Subgroups — #25002477

Order 11008.

830 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	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582765	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582667	C₂	1	4	yes	maximal
4	C₂×C₄₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
7	C₄×#10621877	—	2	2	no	—
8	C₄×#10621877	C₄	1	4	yes	complemented
9	C₂×#13063236	—	2	1	no	—
10	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
11	#14582765	—	2	1	no	—
12	#14582765	—	2	1	no	—
13	#14582765	—	2	2	no	—
14	C₂×C₄₃⋊#95	C₄	1	4	yes	—
15	#14582667	—	2	8	no	—
16	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#423	—	2	2	no	—
19	C₄₃×#423	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
21	C₂³×Dic₄₃	C₂×C₄	1	4	yes	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	C₂×C₄	1	4	yes	—
26	C₂×#10621877	D₄	1	4	yes	—
27	C₂×#10621877	Q₈	1	4	yes	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	—	2	4	no	—
30	C₂×C₄⋊Dic₄₃	—	2	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	4	no	—
33	#13063236	—	2	4	no	—
34	C₄×C₄₃⋊C₈	—	2	4	no	—
35	C₄₃⋊#95	—	2	4	no	—
36	C₄₃⋊#95	—	2	4	no	—
37	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	—	2	2	no	—
40	C₂×M₁₆×C₄₃	—	2	4	no	—
41	C₂×M₁₆×C₄₃	—	2	4	no	—
42	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	D₄	1	2	yes	—
49	C₂×C₄²×C₄₃	D₄	1	2	yes	—
50	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
51	C₄⋊C₈×C₄₃	—	2	8	no	—
52	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	—	4	4	no	—
54	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×C₄×C₄₃	#41	1	2	yes	—
68	C₂²×Dic₄₃	—	2	4	no	—
69	C₂²×Dic₄₃	—	2	8	no	—
70	C₂²×Dic₄₃	—	4	8	no	—
71	M₁₆×C₄₃	—	2	8	no	—
72	M₁₆×C₄₃	—	4	4	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	4	2	no	—
75	C₂×C₈×C₄₃	—	4	2	no	—
76	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
79	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
80	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
82	C₄⋊Dic₄₃	—	4	4	no	—
83	C₄⋊Dic₄₃	—	4	2	no	—
84	C₄⋊Dic₄₃	—	4	2	no	—
85	#10621877	—	2	4	no	—
86	#10621877	—	2	4	no	—
87	#10621877	—	2	4	no	—
88	#10621877	C₄⋊C₄	1	8	yes	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂×C₄₃⋊C₈	—	2	4	no	—
92	C₂³×C₄₃	—	2	4	no	—
93	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
94	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₃	C₄×D₄	1	2	yes	—
97	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₄₃	—	2	4	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	#95	1	4	yes	—
103	C₂×C₄×C₄₃	#95	1	4	yes	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#117	1	2	yes	—
110	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	#121	1	2	yes	—
113	C₂×C₄×C₄₃	#121	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#125	1	2	yes	—
119	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
122	C₂×Dic₄₃	—	4	4	no	—
123	C₂×Dic₄₃	—	4	8	no	—
124	C₂×Dic₄₃	—	8	8	no	—
125	C₈×C₄₃	—	4	4	no	—
126	C₄₃⋊C₈	—	2	8	no	—
127	#10858	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
133	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
134	C₂²×C₄₃	#389	1	2	yes	—
135	C₂²×C₄₃	#390	1	1	yes	characteristic
136	C₂²×C₄₃	#391	1	1	yes	characteristic
137	C₂²×C₄₃	#413	1	4	yes	—
138	C₄×C₄₃	—	2	4	no	—
139	C₄×C₄₃	—	2	4	no	—
140	C₄×C₄₃	#380	1	2	yes	—
141	C₄×C₄₃	#382	1	1	yes	characteristic
142	C₄×C₄₃	#384	1	1	yes	characteristic
143	C₄×C₄₃	#389	1	2	yes	—
144	C₄×C₄₃	#390	1	1	yes	characteristic
145	C₄×C₄₃	#391	1	1	yes	characteristic
146	Dic₄₃	—	8	4	no	—
147	C₂×#423	D₄₃	1	1	yes	characteristic
148	C₂×C₄×M₁₆	—	43	1	no	—
149	C₂×#382	—	43	1	no	—
150	C₂×#343	—	43	4	no	—
151	C₂×C₄₃	—	2	2	no	—
152	C₂×C₄₃	—	2	2	no	—
153	C₂×C₄₃	C₄⋊#95	1	1	yes	characteristic
154	C₂×C₄₃	#1742	1	2	yes	—
155	C₂×C₄₃	#1967	1	2	yes	—
156	C₂×C₄₃	#1998	1	2	yes	—
157	C₂²×M₁₆	—	43	2	no	—
158	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
159	C₂²×C₄⋊C₄	—	43	2	no	—
