Subgroups — C₂²×#14582957 #25960311

Order 11008.

2310 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063284	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582957	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#13063284	C₄	1	12	yes	—
12	C₂×#13063284	C₂²	1	12	yes	—
13	C₂²×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
14	C₂²×C₄⋊Dic₄₃	C₄	1	1	yes	characteristic
15	C₂²×C₂²⋊Dic₄₃	C₄	1	1	yes	characteristic
16	C₂²×C₂²⋊Dic₄₃	C₄	1	1	yes	characteristic
17	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582957	C₂²	1	64	yes	direct product factor
20	C₂×Dic₄₃⋊C₈	C₂²	1	12	yes	complemented
21	C₂×Dic₄₃⋊C₈	C₂²	1	12	yes	complemented
22	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	12	yes	—
23	C₂×#13063235	C₂²	1	12	yes	complemented
24	C₂×C₈×Dic₄₃	C₂²	1	12	yes	complemented
25	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₄₃	C₂²	1	12	yes	—
29	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×Dic₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄₃⋊C₈	—	2	4	no	—
34	C₂²×C₄₃⋊C₈	—	2	4	no	—
35	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
37	#13063284	C₂×C₄	1	48	yes	—
38	#13063284	C₂³	1	16	yes	—
39	C₂×C₄⋊Dic₄₃	C₂×C₄	1	12	yes	—
40	C₂×C₄⋊Dic₄₃	C₂×C₄	1	12	yes	—
41	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	12	yes	—
42	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	12	yes	—
43	C₂×C₄×Dic₄₃	C₂×C₄	1	6	yes	—
44	C₂×C₄×Dic₄₃	C₂×C₄	1	6	yes	—
45	C₂×C₄×Dic₄₃	C₂³	1	6	yes	—
46	C₂×C₄×Dic₄₃	C₂³	1	6	yes	—
47	Dic₄₃⋊C₈	C₂³	1	16	yes	complemented
48	Dic₄₃⋊C₈	C₂³	1	16	yes	complemented
49	C₂²⋊(C₄₃⋊C₈)	C₂³	1	16	yes	—
50	#13063235	C₂³	1	16	yes	complemented
51	C₈×Dic₄₃	C₂³	1	16	yes	complemented
52	C₂⁵×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₃	—	2	4	no	—
54	C₂³×C₄×C₄₃	—	2	4	no	—
55	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₃	C₂×C₄	1	6	yes	—
57	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
58	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	—	2	4	no	—
60	C₂²×C₈×C₄₃	—	2	4	no	—
61	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
62	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
63	C₂²⋊C₈×C₄₃	C₂³	1	16	yes	—
64	C₂⁴×C₄₃	—	2	4	no	—
65	C₂⁴×C₄₃	—	2	4	no	—
66	C₂⁴×C₄₃	—	2	4	no	—
67	C₂⁴×C₄₃	C₂²×C₄	1	6	yes	—
68	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	—	2	24	no	—
70	C₂²×C₄×C₄₃	—	2	12	no	—
71	C₂²×C₄×C₄₃	—	2	12	no	—
72	C₂²×C₄×C₄₃	—	2	4	no	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₄₃	#41	1	4	yes	—
77	C₂²×C₄×C₄₃	#41	1	4	yes	—
78	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
80	C₂²×Dic₄₃	—	2	4	no	—
81	C₂²×Dic₄₃	—	2	4	no	—
82	C₂²×Dic₄₃	—	2	4	no	—
83	C₂²×Dic₄₃	—	2	4	no	—
84	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
