Subgroups — C₂×#20244348 #26220827

Order 12032.

1550 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20244348	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	4	yes	maximal
5	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₉₄	—	2	1	no	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#13094024	—	2	2	no	—
9	C₂×#13094024	—	2	4	no	—
10	C₂×#13094024	C₄	1	8	yes	complemented
11	C₂×#13094024	C₂²	1	4	yes	—
12	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
13	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
16	#15688885	C₂²	1	16	yes	—
17	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#384	—	2	2	no	—
22	C₄₇×#384	C₂²	1	4	yes	—
23	C₂²×Dic₉₄	—	2	8	no	—
24	C₂²×Dic₉₄	—	2	4	no	—
25	C₂²×Dic₉₄	—	4	4	no	—
26	C₂²×Dic₉₄	—	4	4	no	—
27	C₂³×Dic₄₇	—	2	2	no	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	—	2	4	no	—
30	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	#13094024	—	2	16	no	—
33	#13094024	—	2	8	no	—
34	#13094024	C₂×C₄	1	16	yes	complemented
35	C₂×C₄⋊Dic₄₇	—	2	8	no	—
36	C₂×C₄⋊Dic₄₇	—	2	4	no	—
37	C₂×C₄⋊Dic₄₇	—	2	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	4	no	—
39	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
41	C₄×C₄₇⋊C₈	—	2	8	no	—
42	C₂³×C₄×C₄₇	—	2	2	no	—
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
51	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	D₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	4	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	D₈	1	4	yes	—
65	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
67	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₄₇	#41	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	8	no	—
77	C₂²×Dic₄₇	—	2	4	no	—
78	C₂²×Dic₄₇	—	4	4	no	—
79	C₂²×Dic₄₇	—	4	4	no	—
80	C₂×Dic₉₄	—	2	16	no	—
81	C₂×Dic₉₄	—	4	32	no	—
82	C₂×Dic₉₄	—	4	8	no	—
83	C₂×Dic₉₄	—	4	8	no	—
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₄⋊C₄×C₄₇	—	4	4	no	—
86	C₄⋊C₄×C₄₇	—	4	8	no	—
87	C₄⋊C₄×C₄₇	—	4	8	no	—
88	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
90	C₄²×C₄₇	C₂×D₄	1	4	yes	—
91	C₄⋊Dic₄₇	—	2	16	no	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
95	C₂×C₄₇⋊C₈	—	2	16	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	complemented
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂³×C₄₇	#124	1	2	yes	—
101	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
102	C₂³×C₄₇	C₂×D₈	1	2	yes	—
103	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
104	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
105	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
106	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	8	8	no	—
111	C₂×C₄×C₄₇	#102	1	16	yes	—
112	C₂×C₄×C₄₇	#103	1	16	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
118	C₂×C₄×C₄₇	#124	1	2	yes	—
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	16	no	—
132	Dic₉₄	—	4	32	no	—
133	C₄₇⋊C₈	—	2	8	no	—
134	C₂×#1990	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	#391	1	4	yes	—
136	C₂²×C₄₇	C₂×#102	1	4	yes	—
137	C₂²×C₄₇	C₂×#103	1	4	yes	—
138	C₂²×C₄₇	#486	1	4	yes	—
139	C₂²×C₄₇	#487	1	4	yes	—
140	C₂²×C₄₇	#498	1	4	yes	—
141	C₂²×C₄₇	#500	1	4	yes	—
142	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
147	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
149	C₄×C₄₇	—	8	4	no	—
150	C₄×C₄₇	#391	1	4	yes	—
151	C₄×C₄₇	C₂×#102	1	8	yes	—
152	C₄×C₄₇	C₂×#103	1	8	yes	—
153	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
154	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
155	Dic₄₇	—	4	8	no	—
156	C₂²×C₄⋊Q₈	—	47	1	no	—
157	C₂²×C₄×C₈	—	47	1	no	—
158	C₂×#384	D₄₇	1	1	yes	characteristic
159	#1990	—	47	8	no	—
160	C₂×#340	—	47	4	no	—
161	C₂×C₄₇	#1990	1	8	yes	complemented
162	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
163	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
164	C₂×C₄₇	C₂²×#103	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#487	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#498	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#500	1	1	yes	characteristic
169	C₂³×Q₈	—	94	1	no	—
170	C₂³×C₈	—	47	2	no	—
171	C₂×C₄⋊Q₈	—	47	8	no	—
172	C₂×C₄⋊Q₈	—	47	4	no	—
173	C₂×C₄⋊Q₈	—	94	4	no	—
174	C₂×C₄⋊Q₈	—	94	2	no	—
175	C₂²×C₄⋊C₄	—	2	1	no	—
176	C₂²×C₄⋊C₄	—	2	1	no	—
177	C₂²×C₄⋊C₄	—	47	2	no	—
178	C₂²×C₄⋊C₄	—	94	1	no	—
179	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
180	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	C₂×C₄×C₈	—	47	4	no	—
182	C₂×C₄×C₈	—	94	4	no	—
183	#384	—	2	2	no	—
184	#384	C₂×D₄₇	1	4	yes	—
185	#340	—	47	16	no	—
186	C₄₇	C₂×#1990	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	—	94	8	no	—
188	C₂²×Q₈	—	94	4	no	—
189	C₂²×Q₈	—	188	4	no	—
