Subgroups — C₂×#15689832 #20256921

Order 6016.

1200 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689832	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094023	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13094016	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13094015	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13094003	C₂	1	2	yes	maximal, complemented
6	C₂×#13094001	C₂	1	2	yes	maximal, complemented
7	C₂×#13094001	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×D₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₄₇⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₄	C₂²	1	2	yes	—
15	C₂³×Dic₄₇	C₂²	1	2	yes	—
16	C₂³×Dic₄₇	C₂²	1	2	yes	—
17	#13094023	C₂²	1	8	yes	complemented
18	#13094016	C₂²	1	8	yes	complemented
19	#13094015	C₂²	1	8	yes	complemented
20	#13094003	C₂²	1	16	yes	complemented
21	#13094001	C₂²	1	16	yes	complemented
22	#13094001	C₂²	1	16	yes	complemented
23	C₄×Dic₉₄	C₂²	1	8	yes	complemented
24	D₄×Dic₄₇	C₂²	1	8	yes	—
25	D₄×Dic₄₇	C₂²	1	8	yes	—
26	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
32	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
33	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
36	C₂×Dic₄₇⋊C₄	C₂²	1	16	yes	complemented
37	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
38	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
43	C₄×D₄×C₄₇	C₂²	1	8	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
47	C₂⁴×C₄₇	C₂³	1	2	yes	—
48	C₂×D₄×C₄₇	Q₈	1	8	yes	complemented
49	C₂×D₄×C₄₇	C₂³	1	4	yes	—
50	C₂²×C₄×C₄₇	—	4	4	no	—
51	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂³	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₄₇	—	2	8	no	—
57	C₂²×Dic₄₇	—	4	4	no	—
58	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
59	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
60	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂³	1	2	yes	—
62	C₂²×Dic₄₇	C₂³	1	2	yes	—
63	C₂²×Dic₄₇	C₂³	1	8	yes	—
64	C₂²×Dic₄₇	C₂³	1	8	yes	—
65	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
72	C₄²×C₄₇	C₂³	1	4	yes	—
73	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
74	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
75	C₄⋊Dic₄₇	C₂³	1	4	yes	—
76	C₄⋊Dic₄₇	C₂³	1	4	yes	—
77	C₄⋊Dic₄₇	C₂³	1	4	yes	—
78	C₄⋊Dic₄₇	C₂³	1	8	yes	—
79	Dic₄₇⋊C₄	C₂³	1	4	yes	—
80	Dic₄₇⋊C₄	C₂³	1	8	yes	—
81	Dic₄₇⋊C₄	C₂³	1	8	yes	—
82	Dic₄₇⋊C₄	C₂³	1	8	yes	—
83	Dic₄₇⋊C₄	C₂³	1	4	yes	—
84	Dic₄₇⋊C₄	C₂³	1	8	yes	—
85	C₄×Dic₄₇	C₂³	1	4	yes	—
86	C₄×Dic₄₇	C₂³	1	4	yes	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	C₂×Q₈	1	8	yes	—
90	C₂³×C₄₇	C₂⁴	1	4	yes	—
91	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
92	D₄×C₄₇	C₂×Q₈	1	16	yes	complemented
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	4	8	no	—
97	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
98	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
99	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
