Subgroups — #20257473

Order 6016.

1104 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689858	C₂	1	3	yes	maximal, complemented
2	Q₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689833	C₂	1	3	yes	maximal, complemented
4	#15689795	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689784	C₂	1	2	yes	maximal, complemented
6	#15689774	C₂	1	6	yes	maximal, complemented
7	#15689745	C₂	1	6	yes	maximal, complemented
8	#15689673	C₂	1	3	yes	maximal, complemented
9	#15689657	C₂	1	3	yes	maximal, complemented
10	#15689605	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625354	C₂²	1	2	yes	—
13	C₂×#10625350	C₂²	1	6	yes	—
14	#13094024	C₂²	1	3	yes	—
15	#13094023	C₂²	1	6	yes	—
16	#13094021	C₂²	1	3	yes	—
17	#13094018	C₂²	1	6	yes	—
18	#13094016	C₂²	1	3	yes	—
19	#13094013	C₂²	1	6	yes	—
20	#13094012	C₂²	1	6	yes	—
21	#13094012	C₂²	1	3	yes	—
22	#13094009	C₂²	1	12	yes	—
23	#13094007	C₂²	1	6	yes	—
24	#13094003	C₂²	1	1	yes	characteristic
25	#13094002	C₂²	1	12	yes	—
26	#13094000	C₂²	1	6	yes	—
27	#13093999	C₂²	1	6	yes	—
28	#13093998	C₂²	1	6	yes	—
29	#13093998	C₂²	1	3	yes	—
30	Q₈×Dic₄₇	C₂²	1	2	yes	—
31	C₄×Dic₉₄	C₂²	1	3	yes	—
32	C₄×D₁₈₈	C₂²	1	6	yes	—
33	C₄×D₁₈₈	C₂²	1	3	yes	—
34	C₄×D₁₈₈	C₂²	1	6	yes	—
35	#13093986	C₂²	1	6	yes	—
36	C₄×C₂²⋊D₄₇	C₂²	1	6	yes	—
37	#13093981	C₂²	1	6	yes	—
38	C₄⋊C₄×D₄₇	C₂²	1	6	yes	—
39	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₄₇	C₂²	1	3	yes	—
41	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₄₇	C₂²	1	3	yes	—
43	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
45	C₂×C₄²×C₄₇	C₂²	1	3	yes	—
46	C₂×Q₈×C₄₇	—	2	4	no	—
47	C₂×Q₈×C₄₇	C₂³	1	3	yes	—
48	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
51	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₄₇	C₂³	1	6	yes	—
53	#10625354	—	2	8	no	—
54	#10625350	—	4	12	no	—
55	C₂×D₁₈₈	C₂³	1	6	yes	—
56	C₂×D₁₈₈	C₂³	1	3	yes	—
57	C₂×D₁₈₈	C₂³	1	6	yes	—
58	C₂×C₄×D₄₇	C₂³	1	6	yes	—
59	C₂×C₄×D₄₇	C₂³	1	12	yes	—
60	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
61	C₂×Dic₉₄	C₂³	1	6	yes	—
62	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
63	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
64	C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
65	C₄²×C₄₇	C₂³	1	3	yes	—
66	C₄²×C₄₇	C₂³	1	3	yes	—
67	C₄²×C₄₇	C₂³	1	3	yes	—
68	C₄²×C₄₇	C₂³	1	3	yes	—
69	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
70	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
72	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
73	(C₂×D₄₇)⋊C₄	C₂³	1	12	yes	—
74	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
75	C₄⋊Dic₄₇	C₂³	1	6	yes	—
76	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₄₇	C₂³	1	3	yes	—
78	Dic₄₇⋊C₄	C₂³	1	6	yes	—
79	Dic₄₇⋊C₄	C₂³	1	2	yes	—
