Subgroups — C₂×#20244325 #26220720

Order 12032.

1502 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20244325	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688878	C₂	1	4	yes	maximal
5	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
12	#15688878	C₂²	1	16	yes	—
13	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₇	—	2	4	no	—
15	C₂×C₄×Q₈×C₄₇	—	2	4	no	—
16	C₂×C₄×Q₈×C₄₇	C₄	1	4	yes	—
17	C₂×C₄×Q₈×C₄₇	C₂²	1	4	yes	—
18	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄₇⋊C₈	—	2	4	no	—
23	C₂²×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	8	no	—
26	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
30	C₄×C₄₇⋊C₈	—	2	8	no	—
31	C₂²×Q₈×C₄₇	—	2	4	no	—
32	C₂²×Q₈×C₄₇	—	2	4	no	—
33	C₂²×Q₈×C₄₇	C₈	1	8	yes	complemented
34	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₄₇	—	2	16	no	—
41	C₄×Q₈×C₄₇	—	2	8	no	—
42	C₄×Q₈×C₄₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₄²×C₄₇	—	2	4	no	—
48	C₂×C₄²×C₄₇	—	2	4	no	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	D₄	1	4	yes	—
52	C₂×C₄²×C₄₇	D₄	1	4	yes	—
53	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Q₈×C₄₇	—	2	32	no	—
57	C₂×Q₈×C₄₇	—	2	8	no	—
58	C₂×Q₈×C₄₇	—	2	8	no	—
59	C₂×Q₈×C₄₇	C₂×C₈	1	16	yes	complemented
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×C₈	1	8	yes	—
68	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₄₇	#41	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₄₇	—	2	16	no	—
81	C₄⋊C₄×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	—	2	8	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
86	C₄²×C₄₇	C₂×D₄	1	4	yes	—
87	C₄²×C₄₇	C₂×D₄	1	4	yes	—
88	C₂×C₄₇⋊C₈	—	2	16	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	8	no	—
93	C₂×C₄₇⋊C₈	—	4	8	no	—
94	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	complemented
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
97	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
98	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
99	C₂³×C₄₇	#131	1	4	yes	—
100	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	Q₈×C₄₇	—	2	32	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	16	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	—	4	8	no	—
110	C₂×C₄×C₄₇	C₂²⋊C₈	1	16	yes	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
116	C₂×C₄×C₄₇	#124	1	4	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
119	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
125	C₄₇⋊C₈	—	2	8	no	—
126	C₄₇⋊C₈	—	4	4	no	—
127	C₂×#1984	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	#391	1	4	yes	—
129	C₂²×C₄₇	C₂×C₂²⋊C₈	1	4	yes	—
130	C₂²×C₄₇	#408	1	4	yes	—
131	C₂²×C₄₇	C₈×Q₈	1	8	yes	—
132	C₂²×C₄₇	#446	1	8	yes	—
133	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
135	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
136	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#131	1	2	yes	—
138	C₄×C₄₇	—	2	8	no	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	4	4	no	—
141	C₄×C₄₇	#391	1	4	yes	—
142	C₄×C₄₇	C₂×C₂²⋊C₈	1	8	yes	—
143	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
147	C₂²×C₄⋊C₈	—	47	1	no	—
148	C₂²×C₄×C₈	—	47	1	no	—
149	#1984	—	47	8	no	—
150	C₂×#336	—	47	4	no	—
151	C₂×C₄₇	#1984	1	8	yes	complemented
152	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
153	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₄₇	C₂×#408	1	1	yes	characteristic
155	C₂×C₄₇	C₂×C₈×Q₈	1	2	yes	—
156	C₂×C₄₇	C₂×#446	1	2	yes	—
157	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
158	C₂³×C₈	—	47	2	no	—
159	C₂³×C₈	—	47	2	no	—
160	C₂×C₄×Q₈	—	2	4	no	—
161	C₂×C₄×Q₈	—	2	4	no	—
162	C₂×C₄×Q₈	Dic₄₇	1	4	yes	—
163	C₂×C₄×Q₈	C₂×D₄₇	1	4	yes	—
164	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
166	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
167	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	47	4	no	—
169	C₂×C₄⋊C₈	—	94	4	no	—
170	C₂×C₄×C₈	—	47	4	no	—
171	C₂×C₄×C₈	—	94	4	no	—
172	#336	—	47	16	no	—
173	C₄₇	C₂×#1984	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	2	4	no	—
175	C₂²×Q₈	—	2	4	no	—
176	C₂²×Q₈	C₄₇⋊C₈	1	8	yes	complemented
177	C₂²×Q₈	C₂×Dic₄₇	1	4	yes	—
178	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
179	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
