Subgroups — #20257732

Order 6016.

1004 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689872	C₂	1	4	yes	maximal, complemented
2	#15689853	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689839	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689834	C₂	1	2	yes	maximal, complemented
5	#15689808	C₂	1	4	yes	maximal, complemented
6	#15689774	C₂	1	4	yes	maximal, complemented
7	#15689750	C₂	1	4	yes	maximal, complemented
8	#15689747	C₂	1	4	yes	maximal, complemented
9	#15689697	C₂	1	4	yes	maximal, complemented
10	#15689657	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	4	yes	—
13	#13094023	C₂²	1	4	yes	—
14	#13094022	C₂²	1	4	yes	—
15	#13094022	C₂²	1	4	yes	—
16	#13094019	C₂²	1	8	yes	—
17	#13094018	C₂²	1	4	yes	—
18	#13094016	C₂²	1	2	yes	—
19	#13094015	C₂²	1	4	yes	—
20	#13094014	C₂²	1	8	yes	—
21	#13094013	C₂²	1	4	yes	—
22	#13094013	C₂²	1	4	yes	—
23	#13094012	C₂²	1	2	yes	—
24	#13094009	C₂²	1	8	yes	—
25	#13094007	C₂²	1	8	yes	—
26	#13094007	C₂²	1	4	yes	—
27	#13094004	C₂²	1	4	yes	—
28	#13094003	C₂²	1	2	yes	—
29	#13094002	C₂²	1	8	yes	—
30	#13094000	C₂²	1	8	yes	—
31	#13094000	C₂²	1	8	yes	—
32	#13093999	C₂²	1	4	yes	—
33	#13093998	C₂²	1	2	yes	—
34	C₄×Dic₉₄	C₂²	1	2	yes	—
35	#13093989	C₂²	1	4	yes	—
36	C₄×D₁₈₈	C₂²	1	2	yes	—
37	#13093986	C₂²	1	4	yes	—
38	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	—
39	#13093983	C₂²	1	4	yes	—
40	#13093982	C₂²	1	4	yes	—
41	#13093981	C₂²	1	4	yes	—
42	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
43	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
44	C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
45	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
46	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
47	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
50	C₂×Q₈×C₄₇	—	4	4	no	—
51	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
52	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
53	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
58	#10625350	—	4	8	no	—
59	C₂×D₁₈₈	C₂³	1	2	yes	—
60	C₂×D₁₈₈	C₂³	1	4	yes	—
61	C₂×C₄×D₄₇	C₂³	1	4	yes	—
62	C₂×C₄×D₄₇	C₂³	1	8	yes	—
63	C₂×Dic₉₄	C₂³	1	2	yes	—
64	C₂×Dic₉₄	C₂³	1	4	yes	—
65	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
66	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
67	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
68	C₄²×C₄₇	C₂³	1	2	yes	—
69	C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
72	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
73	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
74	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
78	C₄⋊Dic₄₇	C₂³	1	2	yes	—
79	C₄⋊Dic₄₇	C₂³	1	2	yes	—
80	C₄⋊Dic₄₇	C₂³	1	8	yes	—
81	Dic₄₇⋊C₄	C₂³	1	8	yes	—
82	Dic₄₇⋊C₄	C₂³	1	4	yes	—
83	Dic₄₇⋊C₄	C₂³	1	4	yes	—
84	Dic₄₇⋊C₄	C₂³	1	4	yes	—
85	Dic₄₇⋊C₄	C₂³	1	8	yes	—
86	Dic₄₇⋊C₄	C₂³	1	4	yes	—
87	C₄×Dic₄₇	C₂³	1	4	yes	—
88	C₄×Dic₄₇	C₂³	1	4	yes	—
89	C₄×Dic₄₇	C₂³	1	4	yes	—
90	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂²×D₄₇	C₂⁴	1	4	yes	—
92	Q₈×C₄₇	—	4	2	no	—
93	Q₈×C₄₇	—	4	4	no	—
94	Q₈×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
99	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
100	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
103	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
104	C₂²⋊D₄₇	—	2	8	no	—
105	C₂×Dic₄₇	C₂⁴	1	4	yes	—
106	C₂×Dic₄₇	C₂⁴	1	4	yes	—
107	C₂×Dic₄₇	C₂⁴	1	4	yes	—
108	D₁₈₈	—	2	4	no	—
109	D₁₈₈	—	4	4	no	—
110	C₄×D₄₇	—	2	8	no	—
111	C₄×D₄₇	—	4	8	no	—
112	Dic₉₄	—	2	4	no	—
113	Dic₉₄	—	4	4	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
117	C₂×D₄₇	—	2	4	no	—
118	C₂×D₄₇	—	2	4	no	—
119	C₂×D₄₇	—	2	4	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
126	Dic₄₇	—	2	4	no	—
127	Dic₄₇	—	2	4	no	—
128	Dic₄₇	—	2	4	no	—
129	#3544	—	47	1	no	maximal, Sylow
130	C₂×C₄₇	—	2	2	no	—
131	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
133	D₄₇	—	4	4	no	—
134	#563	—	47	4	no	—
135	Q₈²	—	47	1	no	—