80	Dic₄₇⋊C₄	C₂³	1	6	yes	—
81	Dic₄₇⋊C₄	C₂³	1	6	yes	—
82	C₄×Dic₄₇	C₂³	1	6	yes	—
83	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×D₄₇	C₂⁴	1	6	yes	—
85	Q₈×C₄₇	—	2	6	no	—
86	Q₈×C₄₇	#41	1	4	yes	complemented
87	C₂×C₄×C₄₇	—	2	6	no	—
88	C₂×C₄×C₄₇	—	2	6	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
92	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
93	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
94	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
95	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₄₇	—	2	12	no	—
97	C₂×Dic₄₇	C₂⁴	1	2	yes	—
98	C₂×Dic₄₇	C₂⁴	1	6	yes	—
99	C₂×Dic₄₇	C₂⁴	1	2	yes	—
100	D₁₈₈	—	2	12	no	—
101	D₁₈₈	—	2	6	no	—
102	D₁₈₈	—	4	6	no	—
103	C₄×D₄₇	—	2	12	no	—
104	C₄×D₄₇	—	4	12	no	—
105	Dic₉₄	—	2	2	no	—
106	Dic₉₄	—	4	6	no	—
107	C₂²×C₄₇	—	2	1	no	—
108	C₂²×C₄₇	—	2	1	no	—
109	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
110	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
111	C₂×D₄₇	—	2	6	no	—
112	C₂×D₄₇	—	2	6	no	—
113	C₂×D₄₇	—	2	6	no	—
114	C₄×C₄₇	—	2	3	no	—
115	C₄×C₄₇	—	2	3	no	—
116	C₄×C₄₇	—	2	3	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	C₂×#41	1	6	yes	—
119	C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
120	Dic₄₇	—	2	2	no	—
121	Dic₄₇	—	2	6	no	—
122	Dic₄₇	—	2	2	no	—
123	#3522	—	47	1	no	maximal, Sylow
124	C₂×C₄₇	—	2	2	no	—
125	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
126	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
128	D₄₇	—	4	6	no	—
129	#559	—	47	3	no	—
130	Q₈²	—	47	1	no	—
131	#554	—	47	3	no	—
132	#550	—	47	1	no	—
133	#548	—	47	2	no	—
134	#547	—	47	6	no	—
135	#542	—	47	6	no	—
136	#532	—	47	3	no	—
137	#529	—	47	3	no	—
138	#520	—	47	2	no	—
139	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
140	C₄₇	#3522	1	1	yes	characteristic, complemented, Sylow
141	C₂×#41	—	47	2	no	—
142	C₂×#41	—	47	6	no	—
143	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
144	C₄⋊Q₈	—	47	6	no	—
145	C₄⋊Q₈	—	47	3	no	—
146	#127	—	47	3	no	—
147	#126	—	47	6	no	—
148	#125	—	47	3	no	—
149	#124	—	47	6	no	—
150	#124	—	47	3	no	—
151	#124	—	47	6	no	—
152	#123	—	47	6	no	—
153	#123	—	47	12	no	—
154	C₂²⋊Q₈	—	47	6	no	—
155	C₂²⋊Q₈	—	47	1	no	—
156	C₂²⋊Q₈	—	47	12	no	—
157	#121	—	47	3	no	—
158	#121	—	47	6	no	—
159	#121	—	47	6	no	—
160	C₄×Q₈	—	47	2	no	—
161	C₄×Q₈	—	47	3	no	—
162	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₄₇	1	3	yes	—
164	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
165	C₄×Q₈	C₂×D₄₇	1	3	yes	—
166	C₄×D₄	—	47	6	no	—
167	C₄×D₄	—	47	6	no	—
168	C₄×D₄	—	47	6	no	—
169	C₄×D₄	—	47	3	no	—
170	C₄×D₄	—	47	6	no	—
171	#117	—	47	6	no	—
172	C₂×C₄⋊C₄	—	47	6	no	—
173	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
174	C₂×C₄²	C₂×D₄₇	1	3	yes	—
