Subgroups — C₂×#20253898 #27085879

Order 12032.

1942 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689680	C₂	1	1	yes	maximal, characteristic
2	#20253898	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689526	C₂	1	2	yes	maximal, complemented
4	C₂×#15689522	C₂	1	4	yes	maximal, complemented
5	C₂×#15689519	C₂	1	2	yes	maximal, complemented
6	C₂×#15689159	C₂	1	4	yes	maximal
7	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
10	#15689680	—	2	4	no	—
11	#15689680	C₂²	1	8	yes	—
12	C₂×#13094024	C₂²	1	2	yes	—
13	C₂×#13094016	C₂²	1	2	yes	—
14	C₂×#13094003	—	2	2	no	—
15	C₂×C₄×Dic₉₄	—	2	2	no	—
16	C₂×#13093983	C₂²	1	2	yes	—
17	#15689526	C₂²	1	16	yes	complemented
18	#15689522	C₂²	1	32	yes	complemented
19	#15689519	C₂²	1	16	yes	complemented
20	#15689159	C₂²	1	32	yes	—
21	C₂×#13093957	C₂²	1	8	yes	—
22	C₂×#13093955	C₂²	1	8	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	8	yes	—
27	C₄₇×#533	—	2	4	no	—
28	C₄₇×#533	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
30	C₂×#125×C₄₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
32	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
33	C₂×#117×C₄₇	C₂²	1	2	yes	—
34	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	Q₈	1	16	yes	complemented
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
40	#13094024	—	2	4	no	—
41	#13094024	C₂³	1	8	yes	—
42	#13094016	—	2	4	no	—
43	#13094016	C₂³	1	8	yes	—
44	#13094003	—	2	16	no	—
45	C₄×Dic₉₄	—	2	16	no	—
46	#13093983	—	2	4	no	—
47	#13093983	C₂³	1	8	yes	—
48	C₂×C₄⋊Dic₄₇	—	2	2	no	—
49	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
50	C₂×Dic₄₇⋊C₄	—	2	4	no	—
51	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
52	C₂×C₄×Dic₄₇	—	2	2	no	—
53	#13093957	C₂³	1	32	yes	—
54	#13093955	C₂³	1	32	yes	—
55	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
56	C₄×C₄₇⋊C₈	C₂³	1	8	yes	—
57	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
58	C₂²×Q₈×C₄₇	—	2	2	no	—
59	C₂³×C₄×C₄₇	D₄	1	2	yes	—
60	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₇	—	2	4	no	—
62	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	—
63	#125×C₄₇	—	2	4	no	—
64	#125×C₄₇	C₂³	1	8	yes	—
65	C₂²⋊Q₈×C₄₇	—	2	16	no	—
66	C₄×Q₈×C₄₇	—	2	16	no	—
67	#117×C₄₇	—	2	4	no	—
68	#117×C₄₇	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
70	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
72	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
73	C₂×C₄²×C₄₇	—	2	2	no	—
74	C₂×C₄²×C₄₇	D₄	1	2	yes	—
75	C₂×C₄²×C₄₇	D₄	1	4	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
79	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂×Q₈×C₄₇	—	2	8	no	—
81	C₂×Q₈×C₄₇	—	4	8	no	—
82	C₂²×C₄×C₄₇	—	2	8	no	—
83	C₂²×C₄×C₄₇	—	2	4	no	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×Q₈	1	8	yes	—
90	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
92	C₂²×Dic₄₇	—	2	4	no	—
93	C₂×Dic₉₄	—	2	8	no	—
94	C₂×Dic₉₄	—	4	8	no	—
95	C₄⋊C₄×C₄₇	—	2	16	no	—
96	C₄⋊C₄×C₄₇	—	2	8	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	C₂⁴	1	8	yes	—
99	C₂²⋊C₄×C₄₇	—	2	8	no	—
100	C₄²×C₄₇	—	2	8	no	—
101	C₄²×C₄₇	C₂×D₄	1	8	yes	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	C₄²×C₄₇	C₂⁴	1	2	yes	—
105	C₄²×C₄₇	C₂⁴	1	2	yes	—
106	C₂²⋊Dic₄₇	—	2	8	no	—
107	C₄⋊Dic₄₇	—	2	8	no	—
108	C₄⋊Dic₄₇	—	2	4	no	—