136	#555	—	47	1	no	—
137	#554	—	47	2	no	—
138	#552	—	47	4	no	—
139	#547	—	47	4	no	—
140	#544	—	47	4	no	—
141	#542	—	47	4	no	—
142	#536	—	47	4	no	—
143	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
144	#529	—	47	2	no	—
145	C₄₇	#3544	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	47	4	no	—
147	C₂²×Q₈	C₂×D₄₇	1	2	yes	—
148	C₄⋊Q₈	—	47	4	no	—
149	C₄⋊Q₈	—	47	4	no	—
150	C₄⋊Q₈	—	47	4	no	—
151	C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
152	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
153	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
154	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
155	#126	—	47	8	no	—
156	#126	—	47	4	no	—
157	#125	—	47	4	no	—
158	#125	—	47	8	no	—
159	#125	—	47	2	no	—
160	#124	—	47	4	no	—
161	#124	—	47	4	no	—
162	#124	—	47	2	no	—
163	#123	—	47	4	no	—
164	#123	—	47	8	no	—
165	#123	—	47	8	no	—
166	C₂²⋊Q₈	—	47	8	no	—
167	C₂²⋊Q₈	—	47	8	no	—
168	C₂²⋊Q₈	—	47	2	no	—
169	C₂²⋊Q₈	—	47	4	no	—
170	C₂²⋊Q₈	—	47	8	no	—
171	#121	—	47	2	no	—
172	#121	—	47	4	no	—
173	C₄×Q₈	—	47	4	no	—
174	C₄×Q₈	—	47	2	no	—
175	C₄×D₄	—	47	4	no	—
176	C₄×D₄	—	47	4	no	—
177	C₄×D₄	—	47	2	no	—
178	#117	—	47	4	no	—
179	#117	—	47	4	no	—
180	#117	—	47	4	no	—
181	C₂×C₄⋊C₄	—	47	4	no	—
182	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
183	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
184	#41	—	188	8	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	—	47	2	no	—
187	C₂×Q₈	—	47	4	no	—
188	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
189	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
190	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
191	C₂×D₄	—	47	4	no	—
192	C₂×D₄	—	47	2	no	—
193	C₂×D₄	—	47	4	no	—
194	C₂²×C₄	—	47	4	no	—
195	C₂²×C₄	—	47	8	no	—
196	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
197	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
199	C₄⋊C₄	—	47	8	no	—
200	C₄⋊C₄	—	47	4	no	—
201	C₄⋊C₄	—	47	4	no	—
202	C₄⋊C₄	—	47	4	no	—
203	C₄⋊C₄	—	47	4	no	—
204	C₄⋊C₄	—	47	8	no	—
205	C₄⋊C₄	—	47	2	no	—
206	C₄⋊C₄	—	47	2	no	—
207	C₄⋊C₄	—	47	8	no	—
208	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
209	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
210	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
211	C₂²⋊C₄	—	47	4	no	—
212	C₂²⋊C₄	—	47	4	no	—
213	C₂²⋊C₄	—	47	8	no	—
214	C₂²⋊C₄	—	47	4	no	—
215	C₂²⋊C₄	—	47	8	no	—
216	C₂²⋊C₄	—	47	4	no	—
217	C₂²⋊C₄	—	47	4	no	—
218	C₄²	—	47	4	no	—
219	C₄²	—	47	4	no	—
220	C₄²	—	47	4	no	—
221	C₄²	C₂²×D₄₇	1	2	yes	—
222	C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	C₂³	—	47	4	no	—
225	C₂³	C₂³×D₄₇	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	94	4	no	—
230	Q₈	—	188	4	no	—
231	D₄	—	94	8	no	—
232	D₄	—	94	4	no	—
233	D₄	—	188	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	47	4	no	—
238	C₂×C₄	—	47	4	no	—
239	C₂×C₄	—	47	4	no	—
240	C₂×C₄	—	94	8	no	—
241	C₂×C₄	—	188	8	no	—
242	C₂×C₄	C₂³×D₄₇	1	4	yes	—
243	C₂×C₄	C₂³×D₄₇	1	2	yes	—
244	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₄₇	1	4	yes	—
247	C₂×C₄	C₂³×D₄₇	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	94	4	no	—
250	C₂²	—	94	4	no	—
251	C₂²	—	94	4	no	—
252	C₂²	#13094069	1	2	yes	—
253	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	2	no	—
259	C₄	—	94	4	no	—
260	C₄	—	94	4	no	—
261	C₄	—	94	4	no	—
262	C₄	#13094072	1	4	yes	—
263	C₂	—	2	2	no	—
264	C₂	—	188	4	no	—
265	C₂	C₂×#13094069	1	1	yes	characteristic
266	C₂	C₂×#13094072	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#3544
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #15689872	C₂	semidirect
2. #15689853	C₂	semidirect
3. #15689839	C₂	semidirect
4. #15689834	C₂	semidirect
5. #15689808	C₂	semidirect
6. #15689774	C₂	semidirect
7. #15689750	C₂	semidirect
8. #15689747	C₂	semidirect
9. #15689697	C₂	semidirect
10. #15689657	C₂	semidirect
11. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
143. C₂×C₄⋊Q₈	D₄₇	semidirect
145. C₄₇	#3544	semidirect