Subgroups — C₂×#20257883 #27124115

Order 12032.

6338 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20257883	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₄₇⋊#564	C₂	1	4	yes	maximal, complemented
3	C₂×#15689852	C₂	1	4	yes	maximal, complemented
4	C₂×#15689851	C₂	1	4	yes	maximal, complemented
5	C₂×#15689848	C₂	1	4	yes	maximal, complemented
6	C₂×#15689735	C₂	1	4	yes	maximal
7	C₂×#15689733	C₂	1	8	yes	maximal
8	C₂×#15689680	C₂	1	2	yes	maximal
9	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
10	C₄₇⋊#564	C₂²	1	64	yes	complemented
11	#15689852	C₂²	1	64	yes	complemented
12	#15689851	C₂²	1	64	yes	complemented
13	#15689848	C₂²	1	64	yes	complemented
14	#15689735	C₂²	1	64	yes	—
15	#15689733	C₂²	1	128	yes	—
16	#15689680	C₂²	1	32	yes	—
17	C₂×#13094024	C₂²	1	2	yes	—
18	C₂×#13094024	C₂²	1	2	yes	—
19	C₂×#13094024	C₂²	1	4	yes	—
20	C₂×#13094022	C₂²	1	8	yes	—
21	C₂×#13094017	C₂²	1	16	yes	—
22	C₂×#13094016	C₂²	1	4	yes	—
23	C₂×#13094016	C₂²	1	4	yes	—
24	C₂×#13094015	C₂²	1	8	yes	—
25	C₂×#13094015	C₂²	1	16	yes	—
26	C₂×#13094014	C₂²	1	16	yes	—
27	C₂×#13094014	C₂²	1	16	yes	—
28	C₂×#13094003	C₂²	1	4	yes	—
29	C₂×#13094003	C₂²	1	4	yes	—
30	C₂×Q₈×Dic₄₇	C₂²	1	8	yes	—
31	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
32	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
33	C₂×C₄×Dic₉₄	C₂²	1	8	yes	—
34	C₂×#13093989	C₂²	1	8	yes	—
35	C₂×#13093983	C₂²	1	4	yes	—
36	C₄₇×#533	C₂²	1	16	yes	—
37	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
38	C₂×#125×C₄₇	C₂²	1	2	yes	—
39	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	—
40	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
41	C₂×#117×C₄₇	C₂²	1	2	yes	—
42	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
43	C₂²×Dic₉₄	C₂³	1	4	yes	—
44	C₂²×Dic₉₄	C₂³	1	4	yes	—
45	#13094024	C₂³	1	32	yes	—
46	#13094024	C₂³	1	16	yes	—
47	#13094024	C₂³	1	16	yes	—
48	#13094022	C₂³	1	64	yes	—
49	#13094017	C₂³	1	128	yes	—
50	#13094016	C₂³	1	32	yes	—
51	#13094016	C₂³	1	32	yes	—
52	#13094015	C₂³	1	64	yes	—
53	#13094015	C₂³	1	128	yes	—
54	#13094014	C₂³	1	128	yes	—
55	#13094014	C₂³	1	128	yes	—
56	#13094003	C₂³	1	32	yes	—
57	#13094003	C₂³	1	32	yes	—
58	Q₈×Dic₄₇	C₂³	1	64	yes	—
59	C₄×Dic₉₄	C₂³	1	64	yes	—
60	C₄×Dic₉₄	C₂³	1	32	yes	—
61	C₄×Dic₉₄	C₂³	1	32	yes	—
62	#13093989	C₂³	1	64	yes	—
63	#13093983	C₂³	1	32	yes	—
64	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
65	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
66	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
67	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
68	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	—
69	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
70	C₂×Dic₄₇⋊C₄	C₂³	1	8	yes	—
71	C₂×Dic₄₇⋊C₄	C₂³	1	8	yes	—
72	C₂×Dic₄₇⋊C₄	C₂³	1	16	yes	—
73	C₂×C₂²⋊Dic₄₇	C₂³	1	8	yes	—
74	C₂×C₄×Dic₄₇	C₂³	1	8	yes	—
75	C₂×C₄×Dic₄₇	C₂³	1	16	yes	—
76	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
77	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₄₇	C₂³	1	16	yes	—
80	#125×C₄₇	C₂³	1	16	yes	—
