Subgroups — C₂×#20243143 #26213781

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689078	C₂	1	2	yes	maximal
2	#20243143	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688880	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688879	C₂	1	2	yes	maximal
6	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂×#13093983	C₂²	1	2	yes	—
9	#15689078	C₄	1	8	yes	complemented
10	#15689078	C₂²	1	8	yes	—
11	C₂×#13093934	C₄	1	4	yes	complemented
12	C₂×C₈×Dic₄₇	C₄	1	4	yes	complemented
13	C₄₇⋊#338	C₂²	1	4	yes	—
14	#15688880	C₂²	1	4	yes	—
15	#15688879	C₂²	1	8	yes	—
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#405	C₄	1	4	yes	complemented
20	C₄₇×#405	C₂²	1	4	yes	—
21	C₂×#97×C₄₇	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	complemented
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	C₂×C₄	1	4	yes	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	#13093983	—	2	2	no	—
31	#13093983	—	2	2	no	—
32	#13093983	C₂×C₄	1	4	yes	—
33	#13093983	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
37	#13093934	C₂×C₄	1	16	yes	complemented
38	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	#117×C₄₇	—	2	1	no	—
53	#117×C₄₇	—	2	1	no	—
54	#117×C₄₇	C₂×C₄	1	2	yes	—
55	#117×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
59	#97×C₄₇	C₂×C₄	1	8	yes	complemented
60	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
61	C₂⁴×C₄₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	2	4	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	C₄²	1	2	yes	—
79	C₂×C₈×C₄₇	C₄²	1	4	yes	—
80	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₄²	1	4	yes	—
82	C₂×C₈×C₄₇	C₄²	1	2	yes	—
83	C₂×C₈×C₄₇	C₄²	1	2	yes	—
84	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₂²⋊C₄×C₄₇	—	2	2	no	—
93	C₂²⋊C₄×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
96	C₂²⋊Dic₄₇	—	2	4	no	—
97	C₂²⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	C₂²×C₄	1	8	yes	—
102	#10625329	—	2	4	no	—
103	#10625329	—	2	4	no	—
104	#10625329	—	2	4	no	—
105	#10625329	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
109	C₂³×C₄₇	—	4	2	no	—
110	C₂³×C₄₇	#117	1	2	yes	—
111	C₂³×C₄₇	#117	1	2	yes	—
112	C₂³×C₄₇	#117	1	1	yes	characteristic
113	C₂³×C₄₇	#117	1	1	yes	characteristic
114	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#117	1	2	yes	—
121	C₂×C₄×C₄₇	#117	1	2	yes	—
122	C₂×C₄×C₄₇	#117	1	2	yes	—
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	#117	1	2	yes	—
125	C₂×C₄×C₄₇	#117	1	2	yes	—
126	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#117	1	2	yes	—
129	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
135	C₂×Dic₄₇	—	2	8	no	—
136	C₂×Dic₄₇	—	4	8	no	—
137	C₈×C₄₇	—	2	4	no	—
138	C₈×C₄₇	C₂×C₄²	1	4	yes	—
139	C₈×C₄₇	C₂×C₄²	1	2	yes	—
140	C₈×C₄₇	C₂×C₄²	1	2	yes	—
141	C₄₇⋊C₈	—	2	8	no	—
142	C₂×#1747	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	—	4	1	no	—
148	C₂²×C₄₇	#376	1	2	yes	—
149	C₂²×C₄₇	#376	1	1	yes	characteristic
150	C₂²×C₄₇	#376	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#117	1	2	yes	—
152	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	#376	1	1	yes	characteristic
155	C₄×C₄₇	#376	1	2	yes	—
156	C₄×C₄₇	#376	1	1	yes	characteristic
157	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#117	1	2	yes	—
159	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
160	Dic₄₇	—	4	8	no	—
161	C₂×#405	—	47	2	no	—
162	C₂×#405	D₄₇	1	1	yes	characteristic
163	#1747	—	47	8	no	—
164	C₂×#338	—	47	1	no	—
165	C₂×H(3,ℤ/4)	—	47	1	no	—
166	C₂×H(3,ℤ/4)	—	47	2	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	—	2	1	no	—
171	C₂×C₄₇	#1747	1	2	yes	complemented
172	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
173	C₂²×M₁₆	—	47	2	no	—
174	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
176	C₂×#117	—	47	2	no	—
177	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
178	#405	—	47	8	no	—
179	#405	—	47	8	no	—
180	#405	Dic₄₇	1	4	yes	complemented
181	#405	C₂×D₄₇	1	4	yes	—
182	C₂×#97	—	47	4	no	—
183	C₂×#97	Dic₄₇	1	2	yes	complemented
184	C₂×C₄×C₈	—	47	4	no	—
185	C₂×C₄×C₈	Dic₄₇	1	2	yes	complemented
186	#338	—	47	4	no	—
187	H(3,ℤ/4)	—	47	4	no	—
188	H(3,ℤ/4)	—	47	8	no	—
189	C₄₇	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	47	4	no	—
194	C₂×M₁₆	—	47	4	no	—
195	C₂×M₁₆	—	94	2	no	—
196	C₂×M₁₆	—	94	2	no	—
197	C₂×M₁₆	—	94	2	no	—
198	C₂×M₁₆	—	94	2	no	—
199	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
200	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
201	C₂²×C₈	—	47	4	no	—
202	C₂²×C₈	C₄×D₄₇	1	2	yes	—
203	C₂²×C₈	C₄×D₄₇	1	4	yes	—
204	C₂²×C₈	C₄×D₄₇	1	2	yes	—
205	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
