Subgroups — C₂×#15689744 #20256766

Order 6016.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689744	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094024	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13094019	C₂	1	2	yes	maximal, complemented
4	C₂×#13094012	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13094010	C₂	1	2	yes	maximal, complemented
6	C₂×#13094006	C₂	1	2	yes	maximal, complemented
7	C₂×#13094003	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094002	C₂	1	2	yes	maximal, complemented
9	C₂×#13093998	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₄	C₂²	1	2	yes	—
15	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
17	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
18	#13094024	C₂²	1	8	yes	complemented
19	#13094019	C₂²	1	16	yes	complemented
20	#13094012	C₂²	1	8	yes	complemented
21	#13094010	C₂²	1	16	yes	complemented
22	#13094006	C₂²	1	16	yes	complemented
23	#13094003	C₂²	1	8	yes	complemented
24	#13094002	C₂²	1	16	yes	complemented
25	#13093998	C₂²	1	8	yes	—
26	C₄×Dic₉₄	C₂²	1	8	yes	complemented
27	C₄×D₁₈₈	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
31	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
32	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
34	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
35	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
40	#117×C₄₇	C₂²	1	8	yes	complemented
41	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
47	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
48	C₂³×D₄₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₄₇	—	4	4	no	—
58	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
59	C₂²×Dic₄₇	C₂³	1	2	yes	—
60	C₂²×Dic₄₇	C₂³	1	2	yes	—
61	C₂²×Dic₄₇	C₂³	1	2	yes	—
62	C₂×D₁₈₈	—	2	4	no	—
63	C₂×D₁₈₈	C₂³	1	4	yes	—
64	C₂×C₄×D₄₇	—	4	4	no	—
65	C₂×C₄×D₄₇	C₂³	1	8	yes	—
66	C₂×Dic₉₄	—	2	4	no	—
67	C₂×Dic₉₄	—	4	4	no	—
68	C₂×Dic₉₄	C₂³	1	4	yes	—
69	C₂×Dic₉₄	C₂³	1	8	yes	—
70	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
71	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
72	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
74	C₄²×C₄₇	C₂³	1	4	yes	—
75	C₄²×C₄₇	C₂³	1	4	yes	—
76	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
77	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
81	C₄⋊Dic₄₇	C₂³	1	4	yes	—
82	C₄⋊Dic₄₇	C₂³	1	4	yes	—
83	C₄⋊Dic₄₇	C₂³	1	8	yes	—
84	Dic₄₇⋊C₄	C₂³	1	8	yes	—
85	Dic₄₇⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₄₇	C₂³	1	8	yes	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	C₂⁴	1	2	yes	—
91	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂²×D₄₇	—	2	4	no	—
93	C₂²×D₄₇	—	2	4	no	—
94	C₂²×D₄₇	—	2	4	no	—
95	C₂²×D₄₇	C₂⁴	1	4	yes	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	#41	1	4	yes	—
103	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
104	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
105	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
106	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
107	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
108	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
109	C₂²⋊D₄₇	—	4	8	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	C₂⁴	1	4	yes	—
114	C₂×Dic₄₇	C₂⁴	1	4	yes	—
115	C₂×Dic₄₇	C₂⁴	1	4	yes	—
116	D₁₈₈	—	2	8	no	—
117	C₄×D₄₇	—	4	8	no	—
118	Dic₉₄	—	2	8	no	—
119	Dic₉₄	—	4	8	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	4	4	no	—
124	C₂²×C₄₇	C₂×#41	1	2	yes	—
125	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
126	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
127	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
128	C₂×D₄₇	—	2	4	no	—
129	C₂×D₄₇	—	2	4	no	—
130	C₂×D₄₇	—	2	4	no	—
131	C₂×D₄₇	—	4	8	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	C₂×#41	1	4	yes	—
138	Dic₄₇	—	2	4	no	—
139	Dic₄₇	—	2	4	no	—
140	Dic₄₇	—	2	4	no	—
141	C₂×#542	—	47	1	no	maximal, Sylow
142	C₂×C₄₇	—	4	2	no	—
143	C₂×C₄₇	#542	1	4	yes	complemented
144	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
145	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
147	D₄₇	—	4	4	no	—
148	#542	—	47	16	no	—
149	C₂×C₄⋊Q₈	—	47	1	no	—
150	C₂×#126	—	47	2	no	—
151	C₂×#124	—	47	2	no	—
152	C₂×#124	—	47	1	no	—
153	C₂×#123	—	47	2	no	—
154	C₂×C₂²⋊Q₈	—	47	1	no	—
155	C₂×C₂²⋊Q₈	—	47	2	no	—
156	C₂×#121	—	47	1	no	—
157	C₂×C₄×Q₈	—	47	1	no	—
