Subgroups — C₂×#20256253 #27096706

Order 12032.

1918 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689830	C₂	1	2	yes	maximal, complemented
2	#20256253	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689528	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689524	C₂	1	2	yes	maximal, complemented
5	C₂×#15689421	C₂	1	4	yes	maximal, complemented
6	C₂×#15689413	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₈×Dic₄₇	C₂	1	2	yes	maximal
9	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
10	#15689830	—	2	8	no	—
11	#15689830	C₂²	1	16	yes	complemented
12	C₂×#13094024	C₂²	1	2	yes	—
13	C₂×#13094015	C₂²	1	4	yes	—
14	C₂×#13094003	—	2	2	no	—
15	C₂×#13094001	—	2	4	no	—
16	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
17	C₂×D₄×Dic₄₇	C₂²	1	4	yes	—
18	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
19	#15689528	C₂²	1	8	yes	complemented
20	#15689524	C₂²	1	16	yes	complemented
21	#15689421	C₂²	1	32	yes	complemented
22	#15689413	C₂²	1	16	yes	complemented
23	C₈×Dic₉₄	C₂²	1	8	yes	complemented
24	D₈×Dic₄₇	C₂²	1	16	yes	—
25	C₂×#13093958	C₂²	1	2	yes	—
26	C₂×#13093958	C₂²	1	2	yes	—
27	C₂×#13093955	C₂²	1	4	yes	—
28	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
30	C₂×#13093945	C₂²	1	4	yes	—
31	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
32	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
34	C₄×D₈×C₄₇	C₂²	1	8	yes	—
35	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#102×C₄₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×Dic₉₄	—	2	2	no	—
39	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
40	C₂³×Dic₄₇	—	2	4	no	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	#13094024	—	2	4	no	—
43	#13094024	C₂³	1	8	yes	—
44	#13094015	—	2	8	no	—
45	#13094015	C₂³	1	16	yes	—
46	#13094003	—	2	16	no	—
47	#13094001	—	2	32	no	—
48	C₄×Dic₉₄	D₄	1	4	yes	—
49	C₄×Dic₉₄	C₂³	1	4	yes	—
50	D₄×Dic₄₇	—	2	8	no	—
51	D₄×Dic₄₇	C₂³	1	16	yes	—
52	C₂×C₄⋊Dic₄₇	—	2	2	no	—
53	C₂×C₄⋊Dic₄₇	—	2	4	no	—
54	C₂×C₄⋊Dic₄₇	—	2	16	no	—
55	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
57	C₂×Dic₄₇⋊C₄	—	2	4	no	—
58	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
59	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
60	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
61	#13093958	C₂³	1	8	yes	—
62	#13093958	C₂³	1	8	yes	—
63	#13093955	C₂³	1	16	yes	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
65	Dic₄₇⋊C₈	C₂³	1	8	yes	—
66	#13093945	C₂³	1	16	yes	—
67	C₈×Dic₄₇	C₂³	1	8	yes	—
68	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
69	C₂³×C₄×C₄₇	—	2	2	no	—
70	C₂×D₈×C₄₇	Q₈	1	8	yes	complemented
71	C₂×D₈×C₄₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₄₇	—	2	4	no	—
75	C₄×D₄×C₄₇	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	#107×C₄₇	C₂³	1	4	yes	—
80	#102×C₄₇	C₂³	1	8	yes	—
81	C₄×C₈×C₄₇	C₂³	1	4	yes	—
82	C₂⁴×C₄₇	—	2	2	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	2	4	no	—
85	C₂×D₄×C₄₇	C₂×Q₈	1	8	yes	—