100	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
101	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	2	2	no	—
107	C₂×Dic₄₇	—	2	16	no	—
108	C₂×Dic₄₇	—	4	8	no	—
109	C₂×Dic₄₇	#41	1	4	yes	—
110	C₂×Dic₄₇	C₂⁴	1	2	yes	—
111	C₂×Dic₄₇	C₂⁴	1	2	yes	—
112	C₂×Dic₄₇	C₂⁴	1	2	yes	—
113	C₂×Dic₄₇	C₂⁴	1	4	yes	—
114	C₂×Dic₄₇	C₂⁴	1	4	yes	—
115	C₂×Dic₄₇	C₂⁴	1	2	yes	—
116	Dic₉₄	—	2	8	no	—
117	Dic₉₄	—	4	8	no	—
118	C₂²×C₄₇	—	2	16	no	—
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	—	2	4	no	—
121	C₂²×C₄₇	C₂²×Q₈	1	8	yes	—
122	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
123	C₂²×C₄₇	C₂×#41	1	2	yes	—
124	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
125	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	C₂²×Q₈	1	4	yes	—
130	Dic₄₇	—	2	2	no	—
131	Dic₄₇	—	2	4	no	—
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	—	2	2	no	—
135	Dic₄₇	C₂×#41	1	4	yes	—
136	C₂×#554	—	47	1	no	maximal, Sylow
137	C₂×C₄₇	—	2	8	no	—
138	C₂×C₄₇	#554	1	4	yes	complemented
139	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
140	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
141	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
142	#554	—	47	16	no	—
143	C₂×C₄⋊Q₈	—	47	1	no	—
144	C₂×#125	—	47	1	no	—
145	C₂×#125	—	47	1	no	—
146	C₂×C₂²⋊Q₈	—	47	2	no	—
147	C₂×C₂²⋊Q₈	—	47	2	no	—
148	C₂×C₂²⋊Q₈	—	47	2	no	—
149	C₂×C₄×Q₈	—	47	1	no	—
150	C₂×C₄×D₄	—	47	1	no	—
151	C₂×C₄×D₄	—	47	1	no	—
152	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	47	2	no	—
154	C₄₇	C₂×#554	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	47	1	no	—
156	C₂²×Q₈	—	47	2	no	—
157	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
158	C₂³×C₄	—	47	2	no	—
159	C₂³×C₄	—	47	2	no	—
160	C₂³×C₄	C₂×D₄₇	1	2	yes	—
161	C₄⋊Q₈	—	47	8	no	—
162	#125	—	47	8	no	—
163	#125	—	47	8	no	—
164	C₂²⋊Q₈	—	47	16	no	—
165	C₂²⋊Q₈	—	47	16	no	—
166	C₂²⋊Q₈	—	47	16	no	—
167	C₄×Q₈	—	47	8	no	—
168	C₄×D₄	—	47	8	no	—
169	C₄×D₄	—	47	8	no	—
170	C₄×D₄	C₂×D₄₇	1	8	yes	—
171	C₂×C₄⋊C₄	—	47	1	no	—
172	C₂×C₄⋊C₄	—	47	2	no	—
173	C₂×C₄⋊C₄	—	47	2	no	—
174	C₂×C₄⋊C₄	—	47	2	no	—
175	C₂×C₄⋊C₄	—	47	16	no	—
176	C₂×C₄⋊C₄	—	47	1	no	—
177	C₂×C₄⋊C₄	—	47	1	no	—
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	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	—	47	2	no	—
184	C₂×C₂²⋊C₄	—	47	2	no	—
185	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
186	C₂×C₄²	—	47	1	no	—
187	C₂×C₄²	—	47	1	no	—
188	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
189	C₂⁴	C₂²×D₄₇	1	2	yes	—
190	C₂×Q₈	—	47	4	no	—
191	C₂×Q₈	—	47	8	no	—
192	C₂×Q₈	—	94	4	no	—
193	C₂×Q₈	—	188	4	no	—
194	C₂×D₄	Dic₉₄	1	8	yes	complemented
195	C₂×D₄	C₂²×D₄₇	1	4	yes	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	—	47	1	no	—
198	C₂²×C₄	—	47	8	no	—
199	C₂²×C₄	—	47	1	no	—
200	C₂²×C₄	—	47	1	no	—
201	C₂²×C₄	—	47	1	no	—
202	C₂²×C₄	—	47	2	no	—
203	C₂²×C₄	—	47	2	no	—
204	C₂²×C₄	—	47	8	no	—
205	C₂²×C₄	—	94	8	no	—
206	C₂²×C₄	—	188	4	no	—
