Subgroups — C₂×#20244260 #26220109

Order 12032.

1574 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093948	C₂	1	2	yes	maximal
3	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
4	C₂×#15689030	C₂	1	2	yes	maximal
5	#20244260	C₂	1	8	yes	maximal, direct product factor
6	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093956	—	2	4	no	—
11	C₂×#13093956	C₂²	1	4	yes	—
12	C₂×#13093948	—	2	8	no	—
13	C₂×#13093948	C₂²	1	8	yes	—
14	C₂×#13093934	—	2	4	no	—
15	C₂×#13093934	C₂²	1	4	yes	—
16	#15689030	—	2	4	no	—
17	#15689030	C₂²	1	8	yes	—
18	C₂×C₄₇⋊#95	—	2	4	no	—
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₄₇	—	2	4	no	—
21	C₂²×Q₁₆×C₄₇	—	2	4	no	—
22	C₂²×Q₁₆×C₄₇	C₄	1	8	yes	complemented
23	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×Dic₄₇	—	2	2	no	—
26	C₂³×Dic₄₇	D₄	1	2	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093956	—	2	8	no	—
40	#13093948	—	2	16	no	—
41	#13093934	—	2	8	no	—
42	C₄₇⋊#95	—	2	16	no	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×Q₈×C₄₇	—	2	4	no	—
45	C₂²×Q₈×C₄₇	—	2	4	no	—
46	C₂²×Q₈×C₄₇	—	2	4	no	—
47	C₂²×Q₈×C₄₇	—	2	4	no	—
48	C₂²×Q₈×C₄₇	—	2	4	no	—
49	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
50	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₄₇	—	2	2	no	—
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×Q₁₆×C₄₇	—	2	32	no	—
54	C₂×Q₁₆×C₄₇	—	2	8	no	—
55	C₂×Q₁₆×C₄₇	—	2	8	no	—
56	C₂×Q₁₆×C₄₇	C₂×C₄	1	16	yes	complemented
57	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₄₇	D₄	1	4	yes	—
59	C₂²×C₈×C₄₇	D₄	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	—	2	16	no	—
65	C₂×Q₈×C₄₇	—	2	16	no	—
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	2	4	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	4	4	no	—
71	C₂×Q₈×C₄₇	—	4	8	no	—
72	C₂×Q₈×C₄₇	—	4	8	no	—
73	C₂×Q₈×C₄₇	—	4	4	no	—
74	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₇	—	2	8	no	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	4	4	no	—
78	C₂²×C₄×C₄₇	—	4	4	no	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	4	no	—
88	C₂²×Dic₄₇	—	2	8	no	—
89	C₂²×Dic₄₇	—	2	4	no	—
90	C₂²×Dic₄₇	—	4	4	no	—
91	C₂²×Dic₄₇	—	4	8	no	—
92	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
93	Q₁₆×C₄₇	—	2	32	no	—
94	C₂×C₈×C₄₇	C₂²⋊C₄	1	16	yes	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	4	4	no	—
101	C₄×Dic₄₇	—	2	4	no	—
102	C₄×Dic₄₇	—	2	2	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	8	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
108	C₂³×C₄₇	C₄×D₄	1	2	yes	—
109	C₂³×C₄₇	#121	1	2	yes	—
110	C₂³×C₄₇	#127	1	2	yes	—
111	C₂³×C₄₇	#137	1	8	yes	—
112	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₄₇	—	2	8	no	—
114	Q₈×C₄₇	—	2	8	no	—
115	Q₈×C₄₇	—	4	4	no	—
116	Q₈×C₄₇	—	4	4	no	—
117	Q₈×C₄₇	—	4	16	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	4	16	no	—
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
124	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₇	#121	1	4	yes	—
126	C₂×C₄×C₄₇	#121	1	2	yes	—
127	C₂×C₄×C₄₇	#124	1	4	yes	—
128	C₂×C₄×C₄₇	#127	1	2	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	2	8	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	—	4	8	no	—
140	C₂×Dic₄₇	—	8	8	no	—
141	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
142	C₄₇⋊C₈	—	2	4	no	—
143	C₂×#1971	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	#390	1	4	yes	—
145	C₂²×C₄₇	#441	1	8	yes	—
146	C₂²×C₄₇	#470	1	8	yes	—
147	C₂²×C₄₇	#497	1	8	yes	—
148	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#137	1	4	yes	—
152	C₄×C₄₇	—	4	8	no	—
153	C₄×C₄₇	#390	1	4	yes	—
154	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
157	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
158	Dic₄₇	—	4	4	no	—
159	Dic₄₇	—	8	4	no	—
160	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
161	C₂²×#106	—	47	1	no	—
162	C₂²×#103	—	47	2	no	—
163	C₂²×#97	—	47	1	no	—
164	C₂×#391	—	47	2	no	—
165	#1971	—	47	8	no	—
166	C₂×C₄₇	#1971	1	8	yes	complemented
167	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#441	1	2	yes	—
169	C₂×C₄₇	C₂×#470	1	2	yes	—
170	C₂×C₄₇	C₂×#497	1	2	yes	—
171	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
172	C₂²×Q₁₆	—	2	4	no	—
173	C₂²×Q₁₆	—	2	4	no	—
174	C₂²×Q₁₆	Dic₄₇	1	8	yes	complemented
175	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
176	C₂³×C₈	—	47	1	no	—
177	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	—	47	2	no	—
179	C₂²×C₄²	—	47	1	no	—
180	C₂×#106	—	47	4	no	—
181	C₂×#106	—	94	4	no	—
182	C₂×#103	—	47	8	no	—
183	C₂×#103	—	94	8	no	—
184	C₂×#97	—	47	4	no	—
185	C₂×#97	—	94	4	no	—
186	#391	—	47	8	no	—
187	#391	—	94	4	no	—
188	C₂×#95	—	94	4	no	—
