Subgroups — C₂×#20243426 #26215892

Order 12032.

1566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	2	yes	maximal, complemented
2	C₂×#15689087	C₂	1	2	yes	maximal
3	#20243426	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688888	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	4	yes	—
7	C₂×#13093960	—	2	8	no	—
8	C₂×#13093960	C₂²	1	8	yes	complemented
9	#15689087	—	2	4	no	—
10	#15689087	C₂²	1	8	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
12	#15688888	C₂²	1	8	yes	complemented
13	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	—	2	8	no	—
15	C₂³×C₈×C₄₇	C₄	1	4	yes	—
16	C₂³×C₈×C₄₇	C₄	1	4	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₄₇	C₂²	1	4	yes	—
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	4	no	—
21	C₂×#10625329	—	2	8	no	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	C₂³	1	8	yes	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	#13093960	—	2	16	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
27	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	—	2	4	no	—
29	C₂³×C₄×C₄₇	—	2	4	no	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	4	no	—
39	C₂²×C₈×C₄₇	—	2	16	no	—
40	C₂²×C₈×C₄₇	—	2	16	no	—
41	C₂²×C₈×C₄₇	—	2	16	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	16	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	D₄	1	8	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂⁴×C₄₇	—	2	8	no	—
51	C₂⁴×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
53	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂⁴×C₄₇	C₂×Q₈	1	2	yes	—
55	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	8	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	8	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	#41	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	32	no	—
85	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
86	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
90	#10625329	—	2	8	no	—
91	#10625329	—	2	8	no	—
92	#10625329	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	2	8	no	—
94	C₂×C₄₇⋊C₈	—	4	8	no	—
95	C₂³×C₄₇	—	2	32	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	4	no	—
99	C₂³×C₄₇	—	2	4	no	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	—	2	8	no	—
102	C₂³×C₄₇	—	2	8	no	—
103	C₂³×C₄₇	#108	1	8	yes	complemented
104	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#117	1	2	yes	—
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	#121	1	4	yes	—
125	C₂×C₄×C₄₇	#121	1	4	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
131	C₈×C₄₇	—	2	4	no	—
132	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
133	C₄₇⋊C₈	—	4	8	no	—
134	C₂×#1810	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	16	no	—
136	C₂²×C₄₇	—	2	16	no	—
137	C₂²×C₄₇	—	2	16	no	—
138	C₂²×C₄₇	—	2	8	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	#380	1	2	yes	—
144	C₂²×C₄₇	#380	1	2	yes	—
145	C₂²×C₄₇	C₂×#108	1	4	yes	—
146	C₂²×C₄₇	C₂×#108	1	8	yes	complemented
147	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	#380	1	2	yes	—
151	C₄×C₄₇	#380	1	2	yes	—
152	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#121	1	2	yes	—
155	C₂⁴×C₈	D₄₇	1	1	yes	characteristic
156	C₂²×#108	—	47	2	no	—
157	C₂×#407	—	47	2	no	—
158	#1810	—	47	8	no	—
159	C₂×#341	—	47	2	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	8	no	—
163	C₂×C₄₇	#1810	1	4	yes	complemented
164	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
165	C₂×C₄₇	C₂²×#108	1	2	yes	—
166	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
167	C₂²×M₁₆	—	47	4	no	—
168	C₂³×C₈	—	2	8	no	—
169	C₂³×C₈	Dic₄₇	1	4	yes	—
170	C₂³×C₈	Dic₄₇	1	4	yes	—
171	C₂³×C₈	C₂×D₄₇	1	4	yes	—
172	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#108	—	47	8	no	—
175	C₂×#108	—	94	8	no	—
176	#407	—	47	8	no	—
177	#407	—	94	4	no	—
178	C₂×C₂²⋊C₈	—	94	4	no	—
179	#341	—	47	8	no	—
180	C₄₇	C₂×#1810	1	1	yes	characteristic, complemented, Sylow
181	C₂⁵	Dic₉₄	1	1	yes	characteristic
182	C₂³×C₄	—	2	4	no	—
183	C₂³×C₄	—	2	4	no	—
184	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
185	C₂³×C₄	Dic₉₄	1	2	yes	—
186	C₂³×C₄	D₁₈₈	1	2	yes	—
187	C₂³×C₄	D₁₈₈	1	2	yes	—
188	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
189	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
