Subgroups — C₂×#20243150 #26213800

Order 12032.

2286 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂²×#13093935	C₂	1	2	yes	maximal, complemented
3	#20243150	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688878	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	16	yes	complemented
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
9	C₂×#13093935	—	2	8	no	—
10	C₂×#13093935	C₂²	1	8	yes	complemented
11	#15688878	C₂²	1	8	yes	complemented
12	C₂⁴×C₄×C₄₇	C₄	1	2	yes	—
13	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	—	2	8	no	—
15	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	8	no	—
21	C₂²×C₄₇⋊C₈	—	2	8	no	—
22	C₂²×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
23	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
24	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	32	yes	complemented
25	#13093935	—	2	16	no	—
26	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	—	2	8	no	—
28	C₂³×C₄×C₄₇	—	2	8	no	—
29	C₂³×C₄×C₄₇	—	2	8	no	—
30	C₂³×C₄×C₄₇	—	2	4	no	—
31	C₂³×C₄×C₄₇	—	2	4	no	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	—	2	8	no	—
41	C₂×C₄²×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	—	2	16	no	—
43	C₂×C₄²×C₄₇	—	2	16	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	8	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
50	C₂⁴×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₄²	1	4	yes	—
54	C₂⁴×C₄₇	M₁₆	1	8	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	64	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	16	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	16	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	32	no	—
69	C₂²×C₄×C₄₇	—	2	16	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	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	M₁₆	1	8	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₄₇	#41	1	4	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₄²×C₄₇	—	2	16	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₄₇	C₂×D₄	1	8	yes	—
90	C₂×C₄₇⋊C₈	—	2	32	no	—
91	C₂×C₄₇⋊C₈	—	2	8	no	—
92	C₂×C₄₇⋊C₈	—	2	8	no	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	16	yes	complemented
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	2	8	no	—
98	C₂³×C₄₇	—	2	8	no	—
99	C₂³×C₄₇	—	2	16	no	—
100	C₂³×C₄₇	—	2	16	no	—
101	C₂³×C₄₇	#97	1	16	yes	complemented
102	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
103	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂³×C₄₇	#117	1	2	yes	—
106	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
107	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
108	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
109	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	64	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	—	2	16	no	—
119	C₂×C₄×C₄₇	—	2	16	no	—
120	C₂×C₄×C₄₇	—	2	16	no	—
121	C₂×C₄×C₄₇	—	2	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	8	yes	—
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	#117	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×D₄	1	16	yes	—
128	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₄₇⋊C₈	—	2	16	no	—
133	C₂×#1749	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	2	8	no	—
136	C₂²×C₄₇	—	2	8	no	—
137	C₂²×C₄₇	—	2	32	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
140	C₂²×C₄₇	C₂×#97	1	4	yes	—
141	C₂²×C₄₇	C₂×#97	1	8	yes	complemented
142	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
143	C₂²×C₄₇	#407	1	8	yes	—
144	C₂²×C₄₇	#432	1	8	yes	—
145	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
148	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
149	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
150	C₄×C₄₇	—	2	8	no	—
151	C₄×C₄₇	—	2	8	no	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
154	C₄×C₄₇	C₂×C₄×D₄	1	4	yes	—
155	C₂³×C₄²	D₄₇	1	1	yes	characteristic
156	C₂²×C₂²⋊C₈	—	47	2	no	—
157	C₂²×#97	—	47	2	no	—
158	#1749	—	47	8	no	—
159	C₂×#336	—	47	2	no	—
160	C₂×C₄₇	—	2	8	no	—
161	C₂×C₄₇	#1749	1	8	yes	complemented
162	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
164	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#407	1	2	yes	—
166	C₂×C₄₇	C₂×#432	1	2	yes	—
167	C₂⁴×C₄	Dic₄₇	1	2	yes	—
168	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
169	C₂³×C₈	—	47	4	no	—
170	C₂²×C₄²	—	2	8	no	—
171	C₂²×C₄²	Dic₄₇	1	2	yes	—
172	C₂²×C₄²	Dic₄₇	1	4	yes	—
173	C₂²×C₄²	C₂×D₄₇	1	4	yes	—
174	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	C₂×C₂²⋊C₈	—	47	16	no	—
177	C₂×C₂²⋊C₈	—	47	8	no	—
178	C₂×#97	—	47	8	no	—
179	C₂×#97	—	94	8	no	—
180	#336	—	47	8	no	—
181	C₄₇	C₂×#1749	1	1	yes	characteristic, complemented, Sylow
182	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂³×C₄	—	2	8	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	2	8	no	—
