Subgroups — C₂×#20256030 #27095842

Order 12032.

2058 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689852	C₂	1	1	yes	maximal, characteristic, complemented
2	#20256030	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689475	C₂	1	3	yes	maximal, complemented
4	C₂×#15689405	C₂	1	3	yes	maximal, complemented
5	C₂×#15689328	C₂	1	3	yes	maximal
6	C₂×Q₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×#10625335	C₂	1	3	yes	maximal
8	C₂×D₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625335	C₂²	1	3	yes	—
10	#15689852	—	2	4	no	—
11	#15689852	C₂²	1	8	yes	complemented
12	C₂×#13094024	C₂²	1	3	yes	—
13	C₂×#13094015	—	2	3	no	—
14	C₂×Q₈×Dic₄₇	—	2	1	no	—
15	C₂×C₄×Dic₉₄	C₂²	1	3	yes	—
16	#15689475	C₂²	1	24	yes	complemented
17	#15689405	C₂²	1	24	yes	complemented
18	#15689328	C₂²	1	24	yes	—
19	Q₈×C₄₇⋊C₈	C₂²	1	8	yes	complemented
20	C₄×#10625335	C₂²	1	24	yes	—
21	C₂×#13093955	C₂²	1	3	yes	—
22	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	3	yes	—
23	C₂×#13093947	C₂²	1	3	yes	—
24	C₂×#13093947	C₂²	1	6	yes	—
25	C₂×#13093945	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	3	yes	—
27	C₂³×Q₈×C₄₇	—	2	1	no	—
28	D₄×Q₈×C₄₇	—	2	4	no	—
29	D₄×Q₈×C₄₇	C₂²	1	8	yes	—
30	C₂×C₄⋊Q₈×C₄₇	C₂²	1	3	yes	—
31	C₂×C₂²⋊Q₈×C₄₇	—	2	3	no	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
34	C₂²×Dic₉₄	C₂³	1	3	yes	—
35	C₂×#10625335	—	2	12	no	—
36	C₂×#10625335	C₂³	1	12	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	3	yes	—
39	#13094024	—	2	6	no	—
40	#13094024	C₂³	1	12	yes	—
41	#13094015	—	2	24	no	—
42	Q₈×Dic₄₇	—	2	8	no	—
43	C₄×Dic₉₄	—	2	6	no	—
44	C₄×Dic₉₄	C₂³	1	12	yes	—
45	C₂×C₄⋊Dic₄₇	—	2	3	no	—
46	C₂×C₄⋊Dic₄₇	—	2	3	no	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₇⋊C₄	—	2	3	no	—
49	C₂×C₄×Dic₄₇	—	2	3	no	—
50	#13093955	C₂³	1	12	yes	—
51	C₄⋊(C₄₇⋊C₈)	C₂³	1	12	yes	—
52	#13093947	C₂³	1	12	yes	—
53	#13093947	C₂³	1	24	yes	—
54	#13093945	C₂³	1	4	yes	—
55	C₄×C₄₇⋊C₈	C₂³	1	12	yes	—
56	C₂²×Q₈×C₄₇	—	2	8	no	—
57	C₂²×Q₈×C₄₇	—	2	3	no	—
58	C₂²×Q₈×C₄₇	—	4	8	no	—
59	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	—	2	3	no	—
62	C₄⋊Q₈×C₄₇	—	2	6	no	—
63	C₄⋊Q₈×C₄₇	C₂³	1	12	yes	—
64	C₂²⋊Q₈×C₄₇	—	2	24	no	—
65	C₄×Q₈×C₄₇	D₄	1	4	yes	—
66	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
67	C₄×D₄×C₄₇	—	2	6	no	—
68	C₄×D₄×C₄₇	C₂³	1	12	yes	—
69	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
70	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
72	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
73	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
74	C₂⁴×C₄₇	—	2	1	no	—
75	C₂×Q₈×C₄₇	—	2	12	no	—
76	C₂×Q₈×C₄₇	—	4	32	no	—
77	C₂×Q₈×C₄₇	—	4	12	no	—
78	C₂×Q₈×C₄₇	SD₁₆	1	8	yes	complemented