190	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
191	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	C₂×M₁₆	—	47	8	no	—
193	C₂×M₁₆	—	94	4	no	—
194	C₂×M₁₆	—	94	8	no	—
195	C₂×M₁₆	—	94	8	no	—
196	C₂²×C₈	—	2	4	no	—
197	C₂²×C₈	—	2	16	no	—
198	C₂²×C₈	—	2	16	no	—
199	C₂²×C₈	—	2	16	no	—
200	C₂²×C₈	—	94	4	no	—
201	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
202	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
203	C₂²×C₈	C₂×Dic₄₇	1	16	yes	—
204	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
205	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
206	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
207	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
208	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
209	#108	—	94	16	no	—
210	C₂²⋊C₈	—	94	16	no	—
211	C₂⁴	—	2	8	no	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
214	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
215	C₂⁴	C₂×Dic₉₄	1	2	yes	—
216	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
230	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
231	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
232	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
234	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	#10625350	1	4	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
241	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
242	M₁₆	—	94	8	no	—
243	M₁₆	—	94	8	no	—
244	M₁₆	—	188	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	32	no	—
249	C₂×C₈	—	94	8	no	—
250	C₂×C₈	—	188	8	no	—
251	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
252	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
253	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
254	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
255	C₂×C₈	C₂×C₂²⋊D₄₇	1	8	yes	—
256	C₂³	—	2	32	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	2	8	no	—
264	C₂³	#13093960	1	8	yes	complemented
265	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
266	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
267	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
268	C₂³	#13094003	1	4	yes	—
269	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
279	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
283	C₂×C₄	#13093983	1	2	yes	—
284	C₂×C₄	#13093983	1	2	yes	—
285	C₂×C₄	#13093998	1	4	yes	—
286	C₂×C₄	#13093998	1	4	yes	—
287	C₂×C₄	#13094003	1	4	yes	—
288	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
290	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
291	C₂×C₄	C₂×#10625350	1	2	yes	—
292	C₈	—	2	4	no	—
293	C₈	—	188	8	no	—
294	C₈	C₂×C₂²⋊Dic₄₇	1	8	yes	—
295	C₂²	—	2	16	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	16	no	—
302	C₂²	—	2	8	no	—
303	C₂²	#15688983	1	2	yes	—
304	C₂²	#15688983	1	2	yes	—
305	C₂²	C₂×#13093960	1	4	yes	—
306	C₂²	C₂×#13093960	1	8	yes	complemented
307	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
308	C₂²	C₂×#13094003	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	#15688983	1	2	yes	—
311	C₄	#15688983	1	2	yes	—
312	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
313	C₄	C₂×#13093983	1	1	yes	characteristic
314	C₄	C₂×#13093998	1	2	yes	—
315	C₂	—	2	2	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	8	no	—
318	C₂	#20243426	1	4	yes	direct product factor
319	C₂	C₂×#15688983	1	1	yes	characteristic
320	C₂	C₂²×#13093960	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂⁴×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁵×C₄₇	Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂⁴×C₈	D₄₇
hypocenter	C₄₇	C₂×#1810
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093960	C₂	semidirect
3. #20243426	C₂	direct (N × H)
4. C₂×#15688888	C₂	semidirect
8. C₂×#13093960	C₂²	semidirect
12. #15688888	C₂²	semidirect
103. C₂³×C₄₇	#108	semidirect
146. C₂²×C₄₇	C₂×#108	semidirect
163. C₂×C₄₇	#1810	semidirect
180. C₄₇	C₂×#1810	semidirect
264. C₂³	#13093960	semidirect
306. C₂²	C₂×#13093960	semidirect
318. C₂	#20243426	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13093960	2
C₂×#15689087	2
#20243426	2
C₂×#15688888	2
C₂⁴×C₈×C₄₇	2
C₂×#1810	47

Subgroups — C2×#20243426 #26215892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243426 #26215892