Subgroups — C₂×#20246529 #26378857

Order 12032.

4246 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246529	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15689026	C₂	1	3	yes	maximal, complemented
3	C₂×#15689016	C₂	1	6	yes	maximal, complemented
4	C₂×#15689003	C₂	1	3	yes	maximal, complemented
5	C₂×#15688979	C₂	1	2	yes	maximal, complemented
6	C₂²×#120×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
8	C₂²×C₄⋊Dic₄₇	C₂²	1	3	yes	complemented
9	C₂²×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
10	C₂²×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
11	C₂²×C₂²⋊Dic₄₇	C₂²	1	3	yes	complemented
12	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	complemented
13	#15689026	C₂²	1	24	yes	complemented
14	#15689016	C₂²	1	48	yes	complemented
15	#15689003	C₂²	1	24	yes	complemented
16	#15688979	C₂²	1	16	yes	complemented
17	C₂×C₄₇⋊#95	C₂²	1	6	yes	complemented
18	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×D₄×C₄₇	C₂²	1	3	yes	—
20	C₂×#120×C₄₇	C₄	1	16	yes	complemented
21	C₂×#120×C₄₇	C₂²	1	8	yes	—
22	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
23	C₂³×Dic₄₇	—	4	6	no	—
24	C₂³×Dic₄₇	C₂³	1	2	yes	—
25	C₂³×Dic₄₇	C₂³	1	6	yes	—
26	C₂³×Dic₄₇	C₂³	1	4	yes	complemented
27	C₂×C₄⋊Dic₄₇	—	4	6	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	complemented
29	C₂×C₂²⋊Dic₄₇	—	4	12	no	—
30	C₂×C₂²⋊Dic₄₇	—	4	12	no	—
31	C₂×C₂²⋊Dic₄₇	—	4	6	no	—
32	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
34	C₂×C₂²⋊Dic₄₇	C₂³	1	12	yes	complemented
35	C₂×C₄×Dic₄₇	—	4	6	no	—
36	C₂×C₄×Dic₄₇	C₂³	1	12	yes	complemented
37	C₄₇⋊#95	C₂³	1	24	yes	complemented
38	C₂⁵×C₄₇	—	4	6	no	—
39	C₂⁵×C₄₇	—	4	6	no	—
40	C₂⁵×C₄₇	C₂×C₄	1	8	yes	complemented
41	C₂⁵×C₄₇	C₂³	1	3	yes	—
42	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
43	C₂⁵×C₄₇	C₂³	1	4	yes	—
44	C₂²×D₄×C₄₇	—	4	6	no	—
45	C₂²×D₄×C₄₇	—	4	6	no	—
46	C₂²×D₄×C₄₇	C₂×C₄	1	24	yes	—
47	C₂²×D₄×C₄₇	C₂³	1	12	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
49	#120×C₄₇	C₂×C₄	1	64	yes	complemented
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	24	yes	complemented
51	C₂×C₂²⋊C₄×C₄₇	C₂³	1	12	yes	—
52	C₂⁴×C₄₇	—	2	6	no	—
53	C₂⁴×C₄₇	—	2	6	no	—
54	C₂⁴×C₄₇	—	2	6	no	—
55	C₂⁴×C₄₇	—	2	6	no	—
56	C₂⁴×C₄₇	—	2	12	no	—
57	C₂⁴×C₄₇	—	2	12	no	—
58	C₂⁴×C₄₇	—	4	48	no	—
59	C₂⁴×C₄₇	—	4	12	no	—
60	C₂⁴×C₄₇	—	4	12	no	—
61	C₂⁴×C₄₇	—	4	48	no	—
62	C₂⁴×C₄₇	—	4	16	no	—
63	C₂⁴×C₄₇	C₂²×C₄	1	12	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
65	C₂⁴×C₄₇	C₂²×C₄	1	16	yes	complemented
66	C₂⁴×C₄₇	C₂⁴	1	6	yes	—
67	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
68	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂×D₄×C₄₇	—	4	48	no	—
70	C₂×D₄×C₄₇	—	4	12	no	—
71	C₂×D₄×C₄₇	—	4	12	no	—
72	C₂×D₄×C₄₇	C₂²×C₄	1	48	yes	—
73	C₂²×C₄×C₄₇	—	2	6	no	—