207	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
210	C₂²×C₄	C₂²×D₄₇	1	8	yes	—
211	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
212	C₄⋊C₄	—	47	4	no	—
213	C₄⋊C₄	—	47	8	no	—
214	C₄⋊C₄	—	47	8	no	—
215	C₄⋊C₄	—	47	8	no	—
216	C₄⋊C₄	—	47	4	no	—
217	C₄⋊C₄	—	47	8	no	—
218	C₄⋊C₄	—	47	4	no	—
219	C₄⋊C₄	—	47	4	no	—
220	C₄⋊C₄	—	47	4	no	—
221	C₄⋊C₄	—	47	8	no	—
222	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
223	C₂²⋊C₄	—	47	8	no	—
224	C₂²⋊C₄	—	47	8	no	—
225	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
226	C₄²	—	47	4	no	—
227	C₄²	—	47	4	no	—
228	C₄²	C₂²×D₄₇	1	4	yes	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	4	no	—
231	C₂³	C₂×Dic₉₄	1	8	yes	—
232	C₂³	C₂³×D₄₇	1	4	yes	—
233	C₂³	C₂³×D₄₇	1	1	yes	characteristic
234	Q₈	—	94	8	no	—
235	Q₈	—	188	8	no	—
236	D₄	C₂×Dic₉₄	1	16	yes	complemented
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	47	4	no	—
242	C₂×C₄	—	47	2	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	47	2	no	—
245	C₂×C₄	—	47	2	no	—
246	C₂×C₄	—	47	4	no	—
247	C₂×C₄	—	47	4	no	—
248	C₂×C₄	—	94	16	no	—
249	C₂×C₄	—	94	4	no	—
250	C₂×C₄	—	94	2	no	—
251	C₂×C₄	—	94	2	no	—
252	C₂×C₄	—	94	4	no	—
253	C₂×C₄	—	94	2	no	—
254	C₂×C₄	—	188	8	no	—
255	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
256	C₂×C₄	C₂³×D₄₇	1	2	yes	—
257	C₂×C₄	C₂³×D₄₇	1	2	yes	—
258	C₂×C₄	C₂³×D₄₇	1	4	yes	—
259	C₂×C₄	C₂³×D₄₇	1	2	yes	—
260	C₂²	—	2	16	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	C₂²×Dic₉₄	1	8	yes	—
264	C₂²	C₂²×Dic₉₄	1	2	yes	—
265	C₂²	#41×D₄₇	1	2	yes	—
266	C₂²	#13094069	1	2	yes	—
267	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	4	no	—
271	C₄	—	47	4	no	—
272	C₄	—	94	4	no	—
273	C₄	—	94	2	no	—
274	C₄	—	94	2	no	—
275	C₄	—	94	4	no	—
276	C₄	—	94	2	no	—
277	C₄	C₂²×Dic₉₄	1	4	yes	—
278	C₂	—	2	8	no	—
279	C₂	#15689832	1	4	yes	direct product factor
280	C₂	C₂³×Dic₉₄	1	1	yes	characteristic
281	C₂	C₂×#41×D₄₇	1	1	yes	characteristic
282	C₂	C₂×#13094069	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×C₄×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	C₂×#554
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #15689832	C₂	direct (N × H)
2. C₂×#13094023	C₂	semidirect
3. C₂×#13094016	C₂	semidirect
4. C₂×#13094015	C₂	semidirect
5. C₂×#13094003	C₂	semidirect
6. C₂×#13094001	C₂	semidirect
7. C₂×#13094001	C₂	semidirect
8. C₂×C₄×Dic₉₄	C₂	semidirect
11. C₂²×Dic₄₇⋊C₄	C₂	semidirect
17. #13094023	C₂²	semidirect
18. #13094016	C₂²	semidirect
19. #13094015	C₂²	semidirect
20. #13094003	C₂²	semidirect
21. #13094001	C₂²	semidirect
22. #13094001	C₂²	semidirect
23. C₄×Dic₉₄	C₂²	semidirect
36. C₂×Dic₄₇⋊C₄	C₂²	semidirect
48. C₂×D₄×C₄₇	Q₈	semidirect
92. D₄×C₄₇	C₂×Q₈	semidirect
138. C₂×C₄₇	#554	semidirect
154. C₄₇	C₂×#554	semidirect
194. C₂×D₄	Dic₉₄	semidirect
236. D₄	C₂×Dic₉₄	semidirect
279. C₂	#15689832	direct (N × H)

Subgroups — C2×#15689832 #20256921

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689832 #20256921