79	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
80	C₂×D₄×C₄₇	—	2	1	no	—
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	#41	1	4	yes	—
83	C₂×D₄×C₄₇	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₄₇	—	2	3	no	—
85	C₂²×C₄×C₄₇	—	2	12	no	—
86	C₂²×C₄×C₄₇	—	4	12	no	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₄₇	—	2	1	no	—
91	C₂²×Dic₄₇	—	2	3	no	—
92	C₂×Dic₉₄	—	2	6	no	—
93	C₂×Dic₉₄	—	2	6	no	—
94	C₂×Dic₉₄	—	2	3	no	—
95	C₂×Dic₉₄	C₂⁴	1	6	yes	—
96	C₄⋊C₄×C₄₇	—	2	12	no	—
97	C₄⋊C₄×C₄₇	—	2	3	no	—
98	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
99	C₄⋊C₄×C₄₇	C₂⁴	1	6	yes	—
100	C₂²⋊C₄×C₄₇	—	2	12	no	—
101	C₄²×C₄₇	C₂×D₄	1	6	yes	—
102	C₄²×C₄₇	C₂⁴	1	6	yes	—
103	#10625335	—	2	24	no	—
104	C₄⋊Dic₄₇	—	2	12	no	—
105	C₄⋊Dic₄₇	—	2	12	no	—
106	C₄⋊Dic₄₇	—	2	1	no	—
107	C₄⋊Dic₄₇	C₂⁴	1	2	yes	—
108	Dic₄₇⋊C₄	—	2	12	no	—
109	C₄×Dic₄₇	—	2	12	no	—
110	C₂×C₄₇⋊C₈	—	2	6	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
113	C₂×C₄₇⋊C₈	C₂⁴	1	6	yes	—
114	C₂³×C₄₇	—	2	4	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₄₇	—	4	24	no	—
120	Q₈×C₄₇	C₂×SD₁₆	1	16	yes	complemented
121	D₄×C₄₇	—	2	4	no	—
122	D₄×C₄₇	—	2	2	no	—
123	D₄×C₄₇	C₂×#41	1	4	yes	—
124	C₂×C₄×C₄₇	—	2	6	no	—
125	C₂×C₄×C₄₇	—	2	6	no	—
126	C₂×C₄×C₄₇	—	4	6	no	—
127	C₂×C₄×C₄₇	—	4	24	no	—
128	C₂×C₄×C₄₇	C₂×SD₁₆	1	12	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
134	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	6	no	—
137	C₂×Dic₄₇	—	4	6	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	Dic₉₄	—	2	6	no	—
140	Dic₉₄	—	2	6	no	—
141	Dic₉₄	—	4	6	no	—
142	C₄₇⋊C₈	—	2	6	no	—
143	C₄₇⋊C₈	—	2	2	no	—
144	C₂×#3359	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	4	no	—
146	C₂²×C₄₇	—	4	8	no	—
147	C₂²×C₄₇	—	4	2	no	—
148	C₂²×C₄₇	—	4	2	no	—
149	C₂²×C₄₇	#548	1	2	yes	—
150	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
151	C₂²×C₄₇	#578	1	2	yes	—
152	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
153	C₄×C₄₇	—	2	6	no	—
154	C₄×C₄₇	—	4	6	no	—
155	C₄×C₄₇	#548	1	2	yes	—
156	C₄×C₄₇	C₂²×SD₁₆	1	12	yes	—
157	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
158	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₄₇	—	4	6	no	—
160	Dic₄₇	—	4	2	no	—
161	C₂×#557	—	47	1	no	—
162	C₂×D₄×Q₈	D₄₇	1	1	yes	characteristic
163	#3359	—	47	16	no	—
164	C₂×#487	—	47	3	no	—
165	C₂×#475	—	47	3	no	—
166	C₂×#461	—	47	3	no	—
167	C₂×C₈×Q₈	—	47	1	no	—
168	C₂×C₄×SD₁₆	—	47	3	no	—
169	C₂×C₄₇	—	4	4	no	—
170	C₂×C₄₇	#3359	1	4	yes	complemented
171	C₂×C₄₇	C₂×#548	1	1	yes	characteristic
172	C₂×C₄₇	C₂³×SD₁₆	1	1	yes	characteristic
173	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
174	C₂³×Q₈	—	2	1	no	—
175	C₂²×SD₁₆	—	47	3	no	—
176	#557	—	47	8	no	—
