Subgroups — C₂×#20242864 #26198776

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689315	C₂	1	2	yes	maximal, complemented
2	C₂×#15689202	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242864	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688874	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688867	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10625334	—	2	2	no	—
8	C₂²×#10625324	—	2	1	no	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689315	—	2	4	no	—
11	#15689315	C₂²	1	8	yes	complemented
12	#15689202	—	2	2	no	—
13	#15689202	C₂²	1	4	yes	complemented
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093945	—	2	2	no	—
17	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
18	#15688874	C₂²	1	4	yes	complemented
19	#15688867	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₄₇	—	2	1	no	—
21	C₂²×D₈×C₄₇	—	2	1	no	—
22	C₂×#127×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#493	—	2	2	no	—
24	C₄₇×#493	C₂²	1	4	yes	complemented
25	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	—	2	1	no	—
28	C₂×#10625334	—	2	8	no	—
29	C₂×#10625334	—	4	8	no	—
30	C₂×#10625324	—	2	4	no	—
31	C₂×#10625324	—	4	4	no	—
32	C₂²×C₄₇⋊C₈	—	2	1	no	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₄×D₁₈₈	—	2	1	no	—
35	C₄×D₁₈₈	—	2	1	no	—
36	C₄×D₁₈₈	D₄	1	2	yes	—
37	C₄×D₁₈₈	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
44	#13093945	—	2	8	no	—
45	(C₂×D₄₇)⋊C₈	—	2	4	no	—
46	C₂²×D₄×C₄₇	—	2	2	no	—
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂²×D₄×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	4	no	—
50	C₂×D₈×C₄₇	—	2	4	no	—
51	C₂×D₈×C₄₇	—	4	4	no	—
52	C₂×D₈×C₄₇	—	4	4	no	—
53	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	#127×C₄₇	—	2	2	no	—
56	#127×C₄₇	—	2	2	no	—
57	#127×C₄₇	—	2	2	no	—
58	#127×C₄₇	C₂³	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₇	D₄	1	2	yes	—
61	C₄×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂⁴×C₄₇	—	4	2	no	—
63	C₂³×D₄₇	—	2	1	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	—	2	4	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	—	4	2	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂×D₄×C₄₇	—	4	4	no	—
72	C₂×D₄×C₄₇	—	4	4	no	—
73	C₂×D₄×C₄₇	—	4	4	no	—
74	C₂×D₄×C₄₇	—	8	4	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	1	no	—
79	C₂×D₁₈₈	—	2	4	no	—
80	C₂×D₁₈₈	—	2	1	no	—
81	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
82	C₂×C₄×D₄₇	—	2	2	no	—
83	C₂×C₄×D₄₇	—	2	2	no	—
84	C₂×C₄×D₄₇	—	4	2	no	—
85	C₂×C₄×D₄₇	—	4	2	no	—
86	D₈×C₄₇	—	4	8	no	—
87	D₈×C₄₇	—	4	8	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
90	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₇	#41	1	1	yes	characteristic
95	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
96	#10625334	—	4	16	no	—
97	(C₂×D₄₇)⋊C₄	—	2	2	no	—
98	(C₂×D₄₇)⋊C₄	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	1	no	—
100	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	4	2	no	—
104	C₂×C₄₇⋊C₈	—	4	4	no	—
105	#10625324	—	4	8	no	—
106	C₂³×C₄₇	—	4	4	no	—
107	C₂³×C₄₇	—	8	4	no	—
108	C₂³×C₄₇	—	8	4	no	—
109	C₂³×C₄₇	—	8	4	no	—
110	C₂³×C₄₇	#121	1	1	yes	characteristic
111	C₂²×D₄₇	—	2	2	no	—
112	C₂²×D₄₇	—	4	2	no	—
113	C₂²×D₄₇	—	4	2	no	—
114	C₂²×D₄₇	—	4	2	no	—
