Subgroups — C₂×#20244346 #26220788

Order 12032.

1294 conjugacy classes of subgroups.

The subgroup lattice has 330 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	#20244346	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688885	C₂	1	2	yes	maximal
5	C₂×#15688885	C₂	1	2	yes	maximal
6	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₄₇	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	C₄×C₄⋊Dic₄₇	—	2	2	no	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
14	C₂×C₄₇⋊#95	—	2	1	no	—
15	#15688885	C₂²	1	8	yes	—
16	#15688885	C₂²	1	8	yes	—
17	C₂×#125×C₄₇	—	2	2	no	—
18	C₂×#125×C₄₇	—	2	4	no	—
19	C₂×#125×C₄₇	C₄	1	8	yes	complemented
20	C₂×#125×C₄₇	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₄₇	—	2	1	no	—
23	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	—	2	8	no	—
29	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	Q₈	1	8	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	—	4	4	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
37	C₄₇⋊#95	—	2	4	no	—
38	C₂³×C₄×C₄₇	—	2	1	no	—
39	C₂³×C₄×C₄₇	—	2	1	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	#125×C₄₇	—	2	16	no	—
44	#125×C₄₇	—	2	8	no	—
45	#125×C₄₇	C₂×C₄	1	16	yes	complemented
46	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	D₄	1	2	yes	—
60	C₂×C₄²×C₄₇	D₄	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
62	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₄₇	#41	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	2	8	no	—
84	C₂²×Dic₄₇	—	2	4	no	—
85	C₂²×Dic₄₇	—	4	4	no	—
86	C₂²×Dic₄₇	—	4	4	no	—
87	C₄⋊C₄×C₄₇	—	2	16	no	—
88	C₄⋊C₄×C₄₇	—	2	16	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
94	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
95	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
97	C₄²×C₄₇	C₂×D₄	1	4	yes	—
98	C₄⋊Dic₄₇	—	2	8	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	C₂×Q₈	1	8	yes	—
101	C₄×Dic₄₇	—	4	8	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	8	no	—
104	C₂×C₄₇⋊C₈	—	4	8	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
106	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
107	C₂³×C₄₇	#135	1	4	yes	—
108	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
109	C₂³×C₄₇	#137	1	2	yes	—
110	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	8	no	—
122	C₂×C₄×C₄₇	—	4	8	no	—
123	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₄×Q₈	1	8	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₄₇	#124	1	4	yes	—
129	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×Dic₄₇	—	2	8	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	16	no	—
138	C₄₇⋊C₈	—	4	4	no	—
139	C₂×#1989	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	#391	1	4	yes	—
141	C₂²×C₄₇	#416	1	4	yes	—
142	C₂²×C₄₇	#417	1	4	yes	—
143	C₂²×C₄₇	#478	1	8	yes	—
144	C₂²×C₄₇	#479	1	8	yes	—
145	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
147	C₂²×C₄₇	C₂×#135	1	2	yes	—
148	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
150	C₄×C₄₇	—	2	4	no	—
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	4	4	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	#391	1	4	yes	—
155	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
156	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
157	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	Dic₄₇	—	4	8	no	—
159	C₂²×#125	D₄₇	1	1	yes	characteristic
160	C₂²×C₄⋊C₈	—	47	1	no	—
161	C₂×C₄×C₄⋊C₄	—	47	1	no	—
162	#1989	—	47	8	no	—
163	C₂×#340	—	47	2	no	—
164	C₂×#340	—	47	2	no	—
165	C₂×C₄₇	#1989	1	8	yes	complemented
166	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#416	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#417	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#478	1	2	yes	—
170	C₂×C₄₇	C₂×#479	1	2	yes	—
171	C₂³×C₈	—	47	2	no	—
172	C₂×#125	—	2	2	no	—
173	C₂×#125	—	2	4	no	—
174	C₂×#125	Dic₄₇	1	8	yes	complemented
175	C₂×#125	C₂×D₄₇	1	4	yes	—
176	C₂²×C₄⋊C₄	—	2	1	no	—
177	C₂²×C₄⋊C₄	—	2	1	no	—
178	C₂²×C₄⋊C₄	—	47	2	no	—
179	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₄²	—	94	1	no	—
