Subgroups — #20257181

Order 6016.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094071	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094070	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689785	C₂	1	4	yes	maximal, complemented
4	#15689768	C₂	1	12	yes	maximal, complemented
5	#15689699	C₂	1	3	yes	maximal, complemented
6	#15689692	C₂	1	3	yes	maximal, complemented
7	#15689673	C₂	1	3	yes	maximal, complemented
8	C₂×#13094012	C₂	1	3	yes	maximal, complemented
9	C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#13094071	—	2	8	no	—
11	#13094070	—	2	8	no	—
12	C₂×#10625354	C₂²	1	2	yes	—
13	C₂×#10625350	C₂²	1	3	yes	—
14	C₂×#10625350	C₂²	1	3	yes	complemented
15	C₂×#10625352	C₂²	1	6	yes	complemented
16	C₂²×Dic₉₄	C₂²	1	3	yes	complemented
17	C₂×Q₈×D₄₇	C₂²	1	2	yes	complemented
18	C₂²×D₁₈₈	C₂²	1	3	yes	—
19	C₂×D₄×D₄₇	C₂²	1	6	yes	—
20	#13094024	C₂²	1	3	yes	complemented
21	#13094021	C₂²	1	3	yes	complemented
22	#13094012	C₂²	1	12	yes	complemented
23	#13094012	C₂²	1	3	yes	complemented
24	#13094012	C₂²	1	6	yes	complemented
25	#13094012	C₂²	1	1	yes	characteristic, complemented
26	#13094008	C₂²	1	12	yes	complemented
27	#13094004	C₂²	1	12	yes	complemented
28	#13093999	C₂²	1	12	yes	complemented
29	#13093998	C₂²	1	12	yes	complemented
30	#13093994	C₂²	1	12	yes	complemented
31	C₄×D₁₈₈	C₂²	1	12	yes	complemented
32	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	12	yes	complemented
33	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₄₇	C₂²	1	2	yes	complemented
35	C₄×D₄×C₄₇	C₂²	1	6	yes	complemented
36	#117×C₄₇	C₂²	1	3	yes	complemented
37	C₂×C₄²×C₄₇	C₂²	1	3	yes	complemented
38	C₂³×D₄₇	C₂³	1	6	yes	—
39	#41×C₄₇	D₄	1	8	yes	complemented
40	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₄₇	C₂³	1	3	yes	—
42	C₂²×C₄×C₄₇	C₂³	1	6	yes	—
43	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
44	C₂×C₂²⋊D₄₇	C₂³	1	6	yes	—
45	C₂×C₂²⋊D₄₇	C₂³	1	6	yes	—
46	C₂²×Dic₄₇	C₂³	1	6	yes	—
47	#10625354	—	2	8	no	—
48	Q₈×D₄₇	—	2	8	no	—
49	#10625352	—	2	24	no	—
50	D₄×D₄₇	—	2	24	no	—
51	#10625350	—	2	12	no	—
52	#10625350	—	2	12	no	—
53	C₂×D₁₈₈	—	2	12	no	—
54	C₂×D₁₈₈	C₂³	1	6	yes	—
55	C₂×D₁₈₈	C₂³	1	3	yes	—
56	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
57	C₂×D₁₈₈	C₂³	1	12	yes	—
58	C₂×C₄×D₄₇	C₂³	1	6	yes	—
59	C₂×C₄×D₄₇	C₂³	1	6	yes	—
60	C₂×Dic₉₄	—	2	12	no	—
61	C₂×Dic₉₄	C₂³	1	6	yes	—
62	C₂×Dic₉₄	C₂³	1	3	yes	—
63	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₄₇	C₂³	1	6	yes	—
65	C₂²⋊C₄×C₄₇	C₂³	1	6	yes	—
66	C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄²×C₄₇	C₂³	1	3	yes	—
68	C₄²×C₄₇	C₂³	1	6	yes	—
69	(C₂×D₄₇)⋊C₄	C₂³	1	12	yes	—
70	(C₂×D₄₇)⋊C₄	C₂³	1	24	yes	—
71	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
72	C₄⋊Dic₄₇	C₂³	1	12	yes	—
73	C₂³×C₄₇	C₂⁴	1	3	yes	—
74	C₂²×D₄₇	—	2	12	no	—