115	D₄×C₄₇	—	4	4	no	—
116	D₄×C₄₇	—	4	4	no	—
117	D₄×C₄₇	—	4	4	no	—
118	D₄×C₄₇	—	4	4	no	—
119	D₄×C₄₇	—	8	4	no	—
120	D₄×C₄₇	—	8	4	no	—
121	D₄×C₄₇	—	8	4	no	—
122	D₄×C₄₇	—	8	4	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	D₁₈₈	—	2	4	no	—
136	D₁₈₈	—	4	2	no	—
137	C₄×D₄₇	—	4	2	no	—
138	C₄×D₄₇	—	4	2	no	—
139	C₄×D₄₇	—	4	4	no	—
140	C₈×C₄₇	—	2	2	no	—
141	C₈×C₄₇	C₂×D₈	1	4	yes	complemented
142	C₄₇⋊C₈	—	4	2	no	—
143	C₄₇⋊C₈	—	4	4	no	—
144	C₂×#1687	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	8	4	no	—
146	C₂²×C₄₇	—	8	4	no	—
147	C₂²×C₄₇	—	8	4	no	—
148	C₂²×C₄₇	—	16	8	no	—
149	C₂²×C₄₇	#459	1	2	yes	—
150	C₂²×C₄₇	#469	1	2	yes	—
151	C₂²×C₄₇	#472	1	2	yes	—
152	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
153	C₂×D₄₇	—	4	2	no	—
154	C₂×D₄₇	—	4	2	no	—
155	C₂×D₄₇	—	4	2	no	—
156	C₂×D₄₇	—	8	4	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	#459	1	2	yes	—
160	C₄×C₄₇	#469	1	2	yes	—
161	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
163	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	Dic₄₇	—	4	2	no	—
165	C₂×#493	D₄₇	1	1	yes	characteristic, complemented
166	C₂×#459	—	47	2	no	—
167	C₂×#436	—	47	1	no	—
168	#1687	—	47	8	no	—
169	C₂×#334	—	47	1	no	—
170	C₂×#331	—	47	2	no	—
171	C₂×C₄₇	—	16	4	no	—
172	C₂×C₄₇	#1687	1	4	yes	complemented
173	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
174	C₂×C₄₇	C₂×#469	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
176	D₄₇	—	8	2	no	—
177	C₂²×D₈	—	2	1	no	—
178	C₂²×D₈	—	2	1	no	—
179	C₂²×D₈	—	94	2	no	—
180	C₂²×M₁₆	—	94	1	no	—
181	C₂×#127	C₂×D₄₇	1	2	yes	—
182	C₂×C₄×D₄	—	47	1	no	—
183	#493	—	2	2	no	—
184	#493	C₂×D₄₇	1	4	yes	complemented
185	#459	—	47	8	no	—
186	#459	—	94	4	no	—
187	#436	—	47	4	no	—
188	#436	—	94	2	no	—
189	C₂×C₄⋊C₈	—	47	1	no	—
190	C₂×C₄⋊C₈	—	47	2	no	—
191	C₂×#102	—	94	2	no	—
192	C₂×C₂²⋊C₈	—	94	1	no	—
193	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
194	#334	—	47	4	no	—
195	#331	—	47	8	no	—
196	C₄₇	C₂×#1687	1	1	yes	characteristic, complemented, Sylow
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	—	47	1	no	—
201	C₂³×C₄	—	94	1	no	—
202	C₂×D₈	—	2	4	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×D₈	—	4	4	no	—
205	C₂×D₈	—	4	4	no	—
206	C₂×D₈	—	94	8	no	—
207	C₂×D₈	—	188	8	no	—
208	C₂×M₁₆	—	94	4	no	—
209	C₂×M₁₆	—	188	4	no	—
210	C₂²×C₈	—	94	2	no	—
211	C₂²×C₈	—	94	1	no	—
212	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
214	#127	—	2	2	no	—
215	#127	—	2	2	no	—
216	#127	—	2	2	no	—
217	#127	C₂²×D₄₇	1	4	yes	—
218	C₄×D₄	—	47	2	no	—
219	C₄×D₄	—	47	2	no	—
220	C₄×D₄	—	94	1	no	—
221	C₄×D₄	—	94	1	no	—
222	C₂×C₄⋊C₄	—	47	1	no	—
223	C₂×C₂²⋊C₄	—	94	1	no	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	C₄⋊C₈	—	47	2	no	—
226	C₄⋊C₈	—	47	4	no	—
227	C₄⋊C₈	—	94	2	no	—
228	C₄⋊C₈	—	94	1	no	—
229	#102	—	94	8	no	—
230	C₂²⋊C₈	—	94	4	no	—
231	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
232	C₄×C₈	C₂²×D₄₇	1	2	yes	—
233	C₂⁴	—	4	2	no	—
234	C₂⁴	—	94	1	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	8	4	no	—
246	C₂×D₄	—	47	2	no	—
247	C₂×D₄	—	94	1	no	—
248	C₂×D₄	—	94	4	no	—
249	C₂²×C₄	—	94	1	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	—	188	2	no	—
253	C₂²×C₄	—	188	2	no	—
254	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
