Subgroups — C₂×#15689837 #20256927

Order 6016.

1240 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689837	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094023	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13094011	C₂	1	2	yes	maximal, complemented
4	C₂×#13094010	C₂	1	2	yes	maximal, complemented
5	C₂×#13093996	C₂	1	2	yes	maximal, complemented
6	C₂×#13093995	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13093982	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13093980	C₂	1	2	yes	maximal, complemented
12	C₂×#121×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₉₄	C₂²	1	2	yes	—
14	C₂²×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
16	C₂³×Dic₄₇	C₂²	1	2	yes	—
17	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
18	#13094023	C₂²	1	8	yes	complemented
19	#13094011	C₂²	1	16	yes	complemented
20	#13094010	C₂²	1	16	yes	complemented
21	#13093996	C₂²	1	16	yes	complemented
22	#13093995	C₂²	1	8	yes	complemented
23	D₄×Dic₄₇	C₂²	1	8	yes	complemented
24	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	complemented
25	#13093983	C₂²	1	8	yes	complemented
26	#13093982	C₂²	1	8	yes	complemented
27	#13093980	C₂²	1	16	yes	complemented
28	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
30	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
32	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
44	#121×C₄₇	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
47	C₂⁴×C₄₇	C₂³	1	2	yes	—
48	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
49	C₂³×D₄₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₇	—	2	4	no	—
51	C₂×D₄×C₄₇	—	4	4	no	—
52	C₂×D₄×C₄₇	C₂³	1	4	yes	—
53	C₂×D₄×C₄₇	C₂³	1	8	yes	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₄₇	—	2	4	no	—
60	C₂×C₂²⋊D₄₇	—	4	4	no	—
61	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	C₂³	1	2	yes	—
65	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
66	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
67	C₂²×Dic₄₇	C₂³	1	2	yes	—
68	C₂²×Dic₄₇	C₂³	1	8	yes	—
69	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₄₇	—	2	4	no	—
71	C₂×C₄×D₄₇	C₂³	1	4	yes	—
72	C₂×Dic₉₄	—	4	4	no	—
73	C₂×Dic₉₄	C₂³	1	8	yes	—
74	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
75	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
76	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
77	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
78	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
79	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
80	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
82	C₄⋊Dic₄₇	C₂³	1	4	yes	—
83	C₄⋊Dic₄₇	C₂³	1	4	yes	—
84	Dic₄₇⋊C₄	C₂³	1	4	yes	—
85	Dic₄₇⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₄₇	C₂³	1	8	yes	—
87	C₄×Dic₄₇	C₂³	1	8	yes	—
88	C₄×Dic₄₇	C₂³	1	4	yes	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	2	4	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	C₂⁴	1	2	yes	—
96	C₂³×C₄₇	C₂⁴	1	4	yes	—
97	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
98	C₂²×D₄₇	—	2	2	no	—
99	C₂²×D₄₇	—	2	2	no	—
100	C₂²×D₄₇	—	2	2	no	—
101	C₂²×D₄₇	C₂⁴	1	2	yes	—
102	D₄×C₄₇	—	2	8	no	—
103	D₄×C₄₇	—	4	8	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	#41	1	4	yes	—
108	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
109	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
110	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
111	C₂²⋊D₄₇	—	2	8	no	—
112	C₂²⋊D₄₇	—	4	8	no	—
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	—	2	16	no	—
118	C₂×Dic₄₇	#41	1	4	yes	—
119	C₂×Dic₄₇	C₂⁴	1	2	yes	—
120	C₂×Dic₄₇	C₂⁴	1	4	yes	—
121	C₂×Dic₄₇	C₂⁴	1	4	yes	—
122	C₂×Dic₄₇	C₂⁴	1	2	yes	—
123	C₂×Dic₄₇	C₂⁴	1	2	yes	—
124	C₄×D₄₇	—	2	8	no	—
125	Dic₉₄	—	4	8	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	4	8	no	—
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	C₂×#41	1	2	yes	—
135	C₂²×C₄₇	C₂×#41	1	2	yes	—
136	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
137	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
138	C₂×D₄₇	—	2	2	no	—
139	C₂×D₄₇	—	2	2	no	—
140	C₂×D₄₇	—	2	2	no	—
141	C₂×D₄₇	—	4	4	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	4	no	—
144	C₄×C₄₇	C₂×#41	1	4	yes	—
145	Dic₄₇	—	2	2	no	—
146	Dic₄₇	—	2	4	no	—
147	Dic₄₇	—	2	2	no	—
148	Dic₄₇	—	2	4	no	—
149	Dic₄₇	C₂×#41	1	4	yes	—
150	C₂×#555	—	47	1	no	maximal, Sylow
151	C₂×C₄₇	—	4	4	no	—
152	C₂×C₄₇	—	4	2	no	—