189	C₄₇	C₂×#1971	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	2	2	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	2	4	no	—
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	—	2	4	no	—
196	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
197	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
198	C₂³×C₄	—	2	2	no	—
199	C₂³×C₄	—	47	2	no	—
200	C₂³×C₄	—	94	2	no	—
201	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₂×Q₁₆	—	2	32	no	—
203	C₂×Q₁₆	—	2	8	no	—
204	C₂×Q₁₆	—	2	8	no	—
205	C₂×Q₁₆	C₂×Dic₄₇	1	16	yes	complemented
206	C₂²×C₈	—	47	4	no	—
207	C₂²×C₈	—	47	2	no	—
208	C₂²×C₈	—	94	2	no	—
209	C₂²×C₈	—	94	2	no	—
210	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
211	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
212	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
213	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
214	C₂×C₄⋊C₄	—	47	4	no	—
215	C₂×C₄⋊C₄	—	94	4	no	—
216	C₂×C₄⋊C₄	—	94	4	no	—
217	C₂×C₄⋊C₄	—	94	2	no	—
218	C₂×C₄²	—	47	2	no	—
219	C₂×C₄²	—	47	2	no	—
220	C₂×C₄²	—	94	2	no	—
221	C₂×C₄²	—	94	2	no	—
222	#106	—	94	8	no	—
223	#103	—	94	16	no	—
224	#97	—	94	8	no	—
225	#95	—	94	16	no	—
226	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	16	no	—
230	C₂×Q₈	—	2	16	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	8	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	8	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	C₂²×Dic₄₇	1	8	yes	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	47	4	no	—
245	C₂²×C₄	—	94	4	no	—
246	C₂²×C₄	—	94	8	no	—
247	C₂²×C₄	—	94	4	no	—
248	C₂²×C₄	—	188	4	no	—
249	C₂²×C₄	—	188	8	no	—
250	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	D₄×D₄₇	1	2	yes	—
252	C₂²×C₄	#10625352	1	2	yes	—
253	C₂²×C₄	#10625352	1	2	yes	—
254	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
257	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
258	Q₁₆	—	2	32	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	94	8	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	C₂²⋊Dic₄₇	1	16	yes	—
264	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
265	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
266	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
267	C₄⋊C₄	—	94	4	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₄⋊C₄	—	188	4	no	—
270	C₄²	—	94	4	no	—
271	C₄²	—	94	2	no	—
272	C₄²	—	94	2	no	—
273	C₂³	D₄×Dic₄₇	1	2	yes	—
274	C₂³	#13093995	1	2	yes	—
275	C₂³	#13094020	1	2	yes	—
276	C₂³	#13094056	1	8	yes	—
277	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	2	8	no	—
280	Q₈	—	4	16	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	94	8	no	—
289	C₂×C₄	—	376	8	no	—
290	C₂×C₄	—	188	8	no	—
291	C₂×C₄	—	188	4	no	—
292	C₂×C₄	—	188	8	no	—
293	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
294	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
295	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
296	C₂×C₄	#13093995	1	2	yes	—
297	C₂×C₄	#13093995	1	4	yes	—
298	C₂×C₄	#13094011	1	4	yes	—
299	C₂×C₄	#13094020	1	2	yes	—
300	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
305	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
306	C₈	—	94	4	no	—
307	C₈	C₂×C₂²⋊Dic₄₇	1	8	yes	—
308	C₂²	#15689026	1	4	yes	—
309	C₂²	#15689230	1	8	yes	—
310	C₂²	#15689378	1	8	yes	—
311	C₂²	#15689516	1	8	yes	—
312	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
313	C₂²	C₂×#13093995	1	1	yes	characteristic
314	C₂²	C₂×#13094020	1	1	yes	characteristic
315	C₂²	C₂×#13094056	1	4	yes	—
316	C₄	—	4	8	no	—
317	C₄	—	376	4	no	—
318	C₄	—	188	4	no	—
319	C₄	#15689026	1	4	yes	—
320	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
321	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
322	C₄	C₂×#13093995	1	1	yes	characteristic
323	C₄	C₂×#13094011	1	1	yes	characteristic
324	C₂	#20244260	1	8	yes	direct product factor
325	C₂	C₂×#15689026	1	1	yes	characteristic
326	C₂	C₂×#15689230	1	2	yes	—
327	C₂	C₂×#15689378	1	2	yes	—
328	C₂	C₂×#15689516	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#1971
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
5. #20244260	C₂	direct (N × H)
22. C₂²×Q₁₆×C₄₇	C₄	semidirect
56. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
166. C₂×C₄₇	#1971	semidirect
174. C₂²×Q₁₆	Dic₄₇	semidirect
189. C₄₇	C₂×#1971	semidirect
205. C₂×Q₁₆	C₂×Dic₄₇	semidirect
324. C₂	#20244260	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×#13093956	2
C₂²×#13093948	2
C₂²×#13093934	2
C₂×#15689030	2
#20244260	2
C₂³×Q₁₆×C₄₇	2
C₂×#1971	47

Subgroups — C2×#20244260 #26220109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244260 #26220109