74	C₂²×C₄×C₄₇	—	2	6	no	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
77	C₂²×Dic₄₇	—	2	12	no	—
78	C₂²×Dic₄₇	—	2	12	no	—
79	C₂²×Dic₄₇	—	2	12	no	—
80	C₂²×Dic₄₇	—	2	12	no	—
81	C₂²×Dic₄₇	—	4	12	no	—
82	C₂²×Dic₄₇	—	4	16	no	—
83	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
84	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
85	C₂²⋊C₄×C₄₇	C₂²×C₄	1	48	yes	complemented
86	C₂²⋊Dic₄₇	—	4	24	no	—
87	C₂²⋊Dic₄₇	—	4	24	no	—
88	C₂²⋊Dic₄₇	—	4	12	no	—
89	C₄⋊Dic₄₇	—	4	12	no	—
90	C₄×Dic₄₇	—	4	12	no	—
91	C₂³×C₄₇	—	2	24	no	—
92	C₂³×C₄₇	—	2	24	no	—
93	C₂³×C₄₇	—	2	48	no	—
94	C₂³×C₄₇	—	2	6	no	—
95	C₂³×C₄₇	—	2	12	no	—
96	C₂³×C₄₇	—	2	12	no	—
97	C₂³×C₄₇	—	2	6	no	—
98	C₂³×C₄₇	—	2	6	no	—
99	C₂³×C₄₇	—	2	6	no	—
100	C₂³×C₄₇	—	4	48	no	—
101	C₂³×C₄₇	—	4	128	no	—
102	C₂³×C₄₇	—	4	48	no	—
103	C₂³×C₄₇	—	4	16	no	—
104	C₂³×C₄₇	—	4	12	no	—
105	C₂³×C₄₇	—	4	24	no	—
106	C₂³×C₄₇	—	4	12	no	—
107	C₂³×C₄₇	—	4	4	no	—
108	C₂³×C₄₇	—	4	12	no	—
109	C₂³×C₄₇	C₂³×C₄	1	12	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
111	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
112	C₂³×C₄₇	H(5,𝔽₂)	1	6	yes	—
113	C₂³×C₄₇	H(5,𝔽₂)	1	6	yes	—
114	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
115	D₄×C₄₇	—	4	48	no	—
116	C₂×C₄×C₄₇	—	2	24	no	—
117	C₂×C₄×C₄₇	—	2	6	no	—
118	C₂×C₄×C₄₇	—	2	6	no	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
120	C₂×Dic₄₇	—	2	12	no	—
121	C₂×Dic₄₇	—	2	12	no	—
122	C₂×Dic₄₇	—	2	12	no	—
123	C₂×Dic₄₇	—	2	12	no	—
124	C₂×Dic₄₇	—	4	16	no	—
125	C₂×Dic₄₇	—	4	24	no	—
126	C₂×Dic₄₇	—	4	8	no	—
127	C₂×#2375	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	12	no	—
129	C₂²×C₄₇	—	2	12	no	—
130	C₂²×C₄₇	—	2	24	no	—
131	C₂²×C₄₇	—	4	64	no	—
132	C₂²×C₄₇	—	4	16	no	—
133	C₂²×C₄₇	—	4	48	no	—
134	C₂²×C₄₇	—	4	6	no	—
135	C₂²×C₄₇	—	4	6	no	—
136	C₂²×C₄₇	—	4	12	no	—
137	C₂²×C₄₇	—	4	2	no	—
138	C₂²×C₄₇	—	4	6	no	—
139	C₂²×C₄₇	#518	1	12	yes	—
140	C₂²×C₄₇	#560	1	12	yes	—
141	C₂²×C₄₇	H(3,𝔽₄)	1	4	yes	—
142	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	3	yes	—
144	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	3	yes	—
145	C₄×C₄₇	—	2	12	no	—
146	Dic₄₇	—	4	12	no	—
147	Dic₄₇	—	4	4	no	—
148	C₂²×#120	D₄₇	1	1	yes	characteristic
149	#2375	—	47	16	no	—
150	C₂×#390	—	47	3	no	—
151	C₂×#388	—	47	6	no	—
152	C₂×#386	—	47	3	no	—
153	C₂×#379	—	47	2	no	—
154	C₂×C₄₇	—	4	4	no	—
155	C₂×C₄₇	—	4	12	no	—
156	C₂×C₄₇	#2375	1	8	yes	complemented
157	C₂×C₄₇	C₂×#518	1	3	yes	—
158	C₂×C₄₇	C₂×#560	1	3	yes	—
159	C₂×C₄₇	C₂×H(3,𝔽₄)	1	1	yes	characteristic
160	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
161	C₂³×D₄	C₂×D₄₇	1	3	yes	—
162	C₂⁴×C₄	—	47	1	no	—
163	C₂×#120	Dic₄₇	1	16	yes	complemented
164	C₂×#120	C₂×D₄₇	1	8	yes	—
165	C₂²×C₄⋊C₄	—	47	3	no	—
166	C₂²×C₂²⋊C₄	—	47	6	no	—
167	C₂²×C₂²⋊C₄	—	47	6	no	—
168	C₂²×C₂²⋊C₄	—	47	3	no	—
