Subgroups — C₂×#20243166 #26213862

Order 12032.

1918 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	C₂⁴×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13093935	C₂	1	2	yes	maximal, complemented
4	#20243166	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688878	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	—	2	8	no	—
8	C₂³×C₄₇⋊C₈	C₄	1	8	yes	complemented
9	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
12	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
15	C₂×#13093935	—	2	8	no	—
16	C₂×#13093935	C₂²	1	8	yes	complemented
17	#15688878	C₂²	1	8	yes	complemented
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₇	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₂×#95×C₄₇	C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	—	2	32	no	—
28	C₂²×C₄₇⋊C₈	—	2	16	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
33	C₂²×C₄₇⋊C₈	D₄	1	8	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
38	#13093935	—	2	16	no	—
39	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	—	2	4	no	—
41	C₂³×C₄×C₄₇	—	2	4	no	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
50	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₄²×C₄₇	—	2	4	no	—
52	C₂×C₄²×C₄₇	—	2	4	no	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
55	#95×C₄₇	C₂×C₄	1	8	yes	—
56	C₂⁴×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	—	2	4	no	—
58	C₂⁴×C₄₇	C₄²	1	4	yes	—
59	C₂⁴×C₄₇	M₁₆	1	8	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
61	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	8	no	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	2	16	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	8	no	—
70	C₂²×C₄×C₄₇	—	2	8	no	—
71	C₂²×C₄×C₄₇	—	2	8	no	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²⋊C₄×C₄₇	—	2	16	no	—
85	C₄²×C₄₇	—	2	8	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	8	no	—
89	C₂×C₄₇⋊C₈	—	2	16	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	32	no	—
94	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	16	yes	complemented
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
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	—	2	32	no	—
104	C₂³×C₄₇	#97	1	16	yes	complemented
105	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
106	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
107	C₂³×C₄₇	C₄×D₄	1	4	yes	—
108	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
109	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
110	C₂³×C₄₇	#131	1	4	yes	—
111	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	16	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	4	16	no	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₇	#117	1	4	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	2	8	no	—
134	C₄₇⋊C₈	C₂×C₂²⋊C₄	1	8	yes	complemented
135	C₂×#1752	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	4	no	—
137	C₂²×C₄₇	—	2	16	no	—
138	C₂²×C₄₇	—	2	32	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
141	C₂²×C₄₇	C₂×#97	1	4	yes	—
142	C₂²×C₄₇	C₂×#97	1	8	yes	complemented
143	C₂²×C₄₇	#405	1	4	yes	—
144	C₂²×C₄₇	#435	1	16	yes	—
145	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
147	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
148	C₂²×C₄₇	C₂×#131	1	2	yes	—
149	C₄×C₄₇	—	2	4	no	—
150	C₄×C₄₇	—	4	8	no	—
151	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
152	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
155	C₂⁴×C₈	—	47	1	no	—
156	C₂²×C₂²⋊C₈	—	47	1	no	—
157	C₂²×#97	—	47	2	no	—
158	C₂×C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
159	#1752	—	47	8	no	—
160	C₂×#336	—	47	2	no	—
161	C₂×C₄₇	—	2	8	no	—
162	C₂×C₄₇	#1752	1	8	yes	complemented
163	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#435	1	4	yes	—
167	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
168	C₂³×C₈	—	47	1	no	—
169	C₂³×C₈	—	47	8	no	—
170	C₂³×C₈	—	47	4	no	—
171	C₂³×C₈	—	47	2	no	—
172	C₂³×C₈	—	47	1	no	—
173	C₂³×C₈	—	94	8	no	—
174	C₂²×C₂²⋊C₄	Dic₄₇	1	2	yes	—
175	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊C₈	—	47	8	no	—
177	C₂×C₂²⋊C₈	—	47	4	no	—
178	C₂×#97	—	47	8	no	—
179	C₂×#97	—	94	8	no	—
180	C₄×C₂²⋊C₄	Dic₄₇	1	4	yes	—
181	C₄×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
182	C₂×#95	Dic₄₇	1	2	yes	—
183	#336	—	47	8	no	—
184	C₄₇	C₂×#1752	1	1	yes	characteristic, complemented, Sylow
185	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	—	2	4	no	—
188	C₂³×C₄	C₄×D₄₇	1	4	yes	—
189	C₂³×C₄	C₄×D₄₇	1	4	yes	—
190	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