109	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
110	Dic₄₇⋊C₄	—	2	16	no	—
111	C₄×Dic₄₇	—	2	8	no	—
112	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
113	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
114	C₂×C₄₇⋊C₈	C₂×Q₈	1	32	yes	complemented
115	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
116	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
117	C₂³×C₄₇	—	2	4	no	—
118	C₂³×C₄₇	C₄⋊Q₈	1	4	yes	—
119	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
120	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₄₇	—	4	16	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	16	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	4	8	no	—
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
128	C₂×C₄×C₄₇	#135	1	8	yes	—
129	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₄₇	C₂²×Q₈	1	4	yes	—
134	C₂×C₄×C₄₇	C₂²×Q₈	1	8	yes	—
135	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
136	C₂×Dic₄₇	—	2	8	no	—
137	C₂×Dic₄₇	—	4	8	no	—
138	Dic₉₄	—	4	16	no	—
139	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
140	C₄₇⋊C₈	C₂²×Q₈	1	8	yes	—
141	C₂×#3160	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	8	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
145	C₂²×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
146	C₂²×C₄₇	C₂×#135	1	4	yes	—
147	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	4	8	no	—
150	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
151	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
152	C₄×C₄₇	C₂×#135	1	8	yes	—
153	C₄×C₄₇	C₂³×Q₈	1	2	yes	—
154	Dic₄₇	—	4	8	no	—
155	C₂×#533	—	47	1	no	—
156	C₂×#533	D₄₇	1	1	yes	characteristic
157	#3160	—	47	16	no	—
158	C₂×#500	—	47	2	no	—
159	C₂×#499	—	47	4	no	—
160	C₂×#498	—	47	2	no	—
161	C₂×#427	—	47	4	no	—
162	C₂²×C₄×C₈	—	47	1	no	—
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	#3160	1	4	yes	complemented
165	C₂×C₄₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₄₇	C₂²×#135	1	2	yes	—
167	C₂³×C₈	—	47	2	no	—
168	#533	—	2	4	no	—
169	#533	—	47	8	no	—
170	#533	—	94	4	no	—
171	#533	C₂×D₄₇	1	8	yes	—
172	C₂×C₄⋊Q₈	—	47	2	no	—
173	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
174	C₂×#125	—	47	2	no	—
175	C₂×#125	C₂×D₄₇	1	2	yes	—
176	C₂×C₂²⋊Q₈	—	2	2	no	—
177	C₂×C₂²⋊Q₈	—	94	2	no	—
178	C₂×C₄×Q₈	—	2	2	no	—
179	C₂×C₄×Q₈	—	94	2	no	—
180	C₂×#117	—	47	2	no	—
181	C₂×#117	C₂×D₄₇	1	2	yes	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	#500	—	47	16	no	—
184	#499	—	47	32	no	—
185	#498	—	47	16	no	—
186	#427	—	47	32	no	—
187	C₂×#107	—	47	8	no	—
188	C₂×#106	—	47	8	no	—
189	C₂×C₄×C₈	—	47	1	no	—
190	C₂×C₄×C₈	—	47	2	no	—
191	C₂×C₄×C₈	—	47	1	no	—
192	C₂×C₄×C₈	—	47	8	no	—
193	C₄₇	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	—	2	2	no	—
195	C₂²×Q₈	—	94	2	no	—
196	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂²×C₈	—	47	16	no	—
199	C₂²×C₈	—	47	2	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	47	8	no	—
202	C₄⋊Q₈	—	2	4	no	—
203	C₄⋊Q₈	—	47	8	no	—
204	C₄⋊Q₈	—	94	4	no	—
205	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	—
206	#125	—	2	4	no	—
207	#125	—	47	8	no	—
208	#125	—	94	4	no	—
209	#125	C₂²×D₄₇	1	8	yes	—
210	C₂²⋊Q₈	—	2	16	no	—
211	C₂²⋊Q₈	—	94	16	no	—
212	C₄×Q₈	—	2	16	no	—
213	C₄×Q₈	—	94	16	no	—
214	#117	—	2	4	no	—
215	#117	—	47	8	no	—