169	C₂²×C₂²⋊C₄	C₂×D₄₇	1	3	yes	—
170	C₂²×C₄²	—	47	3	no	—
171	#390	—	47	24	no	—
172	#388	—	47	48	no	—
173	#386	—	47	24	no	—
174	#379	—	47	16	no	—
175	C₂×#95	—	47	6	no	—
176	C₄₇	C₂×#2375	1	1	yes	characteristic, complemented, Sylow
177	C₂⁵	—	4	6	no	—
178	C₂⁵	—	4	6	no	—
179	C₂⁵	C₂×Dic₄₇	1	8	yes	complemented
180	C₂⁵	C₂²×D₄₇	1	3	yes	—
181	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
182	C₂⁵	C₂²×D₄₇	1	4	yes	—
183	C₂²×D₄	—	4	6	no	—
184	C₂²×D₄	—	4	6	no	—
185	C₂²×D₄	C₂×Dic₄₇	1	24	yes	—
186	C₂²×D₄	C₂²×D₄₇	1	12	yes	—
187	C₂³×C₄	—	47	6	no	—
188	C₂³×C₄	—	47	2	no	—
189	C₂³×C₄	—	47	4	no	—
190	C₂³×C₄	—	188	6	no	—
191	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
192	#120	C₂×Dic₄₇	1	64	yes	complemented
193	C₂×C₄⋊C₄	—	47	12	no	—
194	C₂×C₄⋊C₄	—	188	6	no	—
195	C₂×C₂²⋊C₄	—	47	24	no	—
196	C₂×C₂²⋊C₄	—	47	24	no	—
197	C₂×C₂²⋊C₄	—	47	12	no	—
198	C₂×C₂²⋊C₄	—	188	12	no	—
199	C₂×C₂²⋊C₄	—	188	6	no	—
200	C₂×C₂²⋊C₄	—	188	12	no	—
201	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	24	yes	complemented
202	C₂×C₂²⋊C₄	C₂²×D₄₇	1	12	yes	—
203	C₂×C₄²	—	47	12	no	—
204	C₂×C₄²	—	188	6	no	—
205	#95	—	47	24	no	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	2	12	no	—
208	C₂⁴	—	2	12	no	—
209	C₂⁴	—	2	6	no	—
210	C₂⁴	—	2	6	no	—
211	C₂⁴	—	2	6	no	—
212	C₂⁴	—	4	48	no	—
213	C₂⁴	—	4	12	no	—
214	C₂⁴	—	4	12	no	—
215	C₂⁴	—	4	48	no	—
216	C₂⁴	—	4	16	no	—
217	C₂⁴	C₂²×Dic₄₇	1	12	yes	—
218	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
219	C₂⁴	C₂²×Dic₄₇	1	16	yes	complemented
220	C₂⁴	C₂³×D₄₇	1	6	yes	—
221	C₂⁴	C₂³×D₄₇	1	2	yes	—
222	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
223	C₂×D₄	—	4	48	no	—
224	C₂×D₄	—	4	12	no	—
225	C₂×D₄	—	4	12	no	—
226	C₂×D₄	C₂²×Dic₄₇	1	48	yes	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	2	6	no	—
229	C₂²×C₄	—	47	4	no	—
230	C₂²×C₄	—	47	12	no	—
231	C₂²×C₄	—	94	12	no	—
232	C₂²×C₄	—	94	12	no	—
233	C₂²×C₄	—	94	12	no	—
234	C₂²×C₄	—	94	12	no	—
235	C₂²×C₄	—	188	12	no	—
236	C₂²×C₄	—	188	16	no	—
237	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
238	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
239	C₄⋊C₄	—	188	12	no	—
240	C₂²⋊C₄	—	188	24	no	—
241	C₂²⋊C₄	—	188	12	no	—
242	C₂²⋊C₄	—	188	24	no	—
243	C₂²⋊C₄	C₂²×Dic₄₇	1	48	yes	complemented
244	C₄²	—	188	12	no	—
245	C₂³	—	2	24	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	2	48	no	—
248	C₂³	—	2	6	no	—
249	C₂³	—	2	12	no	—
250	C₂³	—	2	12	no	—
251	C₂³	—	2	6	no	—
252	C₂³	—	2	6	no	—
253	C₂³	—	2	6	no	—
254	C₂³	—	4	48	no	—
255	C₂³	—	4	128	no	—
256	C₂³	—	4	48	no	—
257	C₂³	—	4	16	no	—
258	C₂³	—	4	12	no	—
259	C₂³	—	4	12	no	—
260	C₂³	—	4	24	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	12	no	—
263	C₂³	C₂³×Dic₄₇	1	12	yes	—
264	C₂³	C₂³×Dic₄₇	1	4	yes	—
265	C₂³	C₂³×Dic₄₇	1	2	yes	—
266	C₂³	#13094069	1	6	yes	—