158	C₂×C₄×D₄	—	47	1	no	—
159	C₂×#117	D₄₇	1	1	yes	characteristic, complemented
160	C₄₇	C₂×#542	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	47	1	no	—
162	C₂²×Q₈	—	47	2	no	—
163	C₂²×D₄	—	47	2	no	—
164	C₂²×D₄	—	47	1	no	—
165	C₂³×C₄	—	47	2	no	—
166	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
167	C₄⋊Q₈	—	47	8	no	—
168	#126	—	47	16	no	—
169	#124	—	47	16	no	—
170	#124	—	47	8	no	—
171	#123	—	47	16	no	—
172	C₂²⋊Q₈	—	47	8	no	—
173	C₂²⋊Q₈	—	47	16	no	—
174	#121	—	47	8	no	—
175	C₄×Q₈	—	47	8	no	—
176	C₄×D₄	—	47	8	no	—
177	#117	C₂×D₄₇	1	8	yes	complemented
178	C₂×C₄⋊C₄	—	47	1	no	—
179	C₂×C₄⋊C₄	—	47	1	no	—
180	C₂×C₄⋊C₄	—	47	2	no	—
181	C₂×C₄⋊C₄	—	47	2	no	—
182	C₂×C₄⋊C₄	—	47	2	no	—
183	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	47	2	no	—
186	C₂×C₂²⋊C₄	—	47	2	no	—
187	C₂×C₂²⋊C₄	—	47	2	no	—
188	C₂×C₂²⋊C₄	—	47	2	no	—
189	C₂×C₂²⋊C₄	—	47	2	no	—
190	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
192	C₂×C₄²	—	47	2	no	—
193	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
195	C₂⁴	—	47	2	no	—
196	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×Q₈	—	47	4	no	—
198	C₂×Q₈	—	47	8	no	—
199	C₂×Q₈	—	94	4	no	—
200	C₂×Q₈	—	188	4	no	—
201	C₂×D₄	—	47	8	no	—
202	C₂×D₄	—	47	4	no	—
203	C₂×D₄	—	94	4	no	—
204	C₂×D₄	—	188	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	47	2	no	—
207	C₂²×C₄	—	47	2	no	—
208	C₂²×C₄	—	47	2	no	—
209	C₂²×C₄	—	47	8	no	—
210	C₂²×C₄	—	188	4	no	—
211	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
212	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₄⋊C₄	—	47	8	no	—
219	C₄⋊C₄	—	47	8	no	—
220	C₄⋊C₄	—	47	4	no	—
221	C₄⋊C₄	—	47	4	no	—
222	C₄⋊C₄	—	47	8	no	—
223	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
224	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
225	C₂²⋊C₄	—	47	8	no	—
226	C₂²⋊C₄	—	47	8	no	—
227	C₂²⋊C₄	—	47	8	no	—
228	C₂²⋊C₄	—	47	8	no	—
229	C₂²⋊C₄	—	47	8	no	—
230	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
231	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
232	C₄²	—	47	8	no	—
233	C₄²	C₂²×D₄₇	1	4	yes	—
234	C₄²	C₂²×D₄₇	1	4	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	47	4	no	—
239	C₂³	—	94	4	no	—
240	C₂³	—	94	4	no	—
241	C₂³	—	94	4	no	—
242	C₂³	C₂³×D₄₇	1	2	yes	—
243	C₂³	C₂³×D₄₇	1	1	yes	characteristic
244	Q₈	—	94	8	no	—
245	Q₈	—	188	8	no	—
246	D₄	—	94	8	no	—
247	D₄	—	188	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	47	4	no	—
255	C₂×C₄	—	47	4	no	—
256	C₂×C₄	—	47	4	no	—
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	—	94	4	no	—
259	C₂×C₄	—	94	4	no	—
260	C₂×C₄	—	188	8	no	—
261	C₂×C₄	#10625350	1	4	yes	—
262	C₂×C₄	C₂³×D₄₇	1	2	yes	—
263	C₂×C₄	C₂³×D₄₇	1	2	yes	—
264	C₂×C₄	C₂³×D₄₇	1	2	yes	—
265	C₂×C₄	C₂³×D₄₇	1	2	yes	—
266	C₂×C₄	C₂³×D₄₇	1	2	yes	—
267	C₂×C₄	C₂³×D₄₇	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	94	4	no	—
273	C₂²	—	94	4	no	—
274	C₂²	—	94	4	no	—
275	C₂²	—	188	8	no	—
276	C₂²	C₂×#10625350	1	2	yes	—
277	C₂²	#13094070	1	2	yes	—
278	C₂²	#13094071	1	2	yes	—
279	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	94	4	no	—
286	C₄	—	94	4	no	—
287	C₄	—	94	4	no	—
288	C₄	C₂×#10625350	1	4	yes	—
289	C₂	—	4	2	no	—
290	C₂	—	188	4	no	—
291	C₂	#15689744	1	4	yes	direct product factor
292	C₂	C₂²×#10625350	1	1	yes	characteristic
293	C₂	C₂×#13094070	1	1	yes	characteristic
294	C₂	C₂×#13094071	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	C₂×#542
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #15689744	C₂	direct (N × H)
2. C₂×#13094024	C₂	semidirect
3. C₂×#13094019	C₂	semidirect
4. C₂×#13094012	C₂	semidirect
5. C₂×#13094010	C₂	semidirect
6. C₂×#13094006	C₂	semidirect
7. C₂×#13094003	C₂	semidirect
8. C₂×#13094002	C₂	semidirect
10. C₂×C₄×Dic₉₄	C₂	semidirect
11. C₂×C₄×D₁₈₈	C₂	semidirect
12. C₂×#117×C₄₇	C₂	semidirect
18. #13094024	C₂²	semidirect
19. #13094019	C₂²	semidirect
20. #13094012	C₂²	semidirect
21. #13094010	C₂²	semidirect
22. #13094006	C₂²	semidirect
23. #13094003	C₂²	semidirect
24. #13094002	C₂²	semidirect
26. C₄×Dic₉₄	C₂²	semidirect
27. C₄×D₁₈₈	C₂²	semidirect
40. #117×C₄₇	C₂²	semidirect
143. C₂×C₄₇	#542	semidirect
159. C₂×#117	D₄₇	semidirect
160. C₄₇	C₂×#542	semidirect
177. #117	C₂×D₄₇	semidirect
291. C₂	#15689744	direct (N × H)

Subgroups — C2×#15689744 #20256766

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689744 #20256766