81	C₂²⋊Q₈×C₄₇	C₂³	1	32	yes	—
82	C₄×Q₈×C₄₇	C₂³	1	32	yes	—
83	#117×C₄₇	C₂³	1	16	yes	—
84	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
89	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₄₇	C₂³	1	8	yes	—
92	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
93	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂×Q₈×C₄₇	—	2	16	no	—
95	C₂×Q₈×C₄₇	C₂⁴	1	16	yes	—
96	C₂²×C₄×C₄₇	—	2	8	no	—
97	C₂²×C₄×C₄₇	—	2	4	no	—
98	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
100	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
101	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
102	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
103	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
104	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
105	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
106	C₂²×Dic₄₇	C₂⁴	1	8	yes	—
107	C₂×Dic₉₄	—	2	16	no	—
108	C₂×Dic₉₄	—	2	16	no	—
109	C₂×Dic₉₄	C₂⁴	1	16	yes	—
110	C₂×Dic₉₄	C₂⁴	1	16	yes	—
111	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
112	C₄⋊C₄×C₄₇	C₂⁴	1	32	yes	—
113	C₄⋊C₄×C₄₇	C₂⁴	1	16	yes	—
114	C₂²⋊C₄×C₄₇	C₂⁴	1	16	yes	—
115	C₄²×C₄₇	C₂⁴	1	8	yes	—
116	C₄²×C₄₇	C₂⁴	1	4	yes	—
117	C₄²×C₄₇	C₂⁴	1	4	yes	—
118	C₄²×C₄₇	C₂⁴	1	16	yes	—
119	C₂²⋊Dic₄₇	C₂⁴	1	32	yes	—
120	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
121	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
122	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
123	C₄⋊Dic₄₇	C₂⁴	1	16	yes	—
124	C₄⋊Dic₄₇	C₂⁴	1	64	yes	—
125	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
126	Dic₄₇⋊C₄	C₂⁴	1	32	yes	—
127	Dic₄₇⋊C₄	C₂⁴	1	32	yes	—
128	Dic₄₇⋊C₄	C₂⁴	1	64	yes	—
129	C₄×Dic₄₇	C₂⁴	1	32	yes	—
130	C₄×Dic₄₇	C₂⁴	1	64	yes	—
131	C₂³×C₄₇	—	2	4	no	—
132	C₂³×C₄₇	—	2	2	no	—
133	C₂³×C₄₇	C₂⁵	1	2	yes	—
134	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
135	Q₈×C₄₇	—	2	32	no	—
136	C₂×C₄×C₄₇	—	2	4	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	16	no	—
139	C₂×C₄×C₄₇	—	2	16	no	—
140	C₂×C₄×C₄₇	—	2	8	no	—
141	C₂×C₄×C₄₇	2₋¹⁺⁴	1	8	yes	—
142	C₂×C₄×C₄₇	2₋¹⁺⁴	1	4	yes	—
143	C₂×C₄×C₄₇	C₂⁵	1	16	yes	—
144	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
145	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
146	C₂×C₄×C₄₇	C₂⁵	1	4	yes	—
147	C₂×C₄×C₄₇	C₂⁵	1	2	yes	—
148	C₂×Dic₄₇	—	2	16	no	—
149	C₂×Dic₄₇	—	2	16	no	—
150	C₂×Dic₄₇	C₂⁵	1	16	yes	—
151	C₂×Dic₄₇	C₂⁵	1	16	yes	—
152	Dic₉₄	—	2	32	no	—
153	Dic₉₄	—	2	32	no	—
154	C₂×#3556	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	4	no	—
156	C₂²×C₄₇	—	2	2	no	—
157	C₂²×C₄₇	—	4	4	no	—
158	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	4	yes	—
159	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
160	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	16	no	—
164	C₄×C₄₇	C₂×2₋¹⁺⁴	1	8	yes	—
165	C₄×C₄₇	C₂×2₋¹⁺⁴	1	4	yes	—
166	Dic₄₇	—	2	16	no	—
167	Dic₄₇	—	2	16	no	—
168	#3556	—	47	32	no	—
169	C₂×#564	—	47	4	no	—
170	C₂×#557	—	47	4	no	—
171	C₂×#557	—	47	4	no	—
172	C₂×#557	—	47	4	no	—
173	C₂×#541	—	47	8	no	—
174	C₂×#541	—	47	4	no	—