206	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
208	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
210	#117	—	2	1	no	—
211	#117	—	2	1	no	—
212	#117	—	47	4	no	—
213	#117	—	47	4	no	—
214	#117	—	94	2	no	—
215	#117	—	94	2	no	—
216	#117	C₂×Dic₄₇	1	2	yes	—
217	#117	C₂²×D₄₇	1	2	yes	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	94	2	no	—
222	C₂×C₄²	—	47	4	no	—
223	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
224	#97	—	47	16	no	—
225	#97	C₂×Dic₄₇	1	8	yes	complemented
226	C₄×C₈	—	47	16	no	—
227	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
228	C₂⁴	#10625350	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	94	4	no	—
231	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
233	C₂²×C₄	#10625350	1	1	yes	characteristic
234	C₂²×C₄	#10625350	1	1	yes	characteristic
235	C₂²×C₄	#10625350	1	1	yes	characteristic
236	C₂²×C₄	#10625350	1	2	yes	—
237	C₂²×C₄	#10625350	1	2	yes	—
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	94	4	no	—
244	M₁₆	—	94	4	no	—
245	M₁₆	—	94	4	no	—
246	M₁₆	—	94	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	47	8	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
253	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
254	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
255	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
256	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
257	C₂×C₈	C₄×Dic₄₇	1	4	yes	—
258	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
260	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
261	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
262	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
263	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	94	4	no	—
267	C₄⋊C₄	—	94	4	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	94	4	no	—
271	C₂²⋊C₄	—	94	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	47	8	no	—
274	C₄²	—	94	4	no	—
275	C₄²	C₂²×Dic₄₇	1	4	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	#13093981	1	2	yes	—
278	C₂³	#13093981	1	2	yes	—
279	C₂³	#13093983	1	1	yes	characteristic
280	C₂³	#13093983	1	1	yes	characteristic
281	C₂³	C₂×#10625350	1	1	yes	characteristic
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	94	8	no	—
285	C₂×C₄	—	188	8	no	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
289	C₂×C₄	#13093981	1	2	yes	—
290	C₂×C₄	#13093981	1	2	yes	—
291	C₂×C₄	#13093981	1	2	yes	—
292	C₂×C₄	#13093981	1	2	yes	—
293	C₂×C₄	#13093981	1	2	yes	—
294	C₂×C₄	#13093981	1	2	yes	—
295	C₂×C₄	#13093983	1	1	yes	characteristic
296	C₂×C₄	#13093983	1	1	yes	characteristic
297	C₂×C₄	#13093983	1	1	yes	characteristic
298	C₂×C₄	#13093983	1	1	yes	characteristic
299	C₂×C₄	#13093983	1	2	yes	—
300	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
301	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625350	1	2	yes	—
304	C₈	—	2	4	no	—
305	C₈	—	94	8	no	—
306	C₈	C₂×C₄×Dic₄₇	1	4	yes	—
307	C₈	C₂×C₄×Dic₄₇	1	2	yes	—
308	C₈	C₂×C₄×Dic₄₇	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	#15688969	1	2	yes	—
315	C₂²	#15688969	1	1	yes	characteristic
316	C₂²	#15688969	1	1	yes	characteristic
317	C₂²	C₂×#13093981	1	2	yes	—
318	C₂²	C₂×#13093983	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	188	8	no	—
321	C₄	#15688969	1	2	yes	—
322	C₄	#15688969	1	1	yes	characteristic
323	C₄	#15688969	1	1	yes	characteristic
324	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
325	C₄	C₂×#13093981	1	2	yes	—
326	C₄	C₂×#13093983	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	#20243143	1	2	yes	direct product factor
332	C₂	C₂×#15688969	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	#376
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	C₂×#1747
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20243143	C₂	direct (N × H)
9. #15689078	C₄	semidirect
11. C₂×#13093934	C₄	semidirect
12. C₂×C₈×Dic₄₇	C₄	semidirect
19. C₄₇×#405	C₄	semidirect
21. C₂×#97×C₄₇	C₄	semidirect
22. C₂×C₄×C₈×C₄₇	C₄	semidirect
37. #13093934	C₂×C₄	semidirect
38. C₈×Dic₄₇	C₂×C₄	semidirect
59. #97×C₄₇	C₂×C₄	semidirect
60. C₄×C₈×C₄₇	C₂×C₄	semidirect
171. C₂×C₄₇	#1747	semidirect
180. #405	Dic₄₇	semidirect
183. C₂×#97	Dic₄₇	semidirect
185. C₂×C₄×C₈	Dic₄₇	semidirect
189. C₄₇	C₂×#1747	semidirect
225. #97	C₂×Dic₄₇	semidirect
227. C₄×C₈	C₂×Dic₄₇	semidirect
331. C₂	#20243143	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×#15689078	2
#20243143	2
C₂×C₄₇⋊#338	2
C₂×#15688880	2
C₂×#15688879	2
C₂×C₄₇×#405	2
C₂×#1747	47

Subgroups — C2×#20243143 #26213781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243143 #26213781