75	C₂²×D₄₇	—	4	12	no	—
76	C₂²×D₄₇	C₂⁴	1	6	yes	—
77	C₂²×D₄₇	C₂⁴	1	2	yes	—
78	Q₈×C₄₇	C₂×D₄	1	4	yes	—
79	D₄×C₄₇	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₄₇	—	4	6	no	—
81	C₂×C₄×C₄₇	C₂×D₄	1	12	yes	—
82	C₂×C₄×C₄₇	C₂⁴	1	6	yes	—
83	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
86	C₂²⋊D₄₇	—	2	12	no	—
87	C₂²⋊D₄₇	—	2	12	no	—
88	C₂×Dic₄₇	—	2	12	no	—
89	C₂×Dic₄₇	C₂⁴	1	2	yes	—
90	C₂×Dic₄₇	C₂⁴	1	6	yes	—
91	D₁₈₈	—	2	6	no	—
92	D₁₈₈	—	2	12	no	—
93	D₁₈₈	—	2	2	no	—
94	D₁₈₈	—	4	12	no	—
95	C₄×D₄₇	—	2	12	no	—
96	C₄×D₄₇	—	2	12	no	—
97	Dic₉₄	—	2	6	no	—
98	Dic₉₄	—	2	12	no	—
99	Dic₉₄	—	2	2	no	—
100	C₂²×C₄₇	—	2	6	no	—
101	C₂²×C₄₇	C₂²×D₄	1	6	yes	—
102	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
103	C₂×D₄₇	—	2	2	no	—
104	C₂×D₄₇	—	2	6	no	—
105	C₂×D₄₇	—	2	4	no	—
106	C₂×D₄₇	—	2	12	no	—
107	C₂×D₄₇	—	8	12	no	—
108	C₄×C₄₇	—	2	6	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	C₂²×D₄	1	2	yes	—
111	C₄×C₄₇	C₂²×D₄	1	6	yes	—
112	Dic₄₇	—	2	6	no	—
113	Dic₄₇	—	2	2	no	—
114	#3500	—	47	1	no	maximal, Sylow
115	C₂×C₄₇	—	2	6	no	—
116	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
117	C₂×C₄₇	#585	1	2	yes	—
118	D₄₇	—	4	2	no	—
119	D₄₇	—	4	6	no	—
120	C₂×2₋¹⁺⁴	—	47	1	no	—
121	C₂×H(5,𝔽₂)	—	47	1	no	—
122	#548	—	47	4	no	—
123	#546	—	47	12	no	—
124	#536	—	47	3	no	—
125	#535	—	47	3	no	—
126	#532	—	47	3	no	—
127	C₂×#124	—	47	3	no	—
128	C₄×#41	D₄₇	1	1	yes	characteristic, complemented
129	C₄₇	#3500	1	1	yes	characteristic, complemented, Sylow
130	2₋¹⁺⁴	—	94	8	no	—
131	H(5,𝔽₂)	—	94	8	no	—
132	C₂×#41	—	47	2	no	—
133	C₂×#41	—	47	6	no	—
134	C₂×#41	—	47	3	no	—
135	C₂×#41	—	47	3	no	—
136	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
137	C₂²×Q₈	—	47	2	no	—
138	C₂²×Q₈	—	47	3	no	—
139	C₂²×D₄	—	47	6	no	—
140	C₂²×D₄	—	47	3	no	—
141	C₄⋊Q₈	—	47	3	no	—
142	#127	—	47	3	no	—
143	#124	—	47	3	no	—
144	#124	—	47	6	no	—
145	#124	—	47	12	no	—
146	#124	—	47	1	no	—
147	#123	—	47	12	no	—
148	C₂²⋊Q₈	—	47	12	no	—
149	#121	—	47	12	no	—
150	#121	—	47	12	no	—
151	#120	—	47	12	no	—
152	C₄×Q₈	C₂×D₄₇	1	2	yes	complemented
153	C₄×D₄	—	47	12	no	—
154	C₄×D₄	C₂×D₄₇	1	6	yes	complemented
155	#117	C₂×D₄₇	1	3	yes	complemented
156	C₂×C₂²⋊C₄	—	47	12	no	—
157	C₂×C₄²	C₂×D₄₇	1	3	yes	complemented
158	C₂⁴	—	47	6	no	—
159	#41	—	94	8	no	—
160	#41	—	94	24	no	—
161	#41	—	94	12	no	—
162	#41	—	94	12	no	—
163	#41	D₁₈₈	1	8	yes	complemented
164	C₂×Q₈	—	47	6	no	—
165	C₂×Q₈	—	47	3	no	—
166	C₂×Q₈	—	47	1	no	—
167	C₂×Q₈	—	94	8	no	—
168	C₂×Q₈	—	94	12	no	—
169	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×D₄	—	47	6	no	—
171	C₂×D₄	—	47	6	no	—
172	C₂×D₄	—	47	6	no	—
173	C₂×D₄	—	47	3	no	—
174	C₂×D₄	—	47	1	no	—
175	C₂×D₄	—	47	12	no	—
176	C₂×D₄	—	94	24	no	—
177	C₂×D₄	—	94	12	no	—
178	C₂×D₄	C₂²×D₄₇	1	3	yes	—
179	C₂²×C₄	—	47	6	no	—
180	C₂²×C₄	—	47	6	no	—