175	C₂×#533	—	47	2	no	—
176	C₂×#533	D₄₇	1	1	yes	characteristic
177	C₂×C₄₇	—	4	2	no	—
178	C₂×C₄₇	#3556	1	4	yes	complemented
179	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	2	yes	—
180	C₂×C₄₇	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
181	#564	—	47	64	no	—
182	#557	—	47	64	no	—
183	#557	—	47	64	no	—
184	#557	—	47	64	no	—
185	#541	—	47	128	no	—
186	#541	—	47	64	no	—
187	#533	—	47	32	no	—
188	#533	C₂×D₄₇	1	16	yes	—
189	C₂×C₄⋊Q₈	—	47	8	no	—
190	C₂×C₄⋊Q₈	—	47	2	no	—
191	C₂×C₄⋊Q₈	—	47	2	no	—
192	C₂×C₄⋊Q₈	—	47	4	no	—
193	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
194	C₂×#126	—	47	16	no	—
195	C₂×#125	—	47	8	no	—
196	C₂×#125	—	47	16	no	—
197	C₂×#125	—	47	16	no	—
198	C₂×#125	—	47	16	no	—
199	C₂×#125	—	47	4	no	—
200	C₂×#125	—	47	4	no	—
201	C₂×#125	C₂×D₄₇	1	2	yes	—
202	C₂×C₂²⋊Q₈	—	47	4	no	—
203	C₂×C₂²⋊Q₈	—	47	4	no	—
204	C₂×C₂²⋊Q₈	C₂×D₄₇	1	4	yes	—
205	C₂×C₄×Q₈	—	47	8	no	—
206	C₂×C₄×Q₈	—	47	8	no	—
207	C₂×C₄×Q₈	—	47	4	no	—
208	C₂×C₄×Q₈	—	47	4	no	—
209	C₂×C₄×Q₈	—	47	8	no	—
210	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
211	C₂×#117	—	47	4	no	—
212	C₂×#117	C₂×D₄₇	1	2	yes	—
213	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
214	C₄₇	C₂×#3556	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₈	—	47	4	no	—
216	C₂²×Q₈	—	47	4	no	—
217	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
218	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
219	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₄⋊Q₈	—	47	64	no	—
221	C₄⋊Q₈	—	47	16	no	—
222	C₄⋊Q₈	—	47	16	no	—
223	C₄⋊Q₈	—	47	32	no	—
224	C₄⋊Q₈	C₂²×D₄₇	1	16	yes	—
225	#126	—	47	128	no	—
226	#125	—	47	64	no	—
227	#125	—	47	128	no	—
228	#125	—	47	128	no	—
229	#125	—	47	128	no	—
230	#125	—	47	32	no	—
231	#125	—	47	32	no	—
232	#125	C₂²×D₄₇	1	16	yes	—
233	C₂²⋊Q₈	—	47	32	no	—
234	C₂²⋊Q₈	—	47	32	no	—
235	C₂²⋊Q₈	C₂²×D₄₇	1	32	yes	—
236	C₄×Q₈	—	47	64	no	—
237	C₄×Q₈	—	47	64	no	—
238	C₄×Q₈	—	47	32	no	—
239	C₄×Q₈	—	47	32	no	—
240	C₄×Q₈	—	47	64	no	—
241	C₄×Q₈	C₂²×D₄₇	1	32	yes	—
242	#117	—	47	32	no	—
243	#117	C₂²×D₄₇	1	16	yes	—
244	C₂×C₄⋊C₄	—	47	4	no	—
245	C₂×C₄⋊C₄	—	47	4	no	—
246	C₂×C₄⋊C₄	—	47	4	no	—
247	C₂×C₄⋊C₄	—	47	4	no	—
248	C₂×C₄⋊C₄	—	47	8	no	—
249	C₂×C₄⋊C₄	—	47	8	no	—
250	C₂×C₄⋊C₄	—	47	16	no	—
251	C₂×C₄⋊C₄	—	47	16	no	—
252	C₂×C₄⋊C₄	—	47	16	no	—
253	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
254	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
255	C₂×C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
256	C₂×C₂²⋊C₄	—	47	8	no	—
257	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
258	C₂×C₄²	—	47	8	no	—
259	C₂×C₄²	—	47	16	no	—
260	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
261	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
262	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
263	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
264	C₂×C₄²	C₂²×D₄₇	1	8	yes	—
265	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
266	C₂×Q₈	—	2	16	no	—
267	C₂×Q₈	—	47	16	no	—
268	C₂×Q₈	—	47	16	no	—