86	C₂×D₄×C₄₇	C₂⁴	1	4	yes	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	—	2	8	no	—
89	C₂²×C₄×C₄₇	—	4	8	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	4	no	—
94	C₂²×Dic₄₇	—	2	16	no	—
95	C₂²×Dic₄₇	—	8	8	no	—
96	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
97	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
98	C₂×Dic₉₄	—	2	4	no	—
99	C₂×Dic₉₄	—	2	8	no	—
100	C₂×Dic₉₄	—	8	4	no	—
101	C₂×Dic₉₄	C₂×D₄	1	4	yes	—
102	D₈×C₄₇	C₂×Q₈	1	16	yes	complemented
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
105	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
106	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	C₂⁴	1	4	yes	—
109	C₂²⋊C₄×C₄₇	—	2	8	no	—
110	C₄²×C₄₇	C₂×D₄	1	2	yes	—
111	C₄²×C₄₇	C₂⁴	1	2	yes	—
112	C₂²⋊Dic₄₇	—	2	16	no	—
113	C₄⋊Dic₄₇	—	2	8	no	—
114	C₄⋊Dic₄₇	—	2	4	no	—
115	C₄⋊Dic₄₇	—	2	16	no	—
116	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
117	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
118	Dic₄₇⋊C₄	—	2	16	no	—
119	Dic₄₇⋊C₄	C₂×D₄	1	8	yes	—
120	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
121	C₄×Dic₄₇	C₂⁴	1	4	yes	—
122	C₂×C₄₇⋊C₈	—	2	4	no	—
123	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
124	C₂³×C₄₇	—	2	8	no	—
125	C₂³×C₄₇	—	2	4	no	—
126	C₂³×C₄₇	—	4	4	no	—
127	C₂³×C₄₇	—	4	4	no	—
128	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	D₄×C₄₇	—	2	8	no	—
130	D₄×C₄₇	—	2	4	no	—
131	D₄×C₄₇	C₂²×Q₈	1	8	yes	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	4	no	—
134	C₂×C₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	4	16	no	—
136	C₂×C₄×C₄₇	#135	1	4	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
140	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
141	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
142	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	2	4	no	—
145	C₂×Dic₄₇	—	2	8	no	—
146	C₂×Dic₄₇	—	4	8	no	—
147	C₂×Dic₄₇	—	8	16	no	—
148	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
149	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
150	Dic₉₄	—	2	8	no	—
151	Dic₉₄	—	8	8	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	C₂²×Q₈	1	4	yes	—
154	C₄₇⋊C₈	—	2	4	no	—
155	C₂×#3380	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	—	2	8	no	—
157	C₂²×C₄₇	—	4	16	no	—
158	C₂²×C₄₇	—	4	4	no	—
159	C₂²×C₄₇	—	4	4	no	—
160	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
161	C₂²×C₄₇	C₂×#135	1	2	yes	—
162	C₂²×C₄₇	#576	1	2	yes	—
163	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	4	4	no	—
166	C₄×C₄₇	D₄×Q₈	1	2	yes	—
167	C₄×C₄₇	C₂×#135	1	4	yes	—
168	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
169	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
170	Dic₄₇	—	2	4	no	—
171	Dic₄₇	—	2	4	no	—
172	Dic₄₇	—	4	8	no	—
173	C₂×#554	—	47	2	no	—
174	#3380	—	47	16	no	—
175	C₂×#500	—	47	1	no	—
176	C₂×#499	—	47	2	no	—
177	C₂×#478	—	47	4	no	—
178	C₂×#476	—	47	2	no	—
179	C₂×C₈×Q₈	—	47	1	no	—
180	C₂×C₄×D₈	—	47	2	no	—