177	#557	—	94	4	no	—
178	D₄×Q₈	—	2	4	no	—
179	D₄×Q₈	C₂×D₄₇	1	8	yes	—
180	C₂×C₄⋊Q₈	—	47	3	no	—
181	C₂×C₄⋊Q₈	C₂×D₄₇	1	3	yes	—
182	C₂×#125	—	94	3	no	—
183	C₂×C₂²⋊Q₈	—	2	3	no	—
184	C₂×C₄×Q₈	—	47	3	no	—
185	C₂×C₄×Q₈	—	94	1	no	—
186	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
188	#487	—	47	24	no	—
189	#475	—	47	24	no	—
190	#461	—	47	24	no	—
191	C₈×Q₈	—	47	8	no	—
192	C₄×SD₁₆	—	47	24	no	—
193	C₂×#106	—	47	3	no	—
194	C₂×C₄⋊C₈	—	47	3	no	—
195	C₂×#103	—	47	6	no	—
196	C₂×#103	—	47	3	no	—
197	C₂×#102	—	47	1	no	—
198	C₂×C₄×C₈	—	47	3	no	—
199	C₄₇	C₂×#3359	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₈	—	2	8	no	—
201	C₂²×Q₈	—	2	3	no	—
202	C₂²×Q₈	—	4	8	no	—
203	C₂²×Q₈	—	47	3	no	—
204	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂²×D₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₂³×C₄	—	2	3	no	—
207	C₂×SD₁₆	—	47	12	no	—
208	C₂×SD₁₆	—	94	12	no	—
209	C₂²×C₈	—	47	3	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₄⋊Q₈	—	2	6	no	—
212	C₄⋊Q₈	—	47	12	no	—
213	C₄⋊Q₈	—	94	6	no	—
214	C₄⋊Q₈	C₂²×D₄₇	1	12	yes	—
215	#125	—	94	24	no	—
216	C₂²⋊Q₈	—	2	24	no	—
217	C₄×Q₈	—	47	12	no	—
218	C₄×Q₈	—	94	8	no	—
219	C₄×Q₈	—	94	6	no	—
220	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
221	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
222	C₄×D₄	—	2	6	no	—
223	C₄×D₄	C₂²×D₄₇	1	12	yes	—
224	C₂×C₄⋊C₄	—	2	3	no	—
225	C₂×C₄⋊C₄	—	47	1	no	—
226	C₂×C₄⋊C₄	—	94	3	no	—
227	C₂×C₄⋊C₄	—	94	3	no	—
228	C₂×C₄⋊C₄	—	94	3	no	—
229	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
230	C₂×C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
231	C₂×C₂²⋊C₄	—	2	3	no	—
232	C₂×C₄²	—	94	3	no	—
233	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
234	#106	—	47	12	no	—
235	C₄⋊C₈	—	47	12	no	—
236	#103	—	47	12	no	—
237	#103	—	47	24	no	—
238	#102	—	47	4	no	—
239	C₄×C₈	—	47	12	no	—
240	C₂⁴	—	2	1	no	—
241	C₂×Q₈	—	2	12	no	—
242	C₂×Q₈	—	4	32	no	—
243	C₂×Q₈	—	4	12	no	—
244	C₂×Q₈	—	47	6	no	—
245	C₂×Q₈	—	94	3	no	—
246	C₂×Q₈	—	94	6	no	—
247	C₂×Q₈	—	94	6	no	—
248	C₂×Q₈	#10625335	1	8	yes	complemented
249	C₂×Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	#10625354	1	4	yes	—
253	C₂×D₄	C₂³×D₄₇	1	2	yes	—
254	C₂²×C₄	—	2	3	no	—
255	C₂²×C₄	—	2	12	no	—
256	C₂²×C₄	—	4	12	no	—
257	C₂²×C₄	—	94	1	no	—
258	C₂²×C₄	—	94	3	no	—
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
260	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	SD₁₆	—	94	24	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	47	6	no	—
265	C₂×C₈	—	94	6	no	—
266	C₂×C₈	—	94	2	no	—
267	C₄⋊C₄	—	2	12	no	—
268	C₄⋊C₄	—	2	3	no	—
269	C₄⋊C₄	—	47	2	no	—
270	C₄⋊C₄	—	94	12	no	—
271	C₄⋊C₄	—	94	1	no	—
272	C₄⋊C₄	—	94	12	no	—
273	C₄⋊C₄	—	94	12	no	—