85	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
86	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
87	C₂²×Dic₄₃	C₂²×C₄	1	6	yes	—
88	C₂×C₈×C₄₃	—	2	12	no	—
89	C₂×C₈×C₄₃	—	2	12	no	—
90	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
91	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
92	C₂²⋊Dic₄₃	C₂²×C₄	1	16	yes	—
93	C₂²⋊Dic₄₃	C₂²×C₄	1	16	yes	—
94	C₄⋊Dic₄₃	C₂²×C₄	1	16	yes	—
95	C₄⋊Dic₄₃	C₂²×C₄	1	16	yes	—
96	C₄×Dic₄₃	C₂²×C₄	1	12	yes	—
97	C₄×Dic₄₃	C₂²×C₄	1	12	yes	—
98	C₄×Dic₄₃	C₂⁴	1	4	yes	—
99	C₄×Dic₄₃	C₂⁴	1	4	yes	—
100	C₂×C₄₃⋊C₈	—	2	12	no	—
101	C₂×C₄₃⋊C₈	—	2	12	no	—
102	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
103	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
104	C₂³×C₄₃	—	2	12	no	—
105	C₂³×C₄₃	—	2	12	no	—
106	C₂³×C₄₃	—	2	12	no	—
107	C₂³×C₄₃	—	4	16	no	—
108	C₂³×C₄₃	#117	1	4	yes	—
109	C₂³×C₄₃	#131	1	4	yes	—
110	C₂³×C₄₃	#131	1	4	yes	—
111	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
112	C₂³×C₄₃	C₂³×C₄	1	3	yes	—
113	C₂×C₄×C₄₃	—	2	12	no	—
114	C₂×C₄×C₄₃	—	2	24	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	#117	1	12	yes	—
118	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₄₃	C₂³×C₄	1	3	yes	—
120	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
121	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
122	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	12	no	—
124	C₂×Dic₄₃	—	2	12	no	—
125	C₂×Dic₄₃	—	2	12	no	—
126	C₂×Dic₄₃	—	2	12	no	—
127	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
128	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
129	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
130	C₂×Dic₄₃	C₂³×C₄	1	4	yes	—
131	C₈×C₄₃	—	2	4	no	—
132	C₈×C₄₃	—	2	4	no	—
133	C₄₃⋊C₈	—	2	4	no	—
134	C₄₃⋊C₈	—	2	4	no	—
135	C₂²×#433	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	2	4	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	4	no	—
139	C₂²×C₄₃	—	4	24	no	—
140	C₂²×C₄₃	#433	1	16	yes	complemented
141	C₂²×C₄₃	C₂×#117	1	6	yes	—
142	C₂²×C₄₃	C₂×#131	1	6	yes	—
143	C₂²×C₄₃	C₂×#131	1	6	yes	—
144	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
145	C₄×C₄₃	—	2	4	no	—
146	C₄×C₄₃	C₂×#117	1	6	yes	—
147	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
148	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
149	Dic₄₃	—	2	4	no	—
150	Dic₄₃	—	2	4	no	—
151	Dic₄₃	—	2	4	no	—
152	Dic₄₃	—	2	4	no	—
153	C₂²×#117	—	43	1	no	—
154	C₂×#433	—	43	24	no	—
155	C₂²×C₄⋊C₈	—	43	1	no	—
156	C₂²×C₄⋊C₈	—	43	1	no	—
157	C₂²×C₂²⋊C₈	—	43	1	no	—
158	C₂²×C₂²⋊C₈	D₄₃	1	1	yes	characteristic
159	C₂²×#97	—	43	1	no	—
160	C₂²×C₄×C₈	—	43	1	no	—
161	C₂×C₄₃	—	4	4	no	—
162	C₂×C₄₃	C₂×#433	1	12	yes	complemented