153	C₂×C₄₇	#555	1	4	yes	complemented
154	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
155	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
156	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
157	D₄₇	—	4	2	no	—
158	#555	—	47	16	no	—
159	C₂×C₄⋊Q₈	—	47	1	no	—
160	C₂×#124	—	47	2	no	—
161	C₂×#124	—	47	2	no	—
162	C₂×#121	—	47	1	no	—
163	C₂×#121	—	47	2	no	—
164	C₂×#121	D₄₇	1	1	yes	characteristic, complemented
165	C₂×C₄×D₄	—	47	1	no	—
166	C₂×C₄×D₄	—	47	1	no	—
167	C₂×#117	—	47	1	no	—
168	C₂×#117	—	47	1	no	—
169	C₂×#117	—	47	2	no	—
170	C₄₇	C₂×#555	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	47	2	no	—
172	C₂²×D₄	—	47	1	no	—
173	C₂²×D₄	—	47	2	no	—
174	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
175	C₂²×D₄	C₂×D₄₇	1	2	yes	—
176	C₂³×C₄	—	47	2	no	—
177	C₂³×C₄	—	47	1	no	—
178	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
179	C₄⋊Q₈	—	47	8	no	—
180	#124	—	47	16	no	—
181	#124	—	47	16	no	—
182	#121	—	47	8	no	—
183	#121	—	47	16	no	—
184	#121	C₂×D₄₇	1	8	yes	complemented
185	C₄×D₄	—	47	8	no	—
186	C₄×D₄	—	47	8	no	—
187	#117	—	47	8	no	—
188	#117	—	47	8	no	—
189	#117	—	47	16	no	—
190	C₂×C₄⋊C₄	—	47	1	no	—
191	C₂×C₄⋊C₄	—	47	1	no	—
192	C₂×C₄⋊C₄	—	47	2	no	—
193	C₂×C₄⋊C₄	—	47	1	no	—
194	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	47	2	no	—
196	C₂×C₂²⋊C₄	—	47	2	no	—
197	C₂×C₂²⋊C₄	—	47	1	no	—
198	C₂×C₂²⋊C₄	—	47	2	no	—
199	C₂×C₂²⋊C₄	—	47	2	no	—
200	C₂×C₂²⋊C₄	—	47	1	no	—
201	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	47	2	no	—
204	C₂×C₄²	—	47	1	no	—
205	C₂⁴	—	47	1	no	—
206	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
207	C₂⁴	C₂²×D₄₇	1	2	yes	—
208	C₂×Q₈	—	47	8	no	—
209	C₂×Q₈	—	188	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	47	4	no	—
213	C₂×D₄	—	47	8	no	—
214	C₂×D₄	—	94	4	no	—
215	C₂×D₄	—	188	4	no	—
216	C₂×D₄	C₂²×D₄₇	1	4	yes	—
217	C₂×D₄	C₂²×D₄₇	1	8	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	47	1	no	—
220	C₂²×C₄	—	47	8	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	47	1	no	—
223	C₂²×C₄	—	47	1	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	4	no	—
226	C₂²×C₄	—	94	8	no	—
227	C₂²×C₄	—	94	4	no	—
228	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
229	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
231	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
232	C₄⋊C₄	—	47	4	no	—
233	C₄⋊C₄	—	47	8	no	—
234	C₄⋊C₄	—	47	4	no	—
235	C₄⋊C₄	—	47	4	no	—
236	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
237	C₂²⋊C₄	—	47	8	no	—
238	C₂²⋊C₄	—	47	8	no	—
239	C₂²⋊C₄	—	47	4	no	—
240	C₂²⋊C₄	—	47	8	no	—
241	C₂²⋊C₄	—	47	8	no	—
242	C₂²⋊C₄	—	47	4	no	—
243	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
244	C₄²	—	47	8	no	—
245	C₄²	—	47	8	no	—
246	C₄²	—	47	4	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	47	2	no	—
254	C₂³	—	94	2	no	—
255	C₂³	—	94	2	no	—
256	C₂³	—	94	2	no	—
257	C₂³	C₂³×D₄₇	1	2	yes	—
258	C₂³	C₂³×D₄₇	1	4	yes	—
259	C₂³	C₂³×D₄₇	1	1	yes	characteristic
260	Q₈	—	188	8	no	—
261	D₄	—	2	8	no	—
262	D₄	—	4	8	no	—
263	D₄	—	94	8	no	—
264	D₄	—	188	8	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	47	4	no	—
269	C₂×C₄	—	47	2	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	47	2	no	—
273	C₂×C₄	—	47	4	no	—
274	C₂×C₄	—	94	16	no	—
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	4	no	—
278	C₂×C₄	—	94	2	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	#10625352	1	4	yes	—
281	C₂×C₄	C₂³×D₄₇	1	2	yes	—
282	C₂×C₄	C₂³×D₄₇	1	4	yes	—
283	C₂×C₄	C₂³×D₄₇	1	2	yes	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	8	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	94	2	no	—
293	C₂²	—	94	2	no	—
294	C₂²	—	94	2	no	—
295	C₂²	—	188	4	no	—
296	C₂²	#41×D₄₇	1	2	yes	—
297	C₂²	C₂×#10625352	1	2	yes	—
298	C₂²	#13094069	1	2	yes	—
299	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	4	no	—
302	C₄	—	47	4	no	—
303	C₄	—	94	4	no	—
304	C₄	—	94	2	no	—
305	C₄	—	94	4	no	—
306	C₄	—	94	2	no	—
307	C₄	C₂×#10625352	1	4	yes	—
308	C₂	—	4	4	no	—
309	C₂	—	4	2	no	—
310	C₂	—	188	2	no	—
311	C₂	#15689837	1	4	yes	direct product factor
312	C₂	C₂×#41×D₄₇	1	1	yes	characteristic
313	C₂	C₂²×#10625352	1	1	yes	characteristic
314	C₂	C₂×#13094069	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×#121×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#121	D₄₇
hypocenter	C₄₇	C₂×#555
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#121