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	—	2	4	no	—
188	C₂³×C₄	C₄×D₄₇	1	8	yes	—
189	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
190	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
191	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	47	16	no	—
197	C₂²×C₈	—	47	8	no	—
198	C₂²×C₈	—	94	8	no	—
199	C₂²×C₈	—	94	8	no	—
200	C₂×C₄²	—	2	8	no	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	—	2	16	no	—
203	C₂×C₄²	—	2	16	no	—
204	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
205	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
206	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	8	yes	—
208	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
210	C₂²⋊C₈	—	47	32	no	—
211	#97	—	94	16	no	—
212	C₂⁴	—	2	2	no	—
213	C₂⁴	—	2	2	no	—
214	C₂⁴	—	2	4	no	—
215	C₂⁴	#10625329	1	8	yes	—
216	C₂⁴	C₄×Dic₄₇	1	4	yes	—
217	C₂⁴	C₂²×Dic₄₇	1	2	yes	—
218	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	64	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	16	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	32	no	—
231	C₂²×C₄	—	2	16	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	#10625329	1	8	yes	—
235	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
236	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
237	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
238	C₂²×C₄	C₂×C₄×D₄₇	1	8	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
240	C₂²×C₄	#10625350	1	4	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	C₂×C₈	—	47	16	no	—
248	C₂×C₈	—	94	32	no	—
249	C₂×C₈	—	94	8	no	—
250	C₂×C₈	—	94	8	no	—
251	C₄²	—	2	16	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
255	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
256	C₂³	—	2	8	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	8	no	—
260	C₂³	—	2	16	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	16	no	—
263	C₂³	#13093935	1	16	yes	complemented
264	C₂³	C₂×C₄×Dic₄₇	1	4	yes	—
265	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
266	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
267	C₂³	#13093983	1	2	yes	—
268	C₂³	C₂×#10625329	1	8	yes	—
269	C₂³	C₂×#10625329	1	4	yes	—
270	C₂³	C₂×#10625329	1	4	yes	—
271	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
272	C₂×C₄	—	2	64	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	16	no	—
278	C₂×C₄	—	2	16	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	16	no	—
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	8	yes	—
287	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
288	C₂×C₄	#13093983	1	2	yes	—
289	C₂×C₄	C₄×C₂²⋊D₄₇	1	16	yes	—
290	C₂×C₄	C₂×#10625329	1	8	yes	—
291	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
293	C₂×C₄	C₂×#10625350	1	2	yes	—
294	C₈	—	94	16	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	32	no	—
299	C₂²	—	2	4	no	—
300	C₂²	C₄×C₂²⋊Dic₄₇	1	4	yes	—
301	C₂²	C₂×#13093935	1	4	yes	—
302	C₂²	C₂×#13093935	1	8	yes	complemented
303	C₂²	C₄×#10625329	1	4	yes	—
304	C₂²	#15689087	1	8	yes	—
305	C₂²	#15689180	1	8	yes	—
306	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
307	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
308	C₂²	C₂×#13093983	1	1	yes	characteristic
309	C₂²	C₂²×#10625329	1	2	yes	—
310	C₂²	C₂²×#10625329	1	2	yes	—
311	C₄	—	2	8	no	—
312	C₄	—	2	8	no	—
313	C₄	—	2	4	no	—
314	C₄	C₄×C₂²⋊Dic₄₇	1	4	yes	—
315	C₄	C₂×C₄×C₂²⋊D₄₇	1	4	yes	—
316	C₂	—	2	8	no	—
317	C₂	#20243150	1	8	yes	direct product factor
318	C₂	C₂×C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
319	C₂	C₂²×#13093935	1	1	yes	characteristic
320	C₂	C₂×C₄×#10625329	1	1	yes	characteristic
321	C₂	C₂×#15689087	1	2	yes	—
322	C₂	C₂×#15689180	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×C₄²×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×C₄²	D₄₇
hypocenter	C₄₇	C₂×#1749
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×#13093935	C₂	semidirect
3. #20243150	C₂	direct (N × H)
4. C₂×#15688878	C₂	semidirect
7. C₂×C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
10. C₂×#13093935	C₂²	semidirect
11. #15688878	C₂²	semidirect
22. C₂²×C₄₇⋊C₈	C₂×C₄	semidirect
24. C₂²⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
93. C₂×C₄₇⋊C₈	C₂²×C₄	semidirect
101. C₂³×C₄₇	#97	semidirect
141. C₂²×C₄₇	C₂×#97	semidirect
161. C₂×C₄₇	#1749	semidirect
181. C₄₇	C₂×#1749	semidirect
263. C₂³	#13093935	semidirect
302. C₂²	C₂×#13093935	semidirect
317. C₂	#20243150	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×C₂²⋊(C₄₇⋊C₈)	2
C₂²×#13093935	2
#20243150	2
C₂×#15688878	2
C₂³×C₄²×C₄₇	2
C₂×#1749	47

Subgroups — C2×#20243150 #26213800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243150 #26213800