Subgroups — #25097035

Order 11008.

1070 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582939	C₂	1	2	yes	maximal
2	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582854	C₂	1	1	yes	maximal, characteristic
4	#19137219	C₂	1	2	yes	maximal
5	#19134760	C₂	1	4	yes	maximal, complemented
6	#19134756	C₂	1	2	yes	maximal, complemented
7	#19134750	C₂	1	2	yes	maximal, complemented
8	C₂²×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	1	yes	characteristic
10	C₂×#13063284	C₂²	1	2	yes	—
11	C₂×#13063284	C₂²	1	1	yes	characteristic
12	#14582939	—	2	8	no	—
13	#14582937	—	2	4	no	—
14	C₂×#13063259	C₂²	1	2	yes	—
15	C₂×#13063258	C₂²	1	4	yes	—
16	C₂×#13063257	C₂²	1	1	yes	characteristic
17	C₂×#13063257	C₂²	1	1	yes	characteristic
18	C₂×#13063256	C₂²	1	4	yes	—
19	#14582854	C₄	1	8	yes	complemented
20	C₂×#13063235	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
22	#14582795	—	2	4	no	—
23	#14582772	—	2	4	no	—
24	#14582768	C₂²	1	4	yes	—
25	#14582764	C₂²	1	4	yes	—
26	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₂×#106×C₄₃	—	2	4	no	—
29	C₂×#106×C₄₃	C₂²	1	2	yes	—
30	C₂×#106×C₄₃	C₂²	1	2	yes	—
31	C₂×#10621877	C₂×C₄	1	4	yes	—
32	C₂×#10621877	D₄	1	4	yes	—
33	C₂×#10621877	Q₈	1	4	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
35	#13063284	—	2	4	no	—
36	#13063284	—	2	4	no	—
37	#13063284	C₂×C₄	1	4	yes	—
38	#13063284	Q₈	1	4	yes	—
39	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
42	C₂×Dic₄₃⋊C₄	—	2	8	no	—
43	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
44	C₂×C₂²⋊Dic₄₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₃	—	2	2	no	—
46	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
47	#13063259	—	2	4	no	—
48	#13063258	—	2	8	no	—
49	#13063257	—	2	2	no	—
50	#13063257	—	2	2	no	—
51	#13063256	—	2	8	no	—
52	#13063235	C₂×C₄	1	8	yes	complemented
53	C₈×Dic₄₃	C₂×C₄	1	8	yes	complemented
54	C₄₃⋊#95	—	2	8	no	—
55	C₂³×C₄×C₄₃	—	2	2	no	—
56	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₄₃	D₄	1	4	yes	—
59	C₂²×C₈×C₄₃	Q₈	1	4	yes	—
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
63	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
64	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
66	#106×C₄₃	—	2	4	no	—
67	#106×C₄₃	—	2	4	no	—
68	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	—	2	4	no	—
70	C₂²×C₄×C₄₃	—	4	4	no	—
71	C₂²×C₄×C₄₃	—	8	4	no	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	2	2	no	—
81	C₂²×Dic₄₃	—	2	2	no	—
82	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₄₃	C₄⋊C₄	1	16	yes	complemented
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
90	C₄⋊C₄×C₄₃	—	2	4	no	—
91	C₄⋊C₄×C₄₃	—	2	4	no	—
92	C₄⋊C₄×C₄₃	—	4	4	no	—
93	C₂²⋊Dic₄₃	—	4	4	no	—
94	C₂²⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	C₄⋊Dic₄₃	—	2	2	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
100	Dic₄₃⋊C₄	—	8	8	no	—
101	C₄×Dic₄₃	—	4	4	no	—
102	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
103	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
104	#10621877	C₄⋊C₄	1	16	yes	complemented
105	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
107	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
108	C₂³×C₄₃	—	2	4	no	—
109	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₄₃	#127	1	2	yes	—
111	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	—	4	4	no	—
114	C₂×C₄×C₄₃	—	4	8	no	—
115	C₂×C₄×C₄₃	—	8	8	no	—
116	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₃	#127	1	2	yes	—
120	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
121	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×Dic₄₃	—	2	8	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	—	4	4	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	C₂×Dic₄₃	C₂×C₄⋊C₄	1	8	yes	—
135	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₄₃⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
138	#20802	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	4	no	—
140	C₂²×C₄₃	—	2	4	no	—
141	C₂²×C₄₃	—	2	4	no	—
142	C₂²×C₄₃	#384	1	4	yes	—
143	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
145	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₄₃	#577	1	4	yes	—
147	C₄×C₄₃	—	8	4	no	—
148	C₄×C₄₃	#384	1	4	yes	—
149	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	Dic₄₃	—	2	8	no	—
154	Dic₄₃	—	8	2	no	—
155	Dic₄₃	—	8	2	no	—
156	C₂×#428	—	43	2	no	—
157	C₂×#427	—	43	1	no	—
158	C₂²×#106	D₄₃	1	1	yes	characteristic
159	C₂×#405	—	43	1	no	—
160	#2317	—	43	2	no	—
161	#1847	—	43	4	no	—
162	#1846	—	43	2	no	—
163	#1844	—	43	2	no	—
164	C₂×C₄₃	—	2	4	no	—
165	C₂×C₄₃	C₂×#384	1	1	yes	characteristic
166	C₂×C₄₃	#2912	1	2	yes	—
167	C₂×C₄₃	#3154	1	2	yes	—
168	C₂×C₄₃	#3163	1	2	yes	—
169	C₂²×M₁₆	—	43	1	no	—
170	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#117	—	43	1	no	—