Complements

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

N	H ≅ G/N	type
1. #15689837	C₂	direct (N × H)
2. C₂×#13094023	C₂	semidirect
3. C₂×#13094011	C₂	semidirect
4. C₂×#13094010	C₂	semidirect
5. C₂×#13093996	C₂	semidirect
6. C₂×#13093995	C₂	semidirect
7. C₂×D₄×Dic₄₇	C₂	semidirect
8. C₂×C₄×C₂²⋊D₄₇	C₂	semidirect
9. C₂×#13093983	C₂	semidirect
10. C₂×#13093982	C₂	semidirect
11. C₂×#13093980	C₂	semidirect
12. C₂×#121×C₄₇	C₂	semidirect
18. #13094023	C₂²	semidirect
19. #13094011	C₂²	semidirect
20. #13094010	C₂²	semidirect
21. #13093996	C₂²	semidirect
22. #13093995	C₂²	semidirect
23. D₄×Dic₄₇	C₂²	semidirect
24. C₄×C₂²⋊D₄₇	C₂²	semidirect
25. #13093983	C₂²	semidirect
26. #13093982	C₂²	semidirect
27. #13093980	C₂²	semidirect
44. #121×C₄₇	C₂²	semidirect
153. C₂×C₄₇	#555	semidirect
164. C₂×#121	D₄₇	semidirect
170. C₄₇	C₂×#555	semidirect
184. #121	C₂×D₄₇	semidirect
311. C₂	#15689837	direct (N × H)

Subgroups — C2×#15689837 #20256927

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689837 #20256927