216	#117	—	94	4	no	—
217	#117	C₂²×D₄₇	1	8	yes	—
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	47	4	no	—
221	C₂×C₄⋊C₄	—	94	4	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	94	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	94	2	no	—
228	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
229	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
233	#107	—	47	32	no	—
234	#106	—	47	32	no	—
235	C₄×C₈	—	47	4	no	—
236	C₄×C₈	—	47	8	no	—
237	C₄×C₈	—	47	4	no	—
238	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	4	8	no	—
241	C₂×Q₈	—	94	8	no	—
242	C₂×Q₈	—	188	8	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	94	4	no	—
246	C₂²×C₄	Q₈×D₄₇	1	8	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
254	C₂×C₈	—	47	8	no	—
255	C₂×C₈	—	47	8	no	—
256	C₂×C₈	—	47	32	no	—
257	C₂×C₈	—	47	4	no	—
258	C₂×C₈	—	47	4	no	—
259	C₄⋊C₄	—	2	16	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	47	8	no	—
263	C₄⋊C₄	—	94	16	no	—
264	C₄⋊C₄	—	94	4	no	—
265	C₄⋊C₄	—	94	8	no	—
266	C₄⋊C₄	C₂³×D₄₇	1	8	yes	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	94	8	no	—
269	C₄²	—	2	8	no	—
270	C₄²	—	94	8	no	—
271	C₄²	C₂×C₂²⋊D₄₇	1	8	yes	—
272	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
274	C₄²	C₂³×D₄₇	1	2	yes	—
275	C₄²	C₂³×D₄₇	1	2	yes	—
276	C₂³	—	2	4	no	—
277	C₂³	#13094022	1	4	yes	—
278	C₂³	C₂²×C₂²⋊D₄₇	1	2	yes	—
279	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
280	Q₈	—	4	16	no	—
281	Q₈	—	188	16	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	94	8	no	—
287	C₂×C₄	—	188	8	no	—
288	C₂×C₄	#13094022	1	8	yes	—
289	C₂×C₄	#13094022	1	4	yes	—
290	C₂×C₄	#13094044	1	8	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₄₇	1	4	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
294	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₄₇	1	8	yes	—
296	C₂×C₄	C₂×Q₈×D₄₇	1	4	yes	—
297	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
298	C₈	—	47	8	no	—
299	C₈	—	47	8	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	8	no	—
302	C₂²	C₂×#13094022	1	4	yes	—
303	C₂²	C₂×#13094022	1	2	yes	—
304	C₂²	C₂×#13094044	1	4	yes	—
305	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	4	8	no	—
308	C₄	—	188	8	no	—
309	C₄	C₂×#13094022	1	2	yes	—
310	C₄	C₂×#13094022	1	4	yes	—
311	C₄	C₂×#13094044	1	8	yes	—
312	C₄	C₂²×Q₈×D₄₇	1	2	yes	—
313	C₂	—	2	4	no	—
314	C₂	#20253898	1	4	yes	direct product factor
315	C₂	C₂²×#13094022	1	1	yes	characteristic
316	C₂	C₂²×#13094044	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#3160
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #20253898	C₂	direct (N × H)
3. C₂×#15689526	C₂	semidirect
4. C₂×#15689522	C₂	semidirect
5. C₂×#15689519	C₂	semidirect
17. #15689526	C₂²	semidirect
18. #15689522	C₂²	semidirect
19. #15689519	C₂²	semidirect
36. C₂²×C₄₇⋊C₈	Q₈	semidirect
114. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
164. C₂×C₄₇	#3160	semidirect
193. C₄₇	C₂×#3160	semidirect
314. C₂	#20253898	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×#15689680	2
#20253898	2
C₂×#15689526	2
C₂×#15689522	2
C₂×#15689519	2
C₂×#15689159	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×C₄₇×#533	2
C₂×#3160	47

Subgroups — C2×#20253898 #27085879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20253898 #27085879