175	#41	—	94	8	no	—
176	#41	—	188	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	47	1	no	—
179	C₂×Q₈	—	47	6	no	—
180	C₂×Q₈	C₂²×D₄₇	1	3	yes	—
181	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×D₄	—	47	6	no	—
183	C₂×D₄	—	47	3	no	—
184	C₂×D₄	—	47	6	no	—
185	C₂×D₄	—	47	6	no	—
186	C₂²×C₄	—	47	6	no	—
187	C₂²×C₄	—	47	12	no	—
188	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
190	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
191	C₄⋊C₄	—	47	6	no	—
192	C₄⋊C₄	—	47	2	no	—
193	C₄⋊C₄	—	47	6	no	—
194	C₄⋊C₄	—	47	6	no	—
195	C₄⋊C₄	—	47	6	no	—
196	C₄⋊C₄	—	47	1	no	—
197	C₄⋊C₄	—	47	3	no	—
198	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
199	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
200	C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
201	C₂²⋊C₄	—	47	2	no	—
202	C₂²⋊C₄	—	47	6	no	—
203	C₂²⋊C₄	—	47	6	no	—
204	C₂²⋊C₄	—	47	6	no	—
205	C₂²⋊C₄	—	47	12	no	—
206	C₂²⋊C₄	—	47	6	no	—
207	C₄²	—	47	6	no	—
208	C₄²	C₂²×D₄₇	1	3	yes	—
209	C₄²	C₂²×D₄₇	1	3	yes	—
210	C₄²	C₂²×D₄₇	1	3	yes	—
211	C₄²	C₂²×D₄₇	1	3	yes	—
212	C₂³	—	47	6	no	—
213	C₂³	C₂³×D₄₇	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	94	2	no	—
216	Q₈	—	188	6	no	—
217	Q₈	#10625350	1	4	yes	complemented
218	D₄	—	94	12	no	—
219	D₄	—	94	12	no	—
220	D₄	—	94	6	no	—
221	D₄	—	188	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	47	6	no	—
227	C₂×C₄	—	47	2	no	—
228	C₂×C₄	—	94	12	no	—
229	C₂×C₄	—	188	12	no	—
230	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂³×D₄₇	1	3	yes	—
232	C₂×C₄	C₂³×D₄₇	1	3	yes	—
233	C₂×C₄	C₂³×D₄₇	1	3	yes	—
234	C₂×C₄	C₂³×D₄₇	1	3	yes	—
235	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	94	6	no	—
239	C₂²	—	94	6	no	—
240	C₂²	—	94	6	no	—
241	C₂²	#13094070	1	2	yes	—
242	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
243	C₄	—	2	3	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	1	no	—
247	C₄	—	94	2	no	—
248	C₄	—	94	6	no	—
249	C₄	—	94	2	no	—
250	C₄	C₂×#10625350	1	6	yes	—
251	C₄	#13094072	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	188	6	no	—
254	C₂	C₂²×#10625350	1	1	yes	characteristic
255	C₂	C₂×#13094070	1	1	yes	characteristic
256	C₂	C₂×#13094072	1	1	yes	characteristic

	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	#3522
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. #15689858	C₂	semidirect
2. Q₈×Dic₉₄	C₂	semidirect
3. #15689833	C₂	semidirect
4. #15689795	C₂	semidirect
5. #15689784	C₂	semidirect
6. #15689774	C₂	semidirect
7. #15689745	C₂	semidirect
8. #15689673	C₂	semidirect
9. #15689657	C₂	semidirect
10. #15689605	C₂	semidirect
11. C₂×C₄×Q₈×C₄₇	C₂	semidirect
86. Q₈×C₄₇	#41	semidirect
139. C₂×C₄×Q₈	D₄₇	semidirect
140. C₄₇	#3522	semidirect
217. Q₈	#10625350	semidirect