163	C₂×C₄₃	C₂²×#117	1	1	yes	characteristic
164	C₂×C₄₃	C₂²×#131	1	1	yes	characteristic
165	C₂×C₄₃	C₂²×#131	1	1	yes	characteristic
166	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
167	C₂³×C₈	—	43	1	no	—
168	C₂³×C₈	—	43	1	no	—
169	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
170	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#117	—	43	12	no	—
172	C₂×#117	—	43	12	no	—
173	C₂²×C₄⋊C₄	—	43	1	no	—
174	C₂²×C₄⋊C₄	—	43	1	no	—
175	C₂²×C₂²⋊C₄	—	43	1	no	—
176	C₂²×C₂²⋊C₄	—	43	1	no	—
177	C₂²×C₄²	—	43	1	no	—
178	C₂²×C₄²	—	43	1	no	—
179	#433	—	43	64	no	—
180	C₂×C₄⋊C₈	—	43	12	no	—
181	C₂×C₄⋊C₈	—	43	12	no	—
182	C₂×C₂²⋊C₈	—	43	12	no	—
183	C₂×C₂²⋊C₈	C₂×D₄₃	1	12	yes	—
184	C₂×#97	—	43	12	no	—
185	C₂×C₄×C₈	—	43	12	no	—
186	C₄₃	C₂²×#433	1	1	yes	characteristic, complemented, Sylow
187	C₂⁵	C₄×D₄₃	1	1	yes	characteristic
188	C₂³×C₄	—	2	4	no	—
189	C₂³×C₄	—	2	4	no	—
190	C₂³×C₄	—	43	1	no	—
191	C₂³×C₄	—	43	1	no	—
192	C₂³×C₄	—	43	1	no	—
193	C₂³×C₄	—	43	1	no	—
194	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₄₃	1	6	yes	—
196	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
197	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	43	6	no	—
201	C₂²×C₈	—	43	6	no	—
202	C₂²×C₈	—	86	4	no	—
203	C₂²×C₈	—	86	4	no	—
204	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
205	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
206	#117	—	43	48	no	—
207	#117	—	43	16	no	—
208	C₂×C₄⋊C₄	—	43	12	no	—
209	C₂×C₄⋊C₄	—	43	12	no	—
210	C₂×C₂²⋊C₄	—	43	12	no	—
211	C₂×C₂²⋊C₄	—	43	12	no	—
212	C₂×C₄²	—	43	6	no	—
213	C₂×C₄²	—	43	6	no	—
214	C₂×C₄²	—	43	6	no	—
215	C₂×C₄²	—	43	6	no	—
216	C₄⋊C₈	—	43	16	no	—
217	C₄⋊C₈	—	43	16	no	—
218	C₂²⋊C₈	—	43	16	no	—
219	C₂²⋊C₈	C₂²×D₄₃	1	16	yes	—
220	#97	—	43	16	no	—
221	C₄×C₈	—	43	16	no	—
222	C₂⁴	—	2	4	no	—
223	C₂⁴	—	2	4	no	—
224	C₂⁴	—	2	4	no	—
225	C₂⁴	C₂×C₄×D₄₃	1	6	yes	—
226	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	24	no	—
229	C₂²×C₄	—	2	12	no	—
230	C₂²×C₄	—	2	12	no	—
231	C₂²×C₄	—	43	6	no	—
232	C₂²×C₄	—	43	6	no	—
233	C₂²×C₄	—	43	6	no	—
234	C₂²×C₄	—	43	6	no	—
235	C₂²×C₄	—	86	4	no	—
236	C₂²×C₄	—	86	4	no	—
237	C₂²×C₄	—	86	4	no	—
238	C₂²×C₄	—	86	4	no	—
239	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
240	C₂²×C₄	C₂×C₄×D₄₃	1	12	yes	—
241	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
242	C₂²×C₄	#10621900	1	4	yes	—
243	C₂²×C₄	#10621900	1	4	yes	—
244	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
245	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
246	C₂×C₈	—	2	12	no	—
247	C₂×C₈	—	2	12	no	—
248	C₂×C₈	—	43	4	no	—
249	C₂×C₈	—	43	4	no	—
250	C₂×C₈	—	86	12	no	—
251	C₂×C₈	—	86	12	no	—
252	C₂×C₈	C₂³×D₄₃	1	4	yes	—
253	C₂×C₈	C₂³×D₄₃	1	4	yes	—
254	C₄⋊C₄	—	43	16	no	—
