Subgroups — C₂×#20244341 #26220782

Order 12032.

1510 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244341	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	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	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₄₇	—	2	2	no	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	#15688885	C₂²	1	16	yes	—
15	C₂³×Q₈×C₄₇	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₄₇	—	2	2	no	—
17	C₂×C₄⋊Q₈×C₄₇	—	2	4	no	—
18	C₂×C₄⋊Q₈×C₄₇	C₄	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₄₇	C₂²	1	4	yes	—
20	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	8	no	—
26	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	Q₈	1	8	yes	complemented
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄×Dic₄₇	—	2	4	no	—
32	C₂×C₄×Dic₄₇	—	4	4	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂²×Q₈×C₄₇	—	2	8	no	—
36	C₂²×Q₈×C₄₇	—	2	4	no	—
37	C₂²×Q₈×C₄₇	—	4	4	no	—
38	C₂²×Q₈×C₄₇	—	4	4	no	—
39	C₂³×C₄×C₄₇	—	2	2	no	—
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₄₇	—	2	16	no	—
44	C₄⋊Q₈×C₄₇	—	2	8	no	—
45	C₄⋊Q₈×C₄₇	C₂×C₄	1	16	yes	complemented
46	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	D₄	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₄₇	—	2	16	no	—
58	C₂×Q₈×C₄₇	—	4	32	no	—
59	C₂×Q₈×C₄₇	—	4	8	no	—
60	C₂×Q₈×C₄₇	—	4	8	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	8	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	D₈	1	4	yes	—
69	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	#41	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	8	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	—	4	4	no	—
82	C₄⋊C₄×C₄₇	—	2	16	no	—
83	C₄⋊C₄×C₄₇	—	2	8	no	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
86	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	C₂×D₄	1	4	yes	—
89	C₄⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	complemented
92	C₄×Dic₄₇	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	8	no	—
95	C₂×C₄₇⋊C₈	—	4	8	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
98	C₂³×C₄₇	C₂×D₈	1	2	yes	—
99	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
100	C₂³×C₄₇	#137	1	4	yes	—
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	Q₈×C₄₇	—	4	32	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	16	no	—
109	C₂×C₄×C₄₇	#102	1	16	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
114	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
115	C₂×C₄×C₄₇	#124	1	4	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
123	C₂×Dic₄₇	—	2	8	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	16	no	—
127	C₄₇⋊C₈	—	4	4	no	—
128	C₂×#1988	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	#391	1	4	yes	—
130	C₂²×C₄₇	C₂×#102	1	4	yes	—
131	C₂²×C₄₇	#419	1	4	yes	—
132	C₂²×C₄₇	#474	1	8	yes	—
133	C₂²×C₄₇	#475	1	8	yes	—
134	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
136	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#137	1	2	yes	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	4	8	no	—
141	C₄×C₄₇	#391	1	4	yes	—
142	C₄×C₄₇	C₂×#102	1	8	yes	—
143	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
144	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	Dic₄₇	—	4	8	no	—
147	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
148	C₂²×C₄⋊C₈	—	47	1	no	—
149	C₂×C₄×C₄⋊C₄	—	47	1	no	—
150	#1988	—	47	8	no	—
151	C₂×#340	—	47	4	no	—
152	C₂×C₄₇	#1988	1	8	yes	complemented
153	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
155	C₂×C₄₇	C₂×#419	1	1	yes	characteristic
156	C₂×C₄₇	C₂×#474	1	2	yes	—
157	C₂×C₄₇	C₂×#475	1	2	yes	—
158	C₂³×Q₈	—	2	1	no	—
159	C₂³×C₈	—	47	2	no	—
160	C₂×C₄⋊Q₈	—	2	2	no	—
161	C₂×C₄⋊Q₈	—	2	4	no	—
162	C₂×C₄⋊Q₈	Dic₄₇	1	8	yes	complemented
163	C₂×C₄⋊Q₈	C₂×D₄₇	1	4	yes	—
164	C₂²×C₄⋊C₄	—	2	1	no	—
165	C₂²×C₄⋊C₄	—	47	2	no	—
166	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
167	C₂²×C₄²	—	94	1	no	—
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	47	4	no	—
170	C₂×C₄⋊C₈	—	94	4	no	—
171	C₄×C₄⋊C₄	—	47	4	no	—
172	C₄×C₄⋊C₄	—	94	2	no	—
173	C₂×#95	—	94	1	no	—
174	#340	—	47	16	no	—
175	C₄₇	C₂×#1988	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	2	8	no	—
177	C₂²×Q₈	—	2	4	no	—
178	C₂²×Q₈	—	4	4	no	—
179	C₂²×Q₈	—	4	4	no	—