255	C₂²×C₄	#10625352	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	D₈	—	4	8	no	—
258	D₈	—	4	8	no	—
259	D₈	—	188	16	no	—
260	M₁₆	—	188	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	94	4	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	188	2	no	—
265	C₂×C₈	—	188	4	no	—
266	C₂×C₈	#10625334	1	4	yes	complemented
267	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
269	C₄⋊C₄	—	47	2	no	—
270	C₄⋊C₄	—	94	1	no	—
271	C₂²⋊C₄	—	94	2	no	—
272	C₂²⋊C₄	—	94	2	no	—
273	C₄²	D₄×D₄₇	1	1	yes	characteristic
274	C₄²	#10625352	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₄²	C₂³×D₄₇	1	1	yes	characteristic
277	C₂³	—	4	4	no	—
278	C₂³	—	8	4	no	—
279	C₂³	—	8	4	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	94	2	no	—
282	C₂³	—	188	2	no	—
283	C₂³	—	188	2	no	—
284	C₂³	—	188	2	no	—
285	C₂³	#13093995	1	1	yes	characteristic
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	8	4	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	D₄	—	94	4	no	—
295	D₄	—	188	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	188	2	no	—
299	C₂×C₄	—	188	4	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	#13093995	1	1	yes	characteristic
303	C₂×C₄	#13093995	1	1	yes	characteristic
304	C₂×C₄	#13093995	1	1	yes	characteristic
305	C₂×C₄	C₂×#10625334	1	2	yes	—
306	C₂×C₄	#13094048	1	4	yes	—
307	C₂×C₄	#13094050	1	2	yes	—
308	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	188	2	no	—
313	C₈	—	188	4	no	—
314	C₈	C₂×#10625334	1	4	yes	complemented
315	C₂²	—	8	4	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	16	8	no	—
319	C₂²	—	376	4	no	—
320	C₂²	—	188	2	no	—
321	C₂²	—	188	2	no	—
322	C₂²	—	188	2	no	—
323	C₂²	#15689315	1	2	yes	—
324	C₂²	#15689373	1	2	yes	—
325	C₂²	#15689390	1	2	yes	—
326	C₂²	C₂×#13093995	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	188	2	no	—
330	C₄	#15689315	1	2	yes	—
331	C₄	#15689373	1	2	yes	—
332	C₄	C₂²×#10625334	1	1	yes	characteristic
333	C₄	C₂×#13094048	1	2	yes	—
334	C₄	C₂×#13094050	1	1	yes	characteristic
335	C₂	—	16	4	no	—
336	C₂	—	376	2	no	—
337	C₂	#20242864	1	4	yes	direct product factor
338	C₂	C₂×#15689315	1	1	yes	characteristic
339	C₂	C₂×#15689373	1	1	yes	characteristic
340	C₂	C₂×#15689390	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	#121
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	C₂×#1687
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#15689315	C₂	semidirect
2. C₂×#15689202	C₂	semidirect
3. #20242864	C₂	direct (N × H)
4. C₂×#15688874	C₂	semidirect
5. C₂×#15688867	C₂	semidirect
6. C₂×C₄₇×#493	C₂	semidirect
11. #15689315	C₂²	semidirect
13. #15689202	C₂²	semidirect
18. #15688874	C₂²	semidirect
19. #15688867	C₂²	semidirect
24. C₄₇×#493	C₂²	semidirect
89. C₂×C₈×C₄₇	D₈	semidirect
141. C₈×C₄₇	C₂×D₈	semidirect
165. C₂×#493	D₄₇	semidirect
172. C₂×C₄₇	#1687	semidirect
184. #493	C₂×D₄₇	semidirect
196. C₄₇	C₂×#1687	semidirect
266. C₂×C₈	#10625334	semidirect
314. C₈	C₂×#10625334	semidirect
337. C₂	#20242864	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×#15689315	2
C₂×#15689202	2
#20242864	2
C₂×#15688874	2
C₂×#15688867	2
C₂×C₄₇×#493	2
C₂×#1687	47

Subgroups — C2×#20242864 #26198776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20242864 #26198776