180	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
182	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	—	47	8	no	—
184	C₂²×C₈	—	47	4	no	—
185	C₂²×C₈	—	47	4	no	—
186	C₂²×C₈	—	94	4	no	—
187	C₂²×C₈	—	94	4	no	—
188	C₂²×C₈	—	94	8	no	—
189	C₂²×C₈	—	94	4	no	—
190	C₄×Q₈	—	2	16	no	—
191	C₄×Q₈	—	2	8	no	—
192	C₄×Q₈	—	2	8	no	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
196	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	—	2	4	no	—
199	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
200	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
202	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
204	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
205	C₄⋊C₈	—	94	8	no	—
206	C₄×C₈	—	94	8	no	—
207	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
208	C₂×Q₈	—	2	32	no	—
209	C₂×Q₈	—	2	8	no	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	C₂×C₄₇⋊C₈	1	16	yes	complemented
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	C₂×C₄₇⋊C₈	1	8	yes	—
218	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
219	C₂²×C₄	#10625329	1	4	yes	—
220	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
221	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
222	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
223	C₂²×C₄	#10625354	1	4	yes	—
224	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
231	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
232	C₂×C₈	—	47	8	no	—
233	C₂×C₈	—	94	16	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	94	8	no	—
238	C₂×C₈	—	188	8	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
245	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
246	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
247	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
248	C₂³	Q₈×Dic₄₇	1	4	yes	—
249	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
250	C₂³	C₂×#10625329	1	2	yes	—
251	C₂³	#13094031	1	4	yes	—
252	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
253	Q₈	—	2	32	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	4	8	no	—
262	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	16	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
266	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
267	C₂×C₄	#13094000	1	8	yes	—
268	C₂×C₄	#13094013	1	4	yes	—
269	C₂×C₄	#13094022	1	4	yes	—
270	C₂×C₄	C₂²×C₄₇⋊C₈	1	8	yes	—
271	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
272	C₂×C₄	C₂×#10625329	1	4	yes	—
273	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
276	C₂×C₄	C₂×#10625354	1	2	yes	—
277	C₈	—	94	8	no	—
278	C₈	—	188	4	no	—
279	C₂²	#15689030	1	4	yes	—
280	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
281	C₂²	#15689090	1	4	yes	—
282	C₂²	Q₈×C₄₇⋊C₈	1	8	yes	—
283	C₂²	#15689254	1	8	yes	—
284	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
285	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
286	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
287	C₂²	C₂²×#10625329	1	1	yes	characteristic
288	C₂²	C₂×#13094031	1	2	yes	—
289	C₄	—	2	8	no	—
290	C₄	—	2	4	no	—
291	C₄	—	4	4	no	—
292	C₄	#15689030	1	4	yes	—
293	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
294	C₄	C₂×#13094000	1	2	yes	—
295	C₄	C₂×#13094013	1	1	yes	characteristic
296	C₄	C₂×#13094022	1	1	yes	characteristic
297	C₂	#20244325	1	8	yes	direct product factor
298	C₂	C₂×#15689030	1	1	yes	characteristic
299	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
300	C₂	C₂×#15689090	1	1	yes	characteristic
301	C₂	C₂×Q₈×C₄₇⋊C₈	1	2	yes	—
302	C₂	C₂×#15689254	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#1984
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
3. #20244325	C₂	direct (N × H)
26. C₂²×C₄₇⋊C₈	Q₈	semidirect
33. C₂²×Q₈×C₄₇	C₈	semidirect
59. C₂×Q₈×C₄₇	C₂×C₈	semidirect
94. C₂×C₄₇⋊C₈	C₂×Q₈	semidirect
151. C₂×C₄₇	#1984	semidirect
173. C₄₇	C₂×#1984	semidirect
176. C₂²×Q₈	C₄₇⋊C₈	semidirect
211. C₂×Q₈	C₂×C₄₇⋊C₈	semidirect
297. C₂	#20244325	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×C₄⋊(C₄₇⋊C₈)	2
C₂²×C₄×C₄₇⋊C₈	2
#20244325	2
C₂×#15688878	2
C₂²×C₄×Q₈×C₄₇	2
C₂×#1984	47

Subgroups — C2×#20244325 #26220720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244325 #26220720