267	C₂³	#13094069	1	6	yes	—
268	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
269	D₄	—	4	48	no	—
270	C₂×C₄	—	2	24	no	—
271	C₂×C₄	—	2	6	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	94	12	no	—
274	C₂×C₄	—	94	12	no	—
275	C₂×C₄	—	94	12	no	—
276	C₂×C₄	—	94	12	no	—
277	C₂×C₄	—	188	16	no	—
278	C₂×C₄	—	188	24	no	—
279	C₂×C₄	—	188	8	no	—
280	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
281	C₂²	—	2	12	no	—
282	C₂²	—	2	12	no	—
283	C₂²	—	2	24	no	—
284	C₂²	—	4	64	no	—
285	C₂²	—	4	48	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	6	no	—
288	C₂²	—	4	12	no	—
289	C₂²	—	4	6	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	6	no	—
292	C₂²	#15689598	1	12	yes	—
293	C₂²	#15689862	1	12	yes	—
294	C₂²	#15689866	1	4	yes	—
295	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
296	C₂²	C₂×#13094069	1	3	yes	—
297	C₂²	C₂×#13094069	1	3	yes	—
298	C₄	—	2	12	no	—
299	C₄	—	188	12	no	—
300	C₄	—	188	4	no	—
301	C₂	—	4	12	no	—
302	C₂	—	4	4	no	—
303	C₂	#20246529	1	8	yes	direct product factor
304	C₂	C₂×#15689598	1	3	yes	—
305	C₂	C₂×#15689862	1	3	yes	—
306	C₂	C₂×#15689866	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting, prime radical	C₂²×#120×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×#120	D₄₇
hypocenter	C₄₇	C₂×#2375
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#120

Complements

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

N	H ≅ G/N	type
1. #20246529	C₂	direct (N × H)
2. C₂×#15689026	C₂	semidirect
3. C₂×#15689016	C₂	semidirect
4. C₂×#15689003	C₂	semidirect
5. C₂×#15688979	C₂	semidirect
7. C₂⁴×Dic₄₇	C₂²	semidirect
8. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
11. C₂²×C₂²⋊Dic₄₇	C₂²	semidirect
12. C₂²×C₄×Dic₄₇	C₂²	semidirect
13. #15689026	C₂²	semidirect
14. #15689016	C₂²	semidirect
15. #15689003	C₂²	semidirect
16. #15688979	C₂²	semidirect
17. C₂×C₄₇⋊#95	C₂²	semidirect
20. C₂×#120×C₄₇	C₄	semidirect
26. C₂³×Dic₄₇	C₂³	semidirect
28. C₂×C₄⋊Dic₄₇	C₂³	semidirect
34. C₂×C₂²⋊Dic₄₇	C₂³	semidirect
36. C₂×C₄×Dic₄₇	C₂³	semidirect
37. C₄₇⋊#95	C₂³	semidirect
40. C₂⁵×C₄₇	C₂×C₄	semidirect
49. #120×C₄₇	C₂×C₄	semidirect
50. C₂×C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
65. C₂⁴×C₄₇	C₂²×C₄	semidirect
85. C₂²⋊C₄×C₄₇	C₂²×C₄	semidirect
156. C₂×C₄₇	#2375	semidirect
163. C₂×#120	Dic₄₇	semidirect
176. C₄₇	C₂×#2375	semidirect
179. C₂⁵	C₂×Dic₄₇	semidirect
192. #120	C₂×Dic₄₇	semidirect
201. C₂×C₂²⋊C₄	C₂×Dic₄₇	semidirect
219. C₂⁴	C₂²×Dic₄₇	semidirect
243. C₂²⋊C₄	C₂²×Dic₄₇	semidirect
303. C₂	#20246529	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20246529	2
C₂×#15689026	2
C₂×#15689016	2
C₂×#15689003	2
C₂×#15688979	2
C₂²×#120×C₄₇	2
C₂×#2375	47

Subgroups — C2×#20246529 #26378857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20246529 #26378857