181	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
182	C₂×C₄₇	—	4	8	no	—
183	C₂×C₄₇	#3380	1	4	yes	complemented
184	C₂×C₄₇	C₂×D₄×Q₈	1	1	yes	characteristic
185	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
186	C₂×C₄₇	C₂×#576	1	1	yes	characteristic
187	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
188	#554	—	47	16	no	—
189	#554	—	94	8	no	—
190	C₂×C₄⋊Q₈	—	47	2	no	—
191	C₂×#125	—	47	4	no	—
192	C₂×C₂²⋊Q₈	—	94	4	no	—
193	C₂×C₂²⋊Q₈	—	94	2	no	—
194	C₂×C₄×Q₈	—	47	1	no	—
195	C₂×C₄×D₄	—	47	4	no	—
196	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
197	C₂²×C₄⋊C₄	—	94	2	no	—
198	#500	—	47	8	no	—
199	#499	—	47	16	no	—
200	#478	—	47	32	no	—
201	#476	—	47	16	no	—
202	C₈×Q₈	—	47	8	no	—
203	C₄×D₈	—	47	16	no	—
204	C₄×D₈	C₂×D₄₇	1	8	yes	—
205	C₂×#107	—	47	2	no	—
206	C₂×#107	—	47	2	no	—
207	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
208	C₂×#106	—	47	4	no	—
209	C₂×C₄⋊C₈	—	47	2	no	—
210	C₂×C₄⋊C₈	—	47	1	no	—
211	C₂×#102	—	47	4	no	—
212	C₂×#102	C₂×D₄₇	1	2	yes	—
213	C₂×C₄×C₈	—	47	2	no	—
214	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
215	C₄₇	C₂×#3380	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	—	47	1	no	—
217	C₂²×Q₈	—	94	2	no	—
218	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
219	C₂³×C₄	—	2	2	no	—
220	C₂³×C₄	—	94	4	no	—
221	C₂×D₈	Dic₉₄	1	8	yes	complemented
222	C₂×D₈	C₂²×D₄₇	1	4	yes	—
223	C₂²×C₈	—	47	2	no	—
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	C₄⋊Q₈	—	47	8	no	—
227	C₄⋊Q₈	—	94	4	no	—
228	#125	—	47	16	no	—
229	#125	—	94	8	no	—
230	C₂²⋊Q₈	—	94	32	no	—
231	C₂²⋊Q₈	—	94	16	no	—
232	C₄×Q₈	—	47	4	no	—
233	C₄×Q₈	—	47	4	no	—
234	C₄×D₄	—	2	4	no	—
235	C₄×D₄	—	47	16	no	—
236	C₄×D₄	—	94	8	no	—
237	C₄×D₄	C₂²×D₄₇	1	8	yes	—
238	C₂×C₄⋊C₄	—	47	1	no	—
239	C₂×C₄⋊C₄	—	47	2	no	—
240	C₂×C₄⋊C₄	—	47	4	no	—
241	C₂×C₄⋊C₄	—	94	16	no	—
242	C₂×C₄⋊C₄	—	94	4	no	—
243	C₂×C₄⋊C₄	—	94	4	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
246	C₂×C₂²⋊C₄	—	2	2	no	—
247	C₂×C₂²⋊C₄	—	94	4	no	—
248	C₂×C₄²	—	47	2	no	—
249	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
250	#107	—	47	8	no	—
251	#107	—	47	8	no	—
252	#107	C₂²×D₄₇	1	4	yes	—
253	#106	—	47	16	no	—
254	C₄⋊C₈	—	47	8	no	—
255	C₄⋊C₈	—	47	4	no	—
256	#102	—	47	16	no	—
257	#102	C₂²×D₄₇	1	8	yes	—
258	C₄×C₈	—	47	8	no	—
259	C₄×C₈	C₂²×D₄₇	1	4	yes	—
260	C₂⁴	—	2	2	no	—
261	C₂×Q₈	—	47	4	no	—
262	C₂×Q₈	—	94	4	no	—
263	C₂×Q₈	—	94	8	no	—
264	C₂×Q₈	—	376	4	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	C₂×Dic₉₄	1	8	yes	—
268	C₂×D₄	C₂³×D₄₇	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	8	no	—
271	C₂²×C₄	—	4	8	no	—
272	C₂²×C₄	—	47	2	no	—
273	C₂²×C₄	—	47	2	no	—
274	C₂²×C₄	—	94	16	no	—
275	C₂²×C₄	—	94	4	no	—
276	C₂²×C₄	—	376	8	no	—
277	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
278	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
279	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	D₈	C₂×Dic₉₄	1	16	yes	complemented
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	47	4	no	—
