Subgroups — C₂×#20241471 #26131136

Order 12032.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
3	#20241471	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
7	C₂×#13093964	—	2	2	no	—
8	C₂×#13093964	—	2	1	no	—
9	C₂×#13093964	C₄	1	4	yes	complemented
10	C₂×#13093964	C₂²	1	2	yes	—
11	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
12	C₂×#13093934	—	2	1	no	—
13	C₂×#13093934	—	2	1	no	—
14	C₂×#13093934	—	2	1	no	—
15	C₂×#13093934	—	2	1	no	—
16	C₂×#13093934	C₄	1	2	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₄₇	—	2	2	no	—
20	C₂×M₃₂×C₄₇	—	2	1	no	—
21	C₂×M₃₂×C₄₇	C₄	1	4	yes	complemented
22	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₄₇	—	2	1	no	—
24	C₂³×Dic₄₇	C₈	1	2	yes	—
25	C₂²×C₄₇⋊C₈	—	2	1	no	—
26	C₂²×C₄₇⋊C₈	—	2	1	no	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	C₈	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₄₇	—	2	1	no	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	#13093964	—	2	4	no	—
37	#13093964	—	2	2	no	—
38	#13093964	C₂×C₄	1	4	yes	complemented
39	C₂×C₄₇⋊C₁₆	—	2	4	no	—
40	C₂×C₄₇⋊C₁₆	—	2	2	no	—
41	#13093934	—	2	2	no	—
42	#13093934	—	2	2	no	—
43	#13093934	—	2	2	no	—
44	#13093934	—	2	2	no	—
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₄₇	D₄	1	2	yes	—
50	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	M₃₂×C₄₇	—	2	4	no	—
57	M₃₂×C₄₇	—	2	2	no	—
58	M₃₂×C₄₇	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₄₇	—	2	4	no	—
60	C₂×C₁₆×C₄₇	—	2	2	no	—
61	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
69	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₄₇	C₄²	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
94	C₄×Dic₄₇	—	2	2	no	—
95	C₄×Dic₄₇	—	2	2	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	2	2	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
103	C₁₆×C₄₇	—	2	4	no	—
104	C₄₇⋊C₁₆	—	2	4	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	C₄×C₈	1	2	yes	—
107	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
109	C₂³×C₄₇	C₄⋊C₈	1	2	yes	—
110	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	#95	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₄₇	#97	1	2	yes	—
115	C₂×C₄×C₄₇	#97	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₈×C₄₇	#95	1	2	yes	—
134	C₈×C₄₇	#95	1	2	yes	—
135	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₄₇⋊C₈	—	2	4	no	—
140	C₂×#1377	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	4	no	—
145	C₂²×C₄₇	#336	1	2	yes	—
146	C₂²×C₄₇	#336	1	2	yes	—
147	C₂²×C₄₇	#349	1	4	yes	—
148	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₄×C₄₇	#336	1	2	yes	—
152	C₄×C₄₇	#336	1	2	yes	—
153	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
155	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₄×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
157	Dic₄₇	—	4	4	no	—
158	C₂²×M₃₂	—	47	1	no	—
159	C₂²×M₃₂	D₄₇	1	1	yes	characteristic
160	C₂²×#97	—	47	1	no	—
161	#1377	—	47	4	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	1	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	#1377	1	4	yes	complemented
166	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#349	1	2	yes	—
168	C₂³×C₈	—	47	1	no	—
169	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
170	C₂²×C₄²	—	47	1	no	—
171	C₂×M₃₂	—	2	2	no	—
172	C₂×M₃₂	—	2	1	no	—
173	C₂×M₃₂	—	47	4	no	—
174	C₂×M₃₂	—	47	2	no	—
175	C₂×M₃₂	—	94	1	no	—
176	C₂×M₃₂	—	94	2	no	—
177	C₂×M₃₂	Dic₄₇	1	4	yes	complemented
178	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
179	C₂²×C₁₆	—	2	1	no	—
180	C₂²×C₁₆	—	94	1	no	—
181	C₂×#97	—	47	2	no	—
182	C₂×#97	—	47	2	no	—
183	C₂×#97	—	94	1	no	—
184	C₂×#97	—	94	1	no	—
185	C₂×#97	—	94	1	no	—
186	C₂×#97	—	94	1	no	—
187	C₄₇	C₂×#1377	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	47	2	no	—
189	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	—	94	1	no	—
194	C₂²×C₈	—	94	1	no	—
195	C₂²×C₈	Dic₉₄	1	2	yes	—
196	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₄₇	1	2	yes	—
198	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
199	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
204	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
205	C₂×C₄²	—	47	2	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	94	1	no	—
208	C₂×C₄²	—	94	1	no	—
209	C₂×C₄²	—	94	1	no	—
210	C₂×C₄²	—	94	1	no	—
211	M₃₂	—	2	4	no	—
212	M₃₂	—	2	2	no	—
213	M₃₂	—	47	4	no	—
214	M₃₂	—	94	4	no	—
215	M₃₂	—	94	2	no	—
216	M₃₂	C₂×Dic₄₇	1	4	yes	complemented