180	C₂³×C₄	—	2	2	no	—
181	C₂³×C₄	—	94	2	no	—
182	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	—	47	4	no	—
186	C₂²×C₈	—	94	8	no	—
187	C₂²×C₈	—	94	4	no	—
188	C₄⋊Q₈	—	2	16	no	—
189	C₄⋊Q₈	—	2	8	no	—
190	C₄⋊Q₈	C₂×Dic₄₇	1	16	yes	complemented
191	C₂×C₄⋊C₄	—	2	8	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	47	8	no	—
196	C₂×C₄⋊C₄	—	47	4	no	—
197	C₂×C₄⋊C₄	—	94	4	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
200	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
201	C₂×C₄²	—	94	4	no	—
202	C₂×C₄²	—	188	4	no	—
203	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
206	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
207	C₄⋊C₈	—	94	8	no	—
208	#95	—	94	4	no	—
209	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×Q₈	—	2	16	no	—
211	C₂×Q₈	—	4	32	no	—
212	C₂×Q₈	—	4	8	no	—
213	C₂×Q₈	—	4	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	94	8	no	—
220	C₂²×C₄	—	94	4	no	—
221	C₂²×C₄	—	188	4	no	—
222	C₂²×C₄	—	188	4	no	—
223	C₂²×C₄	#10625334	1	4	yes	—
224	C₂²×C₄	#10625335	1	4	yes	—
225	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
226	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
227	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
228	C₂²×C₄	#10625354	1	4	yes	—
229	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	8	no	—
237	C₂×C₈	—	188	8	no	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	47	8	no	—
242	C₄⋊C₄	—	94	8	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
245	C₄²	—	188	8	no	—
246	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
247	C₄²	C₂²×Dic₄₇	1	4	yes	—
248	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
249	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
250	C₂³	#13094000	1	4	yes	—
251	C₂³	C₂×#10625334	1	2	yes	—
252	C₂³	C₂×#10625335	1	2	yes	—
253	C₂³	#13094055	1	4	yes	—
254	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
255	Q₈	—	4	32	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	16	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	94	8	no	—
263	C₂×C₄	—	188	4	no	—
264	C₂×C₄	—	188	16	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	#13093945	1	16	yes	—
267	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
268	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
270	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
271	C₂×C₄	#13094000	1	4	yes	—
272	C₂×C₄	#13094013	1	4	yes	—
273	C₂×C₄	#13094022	1	4	yes	—
274	C₂×C₄	C₂×#10625334	1	4	yes	—
275	C₂×C₄	C₂×#10625335	1	4	yes	—
276	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
279	C₂×C₄	C₂×#10625354	1	2	yes	—
280	C₈	—	188	4	no	—
281	C₂²	#15689030	1	4	yes	—
282	C₂²	C₂×#13093945	1	4	yes	—
283	C₂²	#15689132	1	4	yes	—
284	C₂²	#15689400	1	8	yes	—
285	C₂²	#15689405	1	8	yes	—
286	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
287	C₂²	C₂×#13094000	1	2	yes	—
288	C₂²	C₂²×#10625334	1	1	yes	characteristic
289	C₂²	C₂²×#10625335	1	1	yes	characteristic
290	C₂²	C₂×#13094055	1	2	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	4	8	no	—
293	C₄	—	188	8	no	—
294	C₄	#15689030	1	4	yes	—
295	C₄	C₂×#13093945	1	8	yes	—
296	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
297	C₄	C₂×#13094013	1	1	yes	characteristic
298	C₄	C₂×#13094022	1	1	yes	characteristic
299	C₂	#20244341	1	8	yes	direct product factor
300	C₂	C₂×#15689030	1	1	yes	characteristic
301	C₂	C₂²×#13093945	1	1	yes	characteristic
302	C₂	C₂×#15689132	1	1	yes	characteristic
303	C₂	C₂×#15689400	1	2	yes	—
304	C₂	C₂×#15689405	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#1988
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #20244341	C₂	direct (N × H)
18. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
29. C₂×C₄⋊Dic₄₇	Q₈	semidirect
45. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
91. C₄⋊Dic₄₇	C₂×Q₈	semidirect
152. C₂×C₄₇	#1988	semidirect
162. C₂×C₄⋊Q₈	Dic₄₇	semidirect
175. C₄₇	C₂×#1988	semidirect
190. C₄⋊Q₈	C₂×Dic₄₇	semidirect
299. C₂	#20244341	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
C₂×C₄×C₄⋊Dic₄₇	2
#20244341	2
C₂×#15688885	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂×#1988	47

Subgroups — C2×#20244341 #26220782

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244341 #26220782