190	C₂²×Q₈	—	188	4	no	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	—	94	2	no	—
193	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	47	8	no	—
197	C₂²×C₈	—	47	4	no	—
198	C₂²×C₈	—	94	4	no	—
199	C₂²×C₈	—	94	4	no	—
200	C₄⋊Q₈	—	47	16	no	—
201	C₄⋊Q₈	—	94	16	no	—
202	C₄⋊Q₈	—	94	8	no	—
203	C₂×C₄⋊C₄	—	2	8	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	4	4	no	—
208	C₂×C₄⋊C₄	—	4	4	no	—
209	C₂×C₄⋊C₄	—	47	8	no	—
210	C₂×C₄⋊C₄	—	47	4	no	—
211	C₂×C₄⋊C₄	—	94	8	no	—
212	C₂×C₄⋊C₄	—	94	4	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	94	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
216	C₂×C₄²	C₄×D₄₇	1	4	yes	—
217	C₂×C₄²	D₁₈₈	1	4	yes	—
218	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
219	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
220	C₄×C₈	—	94	8	no	—
221	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂×Q₈	—	94	16	no	—
223	C₂×Q₈	—	188	32	no	—
224	C₂×Q₈	—	188	8	no	—
225	C₂×Q₈	—	188	8	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	94	8	no	—
231	C₂²×C₄	—	94	4	no	—
232	C₂²×C₄	—	188	4	no	—
233	C₂²×C₄	—	188	4	no	—
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
235	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
236	C₂²×C₄	#10625334	1	4	yes	—
237	C₂²×C₄	#10625335	1	4	yes	—
238	C₂²×C₄	#10625336	1	4	yes	—
239	C₂²×C₄	#10625337	1	4	yes	—
240	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
241	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
242	C₂²×C₄	#10625352	1	4	yes	—
243	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
247	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
248	C₂×C₈	—	47	8	no	—
249	C₂×C₈	—	94	16	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	8	no	—
255	C₄⋊C₄	—	4	8	no	—
256	C₄⋊C₄	—	47	8	no	—
257	C₄⋊C₄	—	94	8	no	—
258	C₄⋊C₄	—	94	16	no	—
259	C₄⋊C₄	—	94	4	no	—
260	C₄²	(C₂×D₄₇)⋊C₄	1	8	yes	—
261	C₄²	C₂×C₄×D₄₇	1	4	yes	—
262	C₄²	C₂×D₁₈₈	1	4	yes	—
263	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
264	C₂³	#13094011	1	2	yes	—
265	C₂³	#13094022	1	2	yes	—
266	C₂³	C₂×#10625334	1	2	yes	—
267	C₂³	C₂×#10625336	1	2	yes	—
268	C₂³	C₂×#10625335	1	2	yes	—
269	C₂³	C₂×#10625337	1	2	yes	—
270	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	Q₈	—	188	32	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	8	no	—
276	C₂×C₄	—	94	8	no	—
277	C₂×C₄	—	188	16	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	#13093944	1	16	yes	—
281	C₂×C₄	#13093947	1	16	yes	—
282	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
283	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
284	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	4	yes	—
285	C₂×C₄	#13093989	1	8	yes	—
286	C₂×C₄	#13094004	1	8	yes	—
287	C₂×C₄	#13094011	1	2	yes	—
288	C₂×C₄	#13094022	1	2	yes	—
289	C₂×C₄	C₂×#10625334	1	4	yes	—
290	C₂×C₄	C₂×#10625336	1	4	yes	—
291	C₂×C₄	C₂×#10625335	1	4	yes	—
292	C₂×C₄	C₂×#10625337	1	4	yes	—
293	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
296	C₂×C₄	C₂×#10625352	1	2	yes	—
297	C₈	—	94	8	no	—
298	C₂²	#15689028	1	4	yes	—
299	C₂²	C₂×#13093944	1	4	yes	—
300	C₂²	C₂×#13093947	1	4	yes	—
301	C₂²	#15689469	1	4	yes	—
302	C₂²	#15689473	1	4	yes	—
303	C₂²	#15689519	1	4	yes	—
304	C₂²	#15689526	1	4	yes	—
305	C₂²	C₂²×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
306	C₂²	C₂×#13094011	1	1	yes	characteristic
307	C₂²	C₂×#13094022	1	1	yes	characteristic
308	C₂²	C₂²×#10625334	1	1	yes	characteristic
309	C₂²	C₂²×#10625336	1	1	yes	characteristic
310	C₂²	C₂²×#10625335	1	1	yes	characteristic
311	C₂²	C₂²×#10625337	1	1	yes	characteristic
312	C₄	—	8	4	no	—
313	C₄	—	188	8	no	—
314	C₄	#15689028	1	4	yes	—
315	C₄	C₂×#13093944	1	8	yes	—
316	C₄	C₂×#13093947	1	8	yes	—
317	C₄	C₂×#13093989	1	2	yes	—
318	C₄	C₂×#13094004	1	2	yes	—
319	C₂	#20244348	1	8	yes	direct product factor
320	C₂	C₂×#15689028	1	1	yes	characteristic
321	C₂	C₂²×#13093944	1	1	yes	characteristic
322	C₂	C₂²×#13093947	1	1	yes	characteristic
323	C₂	C₂×#15689469	1	1	yes	characteristic
324	C₂	C₂×#15689473	1	1	yes	characteristic
325	C₂	C₂×#15689519	1	1	yes	characteristic
326	C₂	C₂×#15689526	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	C₂×#1990
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
3. #20244348	C₂	direct (N × H)
10. C₂×#13094024	C₄	semidirect
30. C₂²×C₄₇⋊C₈	Q₈	semidirect
34. #13094024	C₂×C₄	semidirect
98. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
161. C₂×C₄₇	#1990	semidirect
186. C₄₇	C₂×#1990	semidirect
319. C₂	#20244348	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13094024	2
C₂²×C₄×C₄₇⋊C₈	2
#20244348	2
C₂×#15688885	2
C₂×C₄₇×#384	2
C₂×#1990	47

Subgroups — C2×#20244348 #26220827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244348 #26220827