274	C₄⋊C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
275	C₄⋊C₄	C₂³×D₄₇	1	6	yes	—
276	C₂²⋊C₄	—	2	12	no	—
277	C₄²	—	94	12	no	—
278	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
279	C₄²	C₂³×D₄₇	1	6	yes	—
280	C₂³	—	2	4	no	—
281	C₂³	—	2	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	4	2	no	—
284	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
285	Q₈	—	4	24	no	—
286	Q₈	—	94	6	no	—
287	Q₈	—	94	6	no	—
288	Q₈	—	188	6	no	—
289	Q₈	C₂×#10625335	1	16	yes	complemented
290	D₄	—	2	4	no	—
291	D₄	—	2	2	no	—
292	D₄	C₂×#10625354	1	4	yes	—
293	C₂×C₄	—	2	6	no	—
294	C₂×C₄	—	2	6	no	—
295	C₂×C₄	—	4	6	no	—
296	C₂×C₄	—	4	24	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	6	no	—
299	C₂×C₄	—	188	6	no	—
300	C₂×C₄	—	188	2	no	—
301	C₂×C₄	C₂×#10625335	1	12	yes	—
302	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
303	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
305	C₂×C₄	C₂×#10625354	1	2	yes	—
306	C₂×C₄	#13094072	1	2	yes	—
307	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
308	C₈	—	94	6	no	—
309	C₈	—	94	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	#15689784	1	2	yes	—
315	C₂²	C₂²×#10625335	1	2	yes	—
316	C₂²	#15689938	1	2	yes	—
317	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₄	—	2	6	no	—
319	C₄	—	4	6	no	—
320	C₄	—	188	6	no	—
321	C₄	—	188	2	no	—
322	C₄	#15689784	1	2	yes	—
323	C₄	C₂²×#10625335	1	12	yes	—
324	C₄	C₂²×#10625354	1	1	yes	characteristic
325	C₄	C₂×#13094072	1	1	yes	characteristic
326	C₂	—	4	4	no	—
327	C₂	#20256030	1	4	yes	direct product factor
328	C₂	C₂×#15689784	1	1	yes	characteristic
329	C₂	C₂³×#10625335	1	1	yes	characteristic
330	C₂	C₂×#15689938	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×D₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₄×C₄₇	C₂³
hypercenter	C₂×D₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#3359
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689852	C₂	semidirect
2. #20256030	C₂	direct (N × H)
3. C₂×#15689475	C₂	semidirect
4. C₂×#15689405	C₂	semidirect
6. C₂×Q₈×C₄₇⋊C₈	C₂	semidirect
11. #15689852	C₂²	semidirect
16. #15689475	C₂²	semidirect
17. #15689405	C₂²	semidirect
19. Q₈×C₄₇⋊C₈	C₂²	semidirect
78. C₂×Q₈×C₄₇	SD₁₆	semidirect
120. Q₈×C₄₇	C₂×SD₁₆	semidirect
170. C₂×C₄₇	#3359	semidirect
199. C₄₇	C₂×#3359	semidirect
248. C₂×Q₈	#10625335	semidirect
289. Q₈	C₂×#10625335	semidirect
327. C₂	#20256030	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×#15689852	2
#20256030	2
C₂×#15689475	2
C₂×#15689405	2
C₂×#15689328	2
C₂×Q₈×C₄₇⋊C₈	2
C₂×C₄×#10625335	2
C₂×D₄×Q₈×C₄₇	2
C₂×#3359	47

Subgroups — C2×#20256030 #27095842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256030 #27095842