191	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
194	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	—	47	4	no	—
196	C₂²×C₈	—	47	8	no	—
197	C₂²×C₈	—	47	2	no	—
198	C₂²×C₈	—	47	4	no	—
199	C₂²×C₈	—	47	16	no	—
200	C₂²×C₈	—	47	8	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	—	47	4	no	—
203	C₂²×C₈	—	94	32	no	—
204	C₂²×C₈	—	94	16	no	—
205	C₂²×C₈	—	94	4	no	—
206	C₂²×C₈	—	94	4	no	—
207	C₂²×C₈	—	94	4	no	—
208	C₂×C₂²⋊C₄	—	2	8	no	—
209	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	—	2	4	no	—
212	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
213	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
214	C₂²⋊C₈	—	47	16	no	—
215	#97	—	94	16	no	—
216	#95	C₂×Dic₄₇	1	8	yes	—
217	C₂⁴	—	2	4	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	#10625329	1	8	yes	—
220	C₂⁴	C₄×Dic₄₇	1	4	yes	—
221	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
222	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	16	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
235	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	D₄×D₄₇	1	4	yes	—
240	C₂²×C₄	#10625352	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
245	C₂×C₈	—	47	16	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	8	no	—
248	C₂×C₈	—	47	8	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	94	32	no	—
251	C₂×C₈	—	94	8	no	—
252	C₂×C₈	—	94	16	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	94	4	no	—
255	C₂×C₈	—	94	4	no	—
256	C₂²⋊C₄	—	2	16	no	—
257	C₄²	—	2	8	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₂³	—	2	8	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	32	no	—
265	C₂³	#13093935	1	16	yes	complemented
266	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
267	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
268	C₂³	D₄×Dic₄₇	1	4	yes	—
269	C₂³	C₂×#10625329	1	8	yes	—
270	C₂³	C₂×#10625329	1	4	yes	—
271	C₂³	#13094031	1	4	yes	—
272	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	16	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	4	16	no	—
283	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
284	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
285	C₂×C₄	C₂²⋊C₄×D₄₇	1	4	yes	—
286	C₂×C₄	#13093980	1	4	yes	—
287	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	8	yes	—
288	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
289	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
292	C₂×C₄	C₂×#10625352	1	2	yes	—
293	C₈	—	47	8	no	—
294	C₈	—	94	4	no	—
295	C₈	—	94	8	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	16	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	32	no	—
300	C₂²	C₂²⋊C₄×Dic₄₇	1	4	yes	—
301	C₂²	C₂×#13093935	1	4	yes	—
302	C₂²	C₂×#13093935	1	8	yes	complemented
303	C₂²	#15689079	1	4	yes	—
304	C₂²	#15689195	1	16	yes	—
305	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
306	C₂²	C₂×D₄×Dic₄₇	1	2	yes	—
307	C₂²	C₂²×#10625329	1	2	yes	—
308	C₂²	C₂×#13094031	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	4	8	no	—
311	C₄	C₂²⋊C₄×Dic₄₇	1	4	yes	—
312	C₄	C₂×C₂²⋊C₄×D₄₇	1	1	yes	characteristic
313	C₄	C₂×#13093980	1	1	yes	characteristic
314	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
315	C₂	—	2	8	no	—
316	C₂	#20243166	1	8	yes	direct product factor
317	C₂	C₂×C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
318	C₂	C₂²×#13093935	1	1	yes	characteristic
319	C₂	C₂×#15689079	1	1	yes	characteristic
320	C₂	C₂×#15689195	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×C₂²⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	C₂×#1752
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂²×#13093935	C₂	semidirect
4. #20243166	C₂	direct (N × H)
5. C₂×#15688878	C₂	semidirect
8. C₂³×C₄₇⋊C₈	C₄	semidirect
13. C₂×C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
16. C₂×#13093935	C₂²	semidirect
17. #15688878	C₂²	semidirect
32. C₂²×C₄₇⋊C₈	C₂×C₄	semidirect
37. C₂²⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
94. C₂×C₄₇⋊C₈	C₂²⋊C₄	semidirect
104. C₂³×C₄₇	#97	semidirect
134. C₄₇⋊C₈	C₂×C₂²⋊C₄	semidirect
142. C₂²×C₄₇	C₂×#97	semidirect
162. C₂×C₄₇	#1752	semidirect
184. C₄₇	C₂×#1752	semidirect
265. C₂³	#13093935	semidirect
302. C₂²	C₂×#13093935	semidirect
316. C₂	#20243166	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄₇⋊C₈	2
C₂²×C₂²⋊(C₄₇⋊C₈)	2
C₂²×#13093935	2
#20243166	2
C₂×#15688878	2
C₂×C₄×C₂²⋊C₄×C₄₇	2
C₂×#1752	47

Subgroups — C2×#20243166 #26213862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243166 #26213862