₄	—	47	1	no	—
182	C₄⋊C₄	—	47	2	no	—
183	C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
184	C₄⋊C₄	C₄×D₄₇	1	2	yes	—
185	C₂²⋊C₄	—	47	2	no	—
186	C₂²⋊C₄	—	47	2	no	—
187	C₂²⋊C₄	—	47	2	no	—
188	C₂²⋊C₄	—	47	2	no	—
189	C₂²⋊C₄	—	47	2	no	—
190	C₄²	—	47	1	no	—
191	C₄²	—	47	1	no	—
192	C₄²	—	47	1	no	—
193	C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	C₄²	C₂²×D₄₇	1	2	yes	—
195	C₂³	—	47	2	no	—
196	C₂³	—	47	1	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	94	2	no	—
199	D₄	—	188	2	no	—
200	C₂×C₄	—	47	1	no	—
201	C₂×C₄	—	47	1	no	—
202	C₂×C₄	—	47	1	no	—
203	C₂×C₄	—	47	2	no	—
204	C₂×C₄	—	94	2	no	—
205	C₂×C₄	—	94	2	no	—
206	C₂×C₄	—	94	1	no	—
207	C₂×C₄	—	94	1	no	—
208	C₂×C₄	—	94	2	no	—
209	C₂×C₄	—	188	1	no	—
210	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
213	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	94	1	no	—
219	C₈	—	94	2	no	—
220	C₈	—	94	1	no	—
221	C₂²	—	94	2	no	—
222	C₂²	—	94	1	no	—
223	C₂²	—	94	1	no	—
224	C₂²	—	94	2	no	—
225	C₂²	—	94	1	no	—
226	C₂²	—	94	2	no	—
227	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	94	1	no	—
232	C₄	—	94	2	no	—
233	C₄	—	94	1	no	—
234	C₄	—	94	1	no	—
235	C₄	#13094032	1	2	yes	—
236	C₄	#13094070	1	1	yes	characteristic
237	C₄	#13094072	1	1	yes	characteristic
238	C₂	—	188	1	no	—
239	C₂	—	188	2	no	—
240	C₂	#15689605	1	1	yes	characteristic
241	C₂	C₂×#13094032	1	1	yes	characteristic
242	C₂	#15689888	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#446	C₂
hypercenter	#446	D₄₇
hypocenter	C₄₇	#2993
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄₇⋊C₈	C₂	semidirect
3. #15689202	C₂	semidirect
4. #15689200	C₂	semidirect
5. #15689196	C₂	semidirect
6. #15689188	C₂	semidirect
7. #15689185	C₂	semidirect
8. #15689174	C₂	semidirect
9. #15689148	C₂	semidirect
10. #15689145	C₂	semidirect
11. C₄₇×#446	C₂	semidirect
123. #446	D₄₇	semidirect
133. C₄₇	#2993	semidirect