217	C₂×C₁₆	—	2	4	no	—
218	C₂×C₁₆	—	2	2	no	—
219	C₂×C₁₆	—	94	4	no	—
220	C₂×C₁₆	—	94	2	no	—
221	#97	—	94	2	no	—
222	#97	—	94	2	no	—
223	#97	—	94	2	no	—
224	#97	—	94	2	no	—
225	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	—	47	4	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	188	4	no	—
230	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₈×D₄₇	1	2	yes	—
232	C₂²×C₄	#10625324	1	2	yes	—
233	C₂²×C₄	#10625324	1	1	yes	characteristic
234	C₂²×C₄	#10625324	1	1	yes	characteristic
235	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
236	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
237	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
246	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
247	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
248	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
249	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
250	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
251	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
252	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
253	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
254	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
255	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
256	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
257	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
259	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₄²	—	94	2	no	—
264	C₄²	—	94	2	no	—
265	C₄²	—	94	2	no	—
266	C₄²	—	94	2	no	—
267	C₁₆	—	2	4	no	—
268	C₁₆	—	94	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₈×Dic₄₇	1	2	yes	—
271	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
272	C₂³	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
273	C₂³	Dic₄₇⋊C₈	1	2	yes	—
274	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
275	C₂×C₄	—	94	4	no	—
276	C₂×C₄	—	188	4	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	C₄₇⋊#95	1	2	yes	—
280	C₂×C₄	C₄₇⋊#95	1	2	yes	—
281	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
282	C₂×C₄	#13093934	1	2	yes	—
283	C₂×C₄	#13093934	1	2	yes	—
284	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
285	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
286	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
287	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
288	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
289	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
290	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
291	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
296	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
297	C₈	—	94	4	no	—
298	C₈	C₄₇⋊#95	1	2	yes	—
299	C₈	C₄₇⋊#95	1	2	yes	—
300	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
301	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
302	C₈	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
303	C₈	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	4	no	—
308	C₂²	#15688877	1	2	yes	—
309	C₂²	#15688877	1	2	yes	—
310	C₂²	#15688904	1	4	yes	—
311	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
312	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
313	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
314	C₄	—	188	4	no	—
315	C₄	#15688877	1	2	yes	—
316	C₄	#15688877	1	2	yes	—
317	C₄	C₂×C₄₇⋊#95	1	1	yes	characteristic
318	C₄	C₂×#13093934	1	1	yes	characteristic
319	C₄	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
320	C₄	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	—	2	1	no	—
324	C₂	#20241471	1	4	yes	direct product factor
325	C₂	C₂×#15688877	1	1	yes	characteristic
326	C₂	C₂×#15688904	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×M₃₂×C₄₇	C₂
socle	C₂³×C₄₇	C₂²⋊C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×M₃₂	D₄₇
hypocenter	C₄₇	C₂×#1377
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20241471	C₂	direct (N × H)
9. C₂×#13093964	C₄	semidirect
21. C₂×M₃₂×C₄₇	C₄	semidirect
38. #13093964	C₂×C₄	semidirect
58. M₃₂×C₄₇	C₂×C₄	semidirect
165. C₂×C₄₇	#1377	semidirect
177. C₂×M₃₂	Dic₄₇	semidirect
187. C₄₇	C₂×#1377	semidirect
216. M₃₂	C₂×Dic₄₇	semidirect
324. C₂	#20241471	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×#13093964	2
C₂²×#13093934	2
#20241471	2
C₂²×M₃₂×C₄₇	2
C₂×#1377	47

Subgroups — C2×#20241471 #26131136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241471 #26131136