Subgroups — C₂×#20241312 #26125337

Order 12032.

682 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	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	#20241312	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
5	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
6	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
7	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
12	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄₇⋊C₁₆	—	2	4	no	—
15	C₂×C₄₇⋊C₁₆	—	4	4	no	—
16	C₂⁵×C₄₇	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₄₇	—	2	1	no	—
18	C₂³×C₄×C₄₇	—	2	1	no	—
19	C₂³×C₄×C₄₇	—	2	1	no	—
20	C₂³×C₄×C₄₇	—	2	1	no	—
21	C₂³×C₄×C₄₇	—	4	2	no	—
22	C₂³×C₄×C₄₇	—	4	2	no	—
23	C₂³×C₄×C₄₇	C₈	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	C₈	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₄₇	—	2	1	no	—
28	C₂²×C₈×C₄₇	—	2	1	no	—
29	C₂²×C₈×C₄₇	—	4	2	no	—
30	C₂²×C₈×C₄₇	—	4	2	no	—
31	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂²×C₈×C₄₇	D₄	1	2	yes	—
35	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²⋊C₈×C₄₇	—	2	4	no	—
37	C₂²⋊C₈×C₄₇	—	2	4	no	—
38	C₂⁴×C₄₇	—	2	1	no	—
39	C₂⁴×C₄₇	—	2	1	no	—
40	C₂⁴×C₄₇	—	2	2	no	—
41	C₂⁴×C₄₇	—	4	2	no	—
42	C₂⁴×C₄₇	—	4	2	no	—
43	C₂⁴×C₄₇	C₁₆	1	4	yes	complemented
44	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂⁴×C₄₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
69	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	4	1	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	4	1	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
81	C₄₇⋊C₁₆	—	4	4	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	4	no	—
85	C₂³×C₄₇	—	2	4	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	2	no	—
88	C₂³×C₄₇	—	4	2	no	—
89	C₂³×C₄₇	—	4	8	no	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	—	4	8	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
99	C₂³×C₄₇	C₂×C₁₆	1	4	yes	complemented
100	C₂³×C₄₇	M₃₂	1	2	yes	—
101	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#99	1	2	yes	—
121	C₂×C₄×C₄₇	#100	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	—	4	2	no	—
127	C₂×#1314	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	—	4	8	no	—
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	4	8	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	#323	1	2	yes	—
142	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
143	C₂²×C₄₇	C₂²⋊C₁₆	1	2	yes	—
144	C₂²×C₄₇	#349	1	2	yes	—
145	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	4	2	no	—
151	C₄×C₄₇	#323	1	2	yes	—
152	C₄×C₄₇	C₂×#99	1	1	yes	characteristic
153	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
154	C₂²×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
155	C₂×C₂²⋊C₁₆	—	47	2	no	—
156	#1314	—	47	4	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	—	2	1	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	4	4	no	—
161	C₂×C₄₇	#1314	1	4	yes	complemented
162	C₂×C₄₇	C₂×#323	1	1	yes	characteristic
163	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#349	1	1	yes	characteristic
165	C₂⁴×C₄	Dic₄₇	1	1	yes	characteristic
166	C₂³×C₈	Dic₄₇	1	1	yes	characteristic
167	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂²×C₁₆	—	94	2	no	—
169	C₂×C₂²⋊C₈	—	2	2	no	—
170	C₂×C₂²⋊C₈	—	2	2	no	—
171	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	—
172	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
173	C₂²⋊C₁₆	—	47	4	no	—
174	C₂²⋊C₁₆	—	94	2	no	—
175	C₄₇	C₂×#1314	1	1	yes	characteristic, complemented, Sylow
176	C₂⁵	C₄₇⋊C₈	1	1	yes	characteristic
177	C₂³×C₄	—	2	1	no	—
178	C₂³×C₄	—	2	1	no	—
179	C₂³×C₄	—	2	1	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	—	4	2	no	—
182	C₂³×C₄	—	4	2	no	—
183	C₂³×C₄	C₄₇⋊C₈	1	1	yes	characteristic
184	C₂³×C₄	C₄₇⋊C₈	1	2	yes	—
185	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	4	2	no	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
192	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
194	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂×C₁₆	—	94	4	no	—
197	C₂×C₁₆	—	188	4	no	—
198	C₂²⋊C₈	—	2	4	no	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂⁴	—	2	1	no	—
201	C₂⁴	—	2	1	no	—
202	C₂⁴	—	2	2	no	—
203	C₂⁴	—	4	2	no	—
204	C₂⁴	—	4	2	no	—
205	C₂⁴	C₄₇⋊C₁₆	1	4	yes	complemented
206	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
207	C₂⁴	C₂×C₄₇⋊C₈	1	2	yes	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
228	C₂²×C₄	#10625329	1	1	yes	characteristic
229	C₂²×C₄	#10625329	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₁₆	—	188	4	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	8	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	8	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
259	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
260	C₂³	C₂×C₄₇⋊C₁₆	1	2	yes	—
261	C₂³	C₂×C₄₇⋊C₁₆	1	4	yes	complemented
262	C₂³	#13093964	1	2	yes	—
263	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
282	C₂×C₄	#13093939	1	2	yes	—
283	C₂×C₄	#13093941	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
285	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	2	2	no	—
288	C₈	—	4	2	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	8	no	—
302	C₂²	#15688847	1	2	yes	—
303	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
304	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	2	yes	—
305	C₂²	#15688905	1	2	yes	—
306	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
307	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
308	C₂²	C₂×#13093964	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	#15688847	1	2	yes	—
313	C₄	C₂×#13093939	1	1	yes	characteristic
314	C₄	C₂×#13093941	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	#20241312	1	4	yes	direct product factor
320	C₂	C₂×#15688847	1	1	yes	characteristic
321	C₂	C₂×C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
322	C₂	C₂×#15688905	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²⋊C₈
prime radical	C₂⁵×C₄₇	C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#1314
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
2. #20241312	C₂	direct (N × H)
6. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
43. C₂⁴×C₄₇	C₁₆	semidirect
99. C₂³×C₄₇	C₂×C₁₆	semidirect
161. C₂×C₄₇	#1314	semidirect
175. C₄₇	C₂×#1314	semidirect
205. C₂⁴	C₄₇⋊C₁₆	semidirect
261. C₂³	C₂×C₄₇⋊C₁₆	semidirect
319. C₂	#20241312	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20241312	2
C₂²×C₂²⋊C₈×C₄₇	2
C₂×#1314	47

Subgroups — C2×#20241312 #26125337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241312 #26125337