255	C₄⋊C₄	—	43	16	no	—
256	C₂²⋊C₄	—	43	16	no	—
257	C₂²⋊C₄	—	43	16	no	—
258	C₄²	—	43	12	no	—
259	C₄²	—	43	12	no	—
260	C₄²	—	43	4	no	—
261	C₄²	—	43	4	no	—
262	C₂³	—	2	12	no	—
263	C₂³	—	2	12	no	—
264	C₂³	—	2	12	no	—
265	C₂³	—	4	16	no	—
266	C₂³	#13063281	1	4	yes	—
267	C₂³	#13063331	1	4	yes	—
268	C₂³	#13063333	1	4	yes	—
269	C₂³	C₂²×C₄×D₄₃	1	4	yes	—
270	C₂³	C₂²×C₄×D₄₃	1	3	yes	—
271	C₂×C₄	—	2	12	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	43	4	no	—
276	C₂×C₄	—	43	4	no	—
277	C₂×C₄	—	43	4	no	—
278	C₂×C₄	—	43	4	no	—
279	C₂×C₄	—	86	12	no	—
280	C₂×C₄	—	86	12	no	—
281	C₂×C₄	—	86	12	no	—
282	C₂×C₄	—	86	12	no	—
283	C₂×C₄	#13063281	1	12	yes	—
284	C₂×C₄	C₂²×C₄×D₄₃	1	4	yes	—
285	C₂×C₄	C₂²×C₄×D₄₃	1	3	yes	—
286	C₂×C₄	C₂×#10621900	1	6	yes	—
287	C₂×C₄	C₂×#10621900	1	6	yes	—
288	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₈	—	2	4	no	—
291	C₈	—	86	4	no	—
292	C₈	—	86	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	4	24	no	—
297	C₂²	#14582957	1	16	yes	direct product factor
298	C₂²	C₂×#13063281	1	6	yes	—
299	C₂²	C₂×#13063331	1	6	yes	—
300	C₂²	C₂×#13063333	1	6	yes	—
301	C₂²	C₂³×C₄×D₄₃	1	1	yes	characteristic
302	C₄	—	2	4	no	—
303	C₄	—	86	4	no	—
304	C₄	—	86	4	no	—
305	C₄	—	86	4	no	—
306	C₄	—	86	4	no	—
307	C₄	C₂×#13063281	1	6	yes	—
308	C₄	C₂²×#10621900	1	1	yes	characteristic
309	C₄	C₂²×#10621900	1	1	yes	characteristic
310	C₂	—	4	4	no	—
311	C₂	C₂×#14582957	1	12	yes	direct product factor
312	C₂	C₂²×#13063281	1	1	yes	characteristic
313	C₂	C₂²×#13063331	1	1	yes	characteristic
314	C₂	C₂²×#13063333	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×C₂²⋊C₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂²×C₄
prime radical	C₂⁵×C₄₃	C₂×C₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×C₂²⋊C₈	D₄₃
hypocenter	C₄₃	C₂²×#433
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14582957	C₂	direct (N × H)
3. C₂²×Dic₄₃⋊C₈	C₂	semidirect
4. C₂²×Dic₄₃⋊C₈	C₂	semidirect
6. C₂²×#13063235	C₂	semidirect
7. C₂²×C₈×Dic₄₃	C₂	semidirect
19. #14582957	C₂²	direct (N × H)
20. C₂×Dic₄₃⋊C₈	C₂²	semidirect
21. C₂×Dic₄₃⋊C₈	C₂²	semidirect
23. C₂×#13063235	C₂²	semidirect
24. C₂×C₈×Dic₄₃	C₂²	semidirect
47. Dic₄₃⋊C₈	C₂³	semidirect
48. Dic₄₃⋊C₈	C₂³	semidirect
50. #13063235	C₂³	semidirect
51. C₈×Dic₄₃	C₂³	semidirect
140. C₂²×C₄₃	#433	semidirect
162. C₂×C₄₃	C₂×#433	semidirect
186. C₄₃	C₂²×#433	semidirect
297. C₂²	#14582957	direct (N × H)
311. C₂	C₂×#14582957	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×#13063284	2
C₂×#14582957	2
C₂²×Dic₄₃⋊C₈	2
C₂²×C₂²⋊(C₄₃⋊C₈)	2
C₂²×#13063235	2
C₂²×C₈×Dic₄₃	2
C₂²×C₂²⋊C₈×C₄₃	2
C₂²×#433	43

Subgroups — C22×#14582957 #25960311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582957 #25960311