269	C₂×Q₈	—	94	16	no	—
270	C₂×Q₈	—	94	16	no	—
271	C₂×Q₈	C₂³×D₄₇	1	16	yes	—
272	C₂²×C₄	—	2	8	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	47	8	no	—
275	C₂²×C₄	—	47	8	no	—
276	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
277	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
278	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
279	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
280	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
281	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
282	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
283	C₄⋊C₄	—	47	32	no	—
284	C₄⋊C₄	—	47	32	no	—
285	C₄⋊C₄	—	47	64	no	—
286	C₄⋊C₄	—	47	64	no	—
287	C₄⋊C₄	—	47	16	no	—
288	C₄⋊C₄	—	47	16	no	—
289	C₄⋊C₄	—	47	16	no	—
290	C₄⋊C₄	—	47	16	no	—
291	C₄⋊C₄	—	47	64	no	—
292	C₄⋊C₄	C₂³×D₄₇	1	16	yes	—
293	C₄⋊C₄	C₂³×D₄₇	1	16	yes	—
294	C₄⋊C₄	C₂³×D₄₇	1	32	yes	—
295	C₂²⋊C₄	—	47	32	no	—
296	C₂²⋊C₄	C₂³×D₄₇	1	16	yes	—
297	C₄²	—	47	32	no	—
298	C₄²	—	47	64	no	—
299	C₄²	C₂³×D₄₇	1	4	yes	—
300	C₄²	C₂³×D₄₇	1	8	yes	—
301	C₄²	C₂³×D₄₇	1	4	yes	—
302	C₄²	C₂³×D₄₇	1	16	yes	—
303	C₂³	—	2	4	no	—
304	C₂³	—	2	2	no	—
305	C₂³	C₂⁴×D₄₇	1	2	yes	—
306	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
307	Q₈	—	2	32	no	—
308	Q₈	—	94	32	no	—
309	Q₈	—	94	32	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	16	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	16	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	47	16	no	—
316	C₂×C₄	—	47	16	no	—
317	C₂×C₄	—	94	16	no	—
318	C₂×C₄	—	94	16	no	—
319	C₂×C₄	#13094071	1	8	yes	—
320	C₂×C₄	#13094072	1	4	yes	—
321	C₂×C₄	C₂⁴×D₄₇	1	16	yes	—
322	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
323	C₂×C₄	C₂⁴×D₄₇	1	2	yes	—
324	C₂×C₄	C₂⁴×D₄₇	1	4	yes	—
325	C₂×C₄	C₂⁴×D₄₇	1	2	yes	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	C₂×#13094071	1	4	yes	—
330	C₂²	C₂×#13094072	1	2	yes	—
331	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	2	16	no	—
334	C₄	—	2	2	no	—
335	C₄	—	94	16	no	—
336	C₄	—	94	16	no	—
337	C₄	C₂×#13094071	1	8	yes	—
338	C₄	C₂×#13094072	1	4	yes	—
339	C₂	—	4	2	no	—
340	C₂	#20257883	1	4	yes	direct product factor
341	C₂	C₂²×#13094071	1	2	yes	—
342	C₂	C₂²×#13094072	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#533	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	C₂×#3556
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #20257883	C₂	direct (N × H)
2. C₂×C₄₇⋊#564	C₂	semidirect
3. C₂×#15689852	C₂	semidirect
4. C₂×#15689851	C₂	semidirect
5. C₂×#15689848	C₂	semidirect
10. C₄₇⋊#564	C₂²	semidirect
11. #15689852	C₂²	semidirect
12. #15689851	C₂²	semidirect
13. #15689848	C₂²	semidirect
178. C₂×C₄₇	#3556	semidirect
214. C₄₇	C₂×#3556	semidirect
340. C₂	#20257883	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20257883	2
C₂×C₄₇⋊#564	2
C₂×#15689852	2
C₂×#15689851	2
C₂×#15689848	2
C₂×#15689735	2
C₂×#15689733	2
C₂×#15689680	2
C₂×C₄₇×#533	2
C₂×#3556	47

Subgroups — C2×#20257883 #27124115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20257883 #27124115