Subgroups — C₂×#20244382 #26220944

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20244382	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688892	C₂	1	4	yes	maximal
5	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689147	—	2	2	no	—
9	#15689147	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
11	C₄×#10625329	—	2	2	no	—
12	C₄×#10625329	C₂²	1	4	yes	—
13	C₂×#13093935	—	2	1	no	—
14	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
15	#15688892	C₂²	1	16	yes	—
16	C₂×#125×C₄₇	—	2	1	no	—
17	C₂×#125×C₄₇	—	2	1	no	—
18	C₂×#125×C₄₇	—	2	2	no	—
19	C₂×#125×C₄₇	—	2	2	no	—
20	C₂×#125×C₄₇	—	2	2	no	—
21	C₂×#125×C₄₇	C₄	1	4	yes	—
22	C₂×#125×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
24	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	8	no	—
30	C₂×#10625329	Q₈	1	8	yes	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
36	#13093935	—	2	4	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	#125×C₄₇	—	2	4	no	—
42	#125×C₄₇	—	2	4	no	—
43	#125×C₄₇	—	2	8	no	—
44	#125×C₄₇	—	2	4	no	—
45	#125×C₄₇	—	2	4	no	—
46	#125×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	8	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	2	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	16	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	D₄	1	2	yes	—
61	C₂×C₄²×C₄₇	D₄	1	2	yes	—
62	C₂×C₄²×C₄₇	D₄	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	4	4	no	—
68	C₂²×C₄×C₄₇	—	4	8	no	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₄₇	#41	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₄₇	—	2	16	no	—
83	C₄⋊C₄×C₄₇	—	2	16	no	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	2	4	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	2	yes	—
93	C₄²×C₄₇	C₂×D₄	1	4	yes	—
94	#10625329	—	2	8	no	—
95	#10625329	—	2	4	no	—
96	#10625329	—	2	8	no	—
97	#10625329	—	4	4	no	—
98	#10625329	C₂×Q₈	1	8	yes	—
99	C₂×C₄₇⋊C₈	—	2	8	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	4	4	no	—
103	C₂³×C₄₇	—	2	4	no	—
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
107	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	8	no	—
114	C₂×C₄×C₄₇	—	4	8	no	—
115	C₂×C₄×C₄₇	—	4	8	no	—
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₄₇	#124	1	2	yes	—
124	C₂×C₄×C₄₇	#124	1	2	yes	—
125	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
131	C₄₇⋊C₈	—	2	8	no	—
132	C₄₇⋊C₈	—	4	4	no	—
133	C₂×#1998	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	8	no	—
137	C₂²×C₄₇	#391	1	2	yes	—
138	C₂²×C₄₇	#391	1	2	yes	—
139	C₂²×C₄₇	#413	1	4	yes	—
140	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	2	4	no	—
144	C₄×C₄₇	—	4	8	no	—
145	C₄×C₄₇	#391	1	2	yes	—
146	C₄×C₄₇	#391	1	2	yes	—
147	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
148	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
149	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₂²×#125	D₄₇	1	1	yes	characteristic
151	C₂×#423	—	47	1	no	—
152	C₂×C₄×M₁₆	—	47	1	no	—
153	#1998	—	47	8	no	—
154	C₂×#343	—	47	4	no	—
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	—	2	2	no	—
157	C₂×C₄₇	#1998	1	4	yes	complemented
158	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#413	1	2	yes	—
160	C₂²×M₁₆	—	47	2	no	—
161	C₂²×M₁₆	—	47	2	no	—
162	C₂×#125	—	2	1	no	—
163	C₂×#125	—	2	1	no	—
164	C₂×#125	—	2	2	no	—
165	C₂×#125	—	2	2	no	—
166	C₂×#125	—	2	2	no	—
167	C₂×#125	Dic₄₇	1	4	yes	—
168	C₂×#125	C₂×D₄₇	1	4	yes	—
169	C₂²×C₄⋊C₄	Dic₄₇	1	4	yes	—
170	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#423	—	47	4	no	—
173	#423	—	94	2	no	—
174	C₂×C₄⋊C₈	—	94	2	no	—
175	C₄×M₁₆	—	47	4	no	—
176	C₄×M₁₆	—	94	2	no	—
177	C₂×#97	—	94	1	no	—
178	C₂×C₄×C₈	—	94	1	no	—
179	#343	—	47	16	no	—
180	C₄₇	C₂×#1998	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
182	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
183	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	47	8	no	—
185	C₂×M₁₆	—	47	4	no	—
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	—	94	4	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	8	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	#125	—	2	4	no	—
194	#125	—	2	4	no	—
195	#125	—	2	8	no	—
196	#125	—	2	4	no	—
197	#125	—	2	4	no	—
198	#125	—	2	8	no	—
199	C₂×C₄⋊C₄	—	2	8	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
207	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
208	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
209	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
213	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
214	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
215	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
216	C₄⋊C₈	—	94	8	no	—
217	#97	—	94	4	no	—
218	C₄×C₈	—	94	4	no	—
219	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	8	no	—
224	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₄₇	1	2	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	8	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
231	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
237	M₁₆	—	47	8	no	—
238	M₁₆	—	94	8	no	—
239	M₁₆	—	94	4	no	—
240	M₁₆	—	94	8	no	—
241	M₁₆	—	188	4	no	—
242	C₂×C₈	—	94	8	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	188	4	no	—
246	C₄⋊C₄	—	2	16	no	—
247	C₄⋊C₄	—	2	16	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
253	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
254	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
255	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
256	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
257	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
258	C₂³	—	2	4	no	—
259	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
260	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
261	C₂³	Q₈×Dic₄₇	1	4	yes	—
262	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
272	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
273	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
275	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
276	C₂×C₄	#13094000	1	4	yes	—
277	C₂×C₄	#13094000	1	4	yes	—
278	C₂×C₄	#13094013	1	2	yes	—
279	C₂×C₄	#13094013	1	2	yes	—
280	C₂×C₄	#13094022	1	2	yes	—
281	C₂×C₄	#13094022	1	2	yes	—
282	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
285	C₂×C₄	C₂×#10625354	1	2	yes	—
286	C₈	—	94	8	no	—
287	C₈	—	188	4	no	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	#15689030	1	2	yes	—
292	C₂²	#15689030	1	2	yes	—
293	C₂²	#15689107	1	4	yes	—
294	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
295	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
296	C₄	—	2	4	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	8	no	—
299	C₄	#15689030	1	2	yes	—
300	C₄	#15689030	1	2	yes	—
301	C₄	C₂×#13094000	1	2	yes	—
302	C₄	C₂×#13094013	1	1	yes	characteristic
303	C₄	C₂×#13094022	1	1	yes	characteristic
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	#20244382	1	4	yes	direct product factor
307	C₂	C₂×#15689030	1	1	yes	characteristic
308	C₂	C₂×#15689107	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	C₂×#1998
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
3. #20244382	C₂	direct (N × H)
157. C₂×C₄₇	#1998	semidirect
180. C₄₇	C₂×#1998	semidirect
306. C₂	#20244382	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×C₄×#10625329	2
#20244382	2
C₂×#15688892	2
C₂²×#125×C₄₇	2
C₂×#1998	47

Subgroups — C2×#20244382 #26220944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244382 #26220944