182	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	47	4	no	—
184	C₂×C₄⋊C₈	—	94	4	no	—
185	C₄×C₄⋊C₄	—	47	4	no	—
186	C₄×C₄⋊C₄	—	94	2	no	—
187	C₂×#95	—	94	1	no	—
188	#340	—	47	8	no	—
189	#340	—	47	8	no	—
190	C₄₇	C₂×#1989	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	—	2	1	no	—
193	C₂³×C₄	—	94	2	no	—
194	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	—	47	4	no	—
198	C₂²×C₈	—	94	8	no	—
199	C₂²×C₈	—	94	4	no	—
200	#125	—	2	16	no	—
201	#125	—	2	8	no	—
202	#125	C₂×Dic₄₇	1	16	yes	complemented
203	C₂×C₄⋊C₄	—	2	8	no	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	47	8	no	—
212	C₂×C₄⋊C₄	—	47	4	no	—
213	C₂×C₄⋊C₄	—	94	4	no	—
214	C₂×C₄⋊C₄	—	94	2	no	—
215	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
217	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
219	C₂×C₄²	—	94	4	no	—
220	C₂×C₄²	—	188	4	no	—
221	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
222	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
223	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
224	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
225	C₄⋊C₈	—	94	8	no	—
226	#95	—	94	4	no	—
227	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	94	8	no	—
239	C₂²×C₄	—	94	4	no	—
240	C₂²×C₄	—	188	4	no	—
241	C₂²×C₄	—	188	4	no	—
242	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
244	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
245	C₂²×C₄	#10625354	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	8	no	—
254	C₂×C₈	—	188	8	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	16	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	47	8	no	—
262	C₄⋊C₄	—	94	8	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
265	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
266	C₄²	—	188	8	no	—
267	C₄²	C₂²⋊Dic₄₇	1	8	yes	—
268	C₄²	C₂²×Dic₄₇	1	4	yes	—
269	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
270	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
271	C₂³	#13094000	1	4	yes	—
272	C₂³	#13094044	1	4	yes	—
273	C₂³	#13094049	1	2	yes	—
274	C₂³	#13094054	1	2	yes	—
275	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	94	8	no	—
289	C₂×C₄	—	188	4	no	—
290	C₂×C₄	—	188	4	no	—
291	C₂×C₄	—	188	16	no	—
292	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
293	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
294	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
295	C₂×C₄	Q₈×Dic₄₇	1	8	yes	—
296	C₂×C₄	#13094000	1	4	yes	—
297	C₂×C₄	#13094013	1	4	yes	—
298	C₂×C₄	#13094022	1	4	yes	—
299	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
302	C₂×C₄	C₂×#10625354	1	2	yes	—
303	C₈	—	188	4	no	—
304	C₂²	#15689030	1	4	yes	—
305	C₂²	#15689119	1	4	yes	—
306	C₂²	#15689124	1	4	yes	—
307	C₂²	#15689420	1	8	yes	—
308	C₂²	#15689427	1	8	yes	—
309	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
310	C₂²	C₂×#13094000	1	2	yes	—
311	C₂²	C₂×#13094044	1	2	yes	—
312	C₂²	C₂×#13094049	1	1	yes	characteristic
313	C₂²	C₂×#13094054	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	2	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	188	8	no	—
319	C₄	#15689030	1	4	yes	—
320	C₄	C₂×Q₈×Dic₄₇	1	2	yes	—
321	C₄	C₂×#13094013	1	1	yes	characteristic
322	C₄	C₂×#13094022	1	1	yes	characteristic
323	C₂	#20244346	1	8	yes	direct product factor
324	C₂	C₂×#15689030	1	1	yes	characteristic
325	C₂	C₂×#15689119	1	1	yes	characteristic
326	C₂	C₂×#15689124	1	1	yes	characteristic
327	C₂	C₂×#15689420	1	2	yes	—
328	C₂	C₂×#15689427	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	C₂×#1989
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
3. #20244346	C₂	direct (N × H)
19. C₂×#125×C₄₇	C₄	semidirect
45. #125×C₄₇	C₂×C₄	semidirect
165. C₂×C₄₇	#1989	semidirect
174. C₂×#125	Dic₄₇	semidirect
190. C₄₇	C₂×#1989	semidirect
202. #125	C₂×Dic₄₇	semidirect
323. C₂	#20244346	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
#20244346	2
C₂×#15688885	2
C₂²×#125×C₄₇	2
C₂×#1989	47

Subgroups — C2×#20244346 #26220788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244346 #26220788