283	C₂×C₈	—	94	4	no	—
284	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
285	C₂×C₈	C₂³×D₄₇	1	2	yes	—
286	C₂×C₈	C₂³×D₄₇	1	2	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	47	8	no	—
289	C₄⋊C₄	—	47	4	no	—
290	C₄⋊C₄	—	47	8	no	—
291	C₄⋊C₄	—	94	16	no	—
292	C₄⋊C₄	—	94	16	no	—
293	C₄⋊C₄	—	94	4	no	—
294	C₄⋊C₄	—	94	8	no	—
295	C₄⋊C₄	C₂³×D₄₇	1	4	yes	—
296	C₂²⋊C₄	—	2	8	no	—
297	C₂²⋊C₄	—	94	16	no	—
298	C₄²	—	47	4	no	—
299	C₄²	—	47	4	no	—
300	C₄²	D₄×D₄₇	1	2	yes	—
301	C₄²	C₂³×D₄₇	1	2	yes	—
302	C₂³	—	2	8	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	4	4	no	—
305	C₂³	—	4	4	no	—
306	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
307	Q₈	—	94	8	no	—
308	Q₈	—	376	8	no	—
309	D₄	—	2	8	no	—
310	D₄	—	2	4	no	—
311	D₄	C₂²×Dic₉₄	1	8	yes	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	16	no	—
316	C₂×C₄	—	47	4	no	—
317	C₂×C₄	—	47	4	no	—
318	C₂×C₄	—	94	4	no	—
319	C₂×C₄	—	94	4	no	—
320	C₂×C₄	—	94	8	no	—
321	C₂×C₄	—	376	16	no	—
322	C₂×C₄	—	188	8	no	—
323	C₂×C₄	#13094043	1	4	yes	—
324	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
325	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
327	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
328	C₂×C₄	#13094071	1	2	yes	—
329	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	94	4	no	—
332	C₈	C₂²×Dic₉₄	1	4	yes	—
333	C₂²	—	2	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	4	4	no	—
336	C₂²	—	4	16	no	—
337	C₂²	D₄×Dic₉₄	1	2	yes	—
338	C₂²	C₂×#13094043	1	2	yes	—
339	C₂²	#15689925	1	2	yes	—
340	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	94	4	no	—
344	C₄	—	94	4	no	—
345	C₄	—	188	8	no	—
346	C₄	D₄×Dic₉₄	1	2	yes	—
347	C₄	C₂×#13094043	1	4	yes	—
348	C₄	C₂³×Dic₉₄	1	1	yes	characteristic
349	C₄	C₂×#13094071	1	1	yes	characteristic
350	C₂	—	4	8	no	—
351	C₂	#20256253	1	4	yes	direct product factor
352	C₂	C₂×D₄×Dic₉₄	1	1	yes	characteristic
353	C₂	C₂²×#13094043	1	1	yes	characteristic
354	C₂	C₂×#15689925	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	C₂×#3380
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689830	C₂	semidirect
2. #20256253	C₂	direct (N × H)
3. C₂×#15689528	C₂	semidirect
4. C₂×#15689524	C₂	semidirect
5. C₂×#15689421	C₂	semidirect
6. C₂×#15689413	C₂	semidirect
7. C₂×C₈×Dic₉₄	C₂	semidirect
11. #15689830	C₂²	semidirect
19. #15689528	C₂²	semidirect
20. #15689524	C₂²	semidirect
21. #15689421	C₂²	semidirect
22. #15689413	C₂²	semidirect
23. C₈×Dic₉₄	C₂²	semidirect
70. C₂×D₈×C₄₇	Q₈	semidirect
102. D₈×C₄₇	C₂×Q₈	semidirect
183. C₂×C₄₇	#3380	semidirect
215. C₄₇	C₂×#3380	semidirect
221. C₂×D₈	Dic₉₄	semidirect
280. D₈	C₂×Dic₉₄	semidirect
351. C₂	#20256253	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×#15689830	2
#20256253	2
C₂×#15689528	2
C₂×#15689524	2
C₂×#15689421	2
C₂×#15689413	2
C₂×C₈×Dic₉₄	2
C₂×D₈×Dic₄₇	2
C₂×C₄×D₈×C₄₇	2
C₂×#3380	47

Subgroups — C2×#20256253 #27096706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256253 #27096706