172	C₂×#117	—	43	2	no	—
173	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
174	#428	—	86	8	no	—
175	#427	—	86	4	no	—
176	C₂×#107	—	43	2	no	—
177	C₂×#107	—	43	4	no	—
178	C₂×#106	—	2	4	no	—
179	C₂×#106	—	43	1	no	—
180	C₂×#106	—	43	1	no	—
181	C₂×#106	—	43	4	no	—
182	C₂×#106	C₂×D₄₃	1	2	yes	—
183	C₂×#106	C₂×D₄₃	1	2	yes	—
184	#405	—	43	8	no	—
185	C₂×#97	—	43	2	no	—
186	C₂×C₄×C₈	—	43	2	no	—
187	#389	—	86	4	no	—
188	#385	—	86	4	no	—
189	#384	—	43	4	no	—
190	#382	—	43	4	no	—
191	C₄₃	#20802	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
194	C₂×M₁₆	—	43	4	no	—
195	C₂×M₁₆	—	43	4	no	—
196	C₂×M₁₆	—	43	4	no	—
197	C₂²×C₈	—	43	2	no	—
198	C₂²×C₈	Dic₈₆	1	4	yes	—
199	C₂²×C₈	C₄×D₄₃	1	4	yes	—
200	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
201	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	#117	—	43	4	no	—
204	#117	—	43	4	no	—
205	#117	—	86	4	no	—
206	#117	—	86	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	43	2	no	—
211	C₂×C₄⋊C₄	—	43	2	no	—
212	C₂×C₄⋊C₄	—	43	2	no	—
213	C₂×C₄⋊C₄	—	86	8	no	—
214	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
215	C₂×C₂²⋊C₄	—	43	2	no	—
216	C₂×C₂²⋊C₄	—	86	2	no	—
217	C₂×C₄²	—	43	2	no	—
218	C₂×C₄²	—	86	2	no	—
219	#107	—	86	8	no	—
220	#107	—	86	4	no	—
221	#106	—	2	4	no	—
222	#106	—	2	4	no	—
223	#106	—	86	8	no	—
224	#106	—	86	2	no	—
225	#106	—	86	2	no	—
226	#97	—	43	8	no	—
227	C₄×C₈	—	43	8	no	—
228	#95	—	86	8	no	—
229	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	4	no	—
232	C₂²×C₄	—	8	4	no	—
233	C₂²×C₄	—	43	4	no	—
234	C₂²×C₄	—	86	2	no	—
235	C₂²×C₄	—	86	2	no	—
236	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
238	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
239	C₂²×C₄	D₄×D₄₃	1	2	yes	—
240	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
241	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
243	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
244	M₁₆	—	43	16	no	—
245	C₂×C₈	—	43	4	no	—
246	C₂×C₈	—	43	4	no	—
247	C₂×C₈	—	43	4	no	—
248	C₂×C₈	Dic₄₃⋊C₄	1	16	yes	complemented
249	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
250	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
252	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
253	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	43	8	no	—
259	C₄⋊C₄	—	86	2	no	—
260	C₄⋊C₄	—	86	2	no	—
261	C₄⋊C₄	—	86	2	no	—
262	C₄⋊C₄	—	86	2	no	—
263	C₄⋊C₄	—	344	8	no	—
264	C₂²⋊C₄	—	43	8	no	—
265	C₂²⋊C₄	—	172	4	no	—
266	C₄²	—	43	4	no	—
267	C₄²	—	43	4	no	—
268	C₄²	—	172	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₄⋊C₄×D₄₃	1	2	yes	—
271	C₂³	#13063321	1	2	yes	—
272	C₂³	#13063324	1	2	yes	—
273	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	8	8	no	—
277	C₂×C₄	—	43	8	no	—
278	C₂×C₄	—	86	8	no	—
279	C₂×C₄	—	172	2	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	—	172	4	no	—
282	C₂×C₄	—	172	4	no	—
283	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
284	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
285	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
286	C₂×C₄	#13063321	1	2	yes	—
287	C₂×C₄	#13063323	1	4	yes	—
288	C₂×C₄	#13063324	1	4	yes	—
289	C₂×C₄	#13063324	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
293	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
296	C₈	—	43	8	no	—
297	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
298	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#14582768	1	4	yes	—
303	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
304	C₂²	C₂×#13063321	1	1	yes	characteristic
305	C₂²	C₂×#13063324	1	1	yes	characteristic
306	C₂²	#14583705	1	4	yes	—
307	C₄	—	8	4	no	—
308	C₄	—	86	8	no	—
309	C₄	—	344	2	no	—
310	C₄	—	344	2	no	—
311	C₄	#14582768	1	4	yes	—
312	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
313	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
314	C₄	C₂×#13063323	1	1	yes	characteristic
315	C₄	C₂×#13063324	1	1	yes	characteristic
316	C₂	—	2	4	no	—
317	C₂	C₂×#14582768	1	1	yes	characteristic
318	C₂	#19143186	1	2	yes	—
319	C₂	#19145065	1	2	yes	—
320	C₂	#19145123	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×#106×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×#106	D₄₃
hypocenter	C₄₃	#20802
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
5. #19134760	C₂	semidirect
6. #19134756	C₂	semidirect
7. #19134750	C₂	semidirect
19. #14582854	C₄	semidirect
52. #13063235	C₂×C₄	semidirect
53. C₈×Dic₄₃	C₂×C₄	semidirect
83. C₂×C₈×C₄₃	C₄⋊C₄	semidirect
104. #10621877	C₄⋊C₄	semidirect
191. C₄₃	#20802	semidirect
248. C₂×C₈	Dic₄₃⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582939	2
C₂×#14582937	2
C₂×#14582854	2
#19137219	2
#19134760	2
#19134756	2
#19134750	2
C₂²×#106×C₄₃	2
#20802	43