181	C₂²×C₄	—	47	6	no	—
182	C₂²×C₄	C₂²×D₄₇	1	6	yes	—
183	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
184	C₄⋊C₄	—	47	12	no	—
185	C₄⋊C₄	C₂²×D₄₇	1	6	yes	—
186	C₂²⋊C₄	—	47	12	no	—
187	C₂²⋊C₄	—	47	24	no	—
188	C₂²⋊C₄	—	47	4	no	—
189	C₂²⋊C₄	C₂²×D₄₇	1	6	yes	—
190	C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	C₄²	C₂²×D₄₇	1	3	yes	—
192	C₄²	C₂²×D₄₇	1	6	yes	—
193	C₂³	—	47	6	no	—
194	C₂³	—	47	2	no	—
195	C₂³	—	94	12	no	—
196	C₂³	—	188	12	no	—
197	C₂³	C₂³×D₄₇	1	3	yes	—
198	Q₈	—	94	12	no	—
199	Q₈	—	94	6	no	—
200	Q₈	—	94	2	no	—
201	Q₈	C₂×D₁₈₈	1	4	yes	—
202	D₄	—	94	12	no	—
203	D₄	—	94	12	no	—
204	D₄	—	94	12	no	—
205	D₄	—	94	6	no	—
206	D₄	—	94	2	no	—
207	D₄	—	188	12	no	—
208	D₄	C₂×D₁₈₈	1	12	yes	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	—	47	6	no	—
211	C₂×C₄	—	47	2	no	—
212	C₂×C₄	—	94	12	no	—
213	C₂×C₄	—	94	12	no	—
214	C₂×C₄	—	94	12	no	—
215	C₂×C₄	C₂×D₁₈₈	1	12	yes	—
216	C₂×C₄	C₂³×D₄₇	1	2	yes	—
217	C₂×C₄	C₂³×D₄₇	1	6	yes	—
218	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₄₇	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	94	2	no	—
222	C₂²	—	94	6	no	—
223	C₂²	—	94	4	no	—
224	C₂²	—	94	12	no	—
225	C₂²	—	376	12	no	—
226	C₂²	C₂²×D₁₈₈	1	6	yes	—
227	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	6	no	—
230	C₄	—	94	2	no	—
231	C₄	—	94	6	no	—
232	C₄	C₂²×D₁₈₈	1	6	yes	—
233	C₄	C₂²×D₁₈₈	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	188	6	no	—
236	C₂	—	188	2	no	—
237	C₂	C₂³×D₁₈₈	1	1	yes	characteristic
238	C₂	#15689959	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄×#41×C₄₇	C₂
hypercenter	C₄×#41	D₄₇
hypocenter	C₄₇	#3500
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13094071	C₂	semidirect
2. C₂×#13094070	C₂	semidirect
3. #15689785	C₂	semidirect
4. #15689768	C₂	semidirect
5. #15689699	C₂	semidirect
6. #15689692	C₂	semidirect
7. #15689673	C₂	semidirect
8. C₂×#13094012	C₂	semidirect
9. C₄×#41×C₄₇	C₂	semidirect
14. C₂×#10625350	C₂²	semidirect
15. C₂×#10625352	C₂²	semidirect
16. C₂²×Dic₉₄	C₂²	semidirect
17. C₂×Q₈×D₄₇	C₂²	semidirect
20. #13094024	C₂²	semidirect
21. #13094021	C₂²	semidirect
22. #13094012	C₂²	semidirect
23. #13094012	C₂²	semidirect
24. #13094012	C₂²	semidirect
25. #13094012	C₂²	semidirect
26. #13094008	C₂²	semidirect
27. #13094004	C₂²	semidirect
28. #13093999	C₂²	semidirect
29. #13093998	C₂²	semidirect
30. #13093994	C₂²	semidirect
31. C₄×D₁₈₈	C₂²	semidirect
32. C₂×(C₂×D₄₇)⋊C₄	C₂²	semidirect
34. C₄×Q₈×C₄₇	C₂²	semidirect
35. C₄×D₄×C₄₇	C₂²	semidirect
36. #117×C₄₇	C₂²	semidirect
37. C₂×C₄²×C₄₇	C₂²	semidirect
39. #41×C₄₇	D₄	semidirect
128. C₄×#41	D₄₇	semidirect
129. C₄₇	#3500	semidirect
152. C₄×Q₈	C₂×D₄₇	semidirect
154. C₄×D₄	C₂×D₄₇	semidirect
155. #117	C₂×D₄₇	semidirect
157. C₂×C₄²	C₂×D₄₇	semidirect
163. #41	D₁₈₈	semidirect