160	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
161	#423	—	2	2	no	—
162	#423	Dic₄₃	1	4	yes	complemented
163	C₂×C₄⋊C₈	—	2	2	no	—
164	C₄×M₁₆	—	43	4	no	—
165	C₄×M₁₆	—	86	2	no	—
166	C₂×#97	—	86	1	no	—
167	C₂×C₄×C₈	—	86	1	no	—
168	#382	—	86	1	no	—
169	#382	—	86	1	no	—
170	#382	—	86	2	no	—
171	C₂×#95	—	43	4	no	—
172	#343	—	86	8	no	—
173	C₄₃	#10858	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	43	4	no	—
175	C₂³×C₄	C₄×D₄₃	1	2	yes	—
176	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	43	4	no	—
181	C₂×M₁₆	—	43	4	no	—
182	C₂×M₁₆	—	43	4	no	—
183	C₂×M₁₆	—	86	2	no	—
184	C₂×M₁₆	—	86	2	no	—
185	C₂×M₁₆	—	86	2	no	—
186	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	86	2	no	—
189	C₂×C₄⋊C₄	—	86	2	no	—
190	C₂×C₄⋊C₄	—	86	4	no	—
191	C₂×C₄⋊C₄	—	86	4	no	—
192	C₂×C₄⋊C₄	—	86	2	no	—
193	C₂×C₄²	Dic₈₆	1	2	yes	—
194	C₂×C₄²	C₄×D₄₃	1	2	yes	—
195	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
196	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
198	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
200	C₄⋊C₈	—	2	8	no	—
201	#97	—	86	4	no	—
202	C₄×C₈	—	86	4	no	—
203	#95	—	86	4	no	—
204	#95	—	86	4	no	—
205	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	86	8	no	—
208	C₂²×C₄	—	86	4	no	—
209	C₂²×C₄	—	172	8	no	—
210	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
211	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
212	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
213	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
214	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₄₃	1	2	yes	—
218	C₂²×C₄	#10621900	1	2	yes	—
219	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
220	C₂²×C₄	#10621902	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
224	M₁₆	—	2	8	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	43	8	no	—
227	M₁₆	—	86	4	no	—
228	M₁₆	—	86	4	no	—
229	M₁₆	—	86	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	86	4	no	—
234	C₂×C₈	—	86	4	no	—
235	C₂×C₈	—	86	4	no	—
236	C₄⋊C₄	—	172	4	no	—
237	C₄⋊C₄	—	172	2	no	—
238	C₄⋊C₄	—	172	2	no	—
239	C₄²	Dic₄₃⋊C₄	1	4	yes	—
240	C₄²	C₄⋊Dic₄₃	1	2	yes	—
241	C₄²	C₄⋊Dic₄₃	1	2	yes	—
242	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
243	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
244	C₄²	C₂²⋊Dic₄₃	1	4	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
247	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
248	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
249	C₂³	#13063287	1	2	yes	—
250	C₂³	#13063290	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	344	8	no	—
256	C₂×C₄	—	172	4	no	—
257	C₂×C₄	—	172	8	no	—
258	C₂×C₄	C₄₃⋊#95	1	4	yes	—
259	C₂×C₄	C₄₃⋊#95	1	4	yes	—
260	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
262	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
263	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
265	C₂×C₄	#13063283	1	2	yes	—
266	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
267	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
268	C₂×C₄	#13063296	1	2	yes	—
269	C₂×C₄	#13063300	1	2	yes	—
270	C₂×C₄	#13063301	1	2	yes	—
271	C₂×C₄	#13063305	1	2	yes	—
272	C₂×C₄	#13063312	1	1	yes	characteristic
273	C₂×C₄	#13063314	1	1	yes	characteristic
274	C₂×C₄	#13063316	1	2	yes	—
275	C₂×C₄	#13063321	1	1	yes	characteristic
276	C₂×C₄	#13063323	1	1	yes	characteristic
277	C₂×C₄	#13063324	1	2	yes	—
278	C₈	—	4	4	no	—
279	C₈	—	86	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
285	C₂²	C₄⋊C₄×Dic₄₃	1	2	yes	—
286	C₂²	#14582795	1	2	yes	—
287	C₂²	#14582800	1	1	yes	characteristic
288	C₂²	#14582804	1	1	yes	characteristic
289	C₂²	#14582881	1	4	yes	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	344	4	no	—
293	C₄	#14582756	1	2	yes	—
294	C₄	#14582764	1	1	yes	characteristic
295	C₄	#14582768	1	1	yes	characteristic
296	C₄	#14582798	1	2	yes	—
297	C₄	#14582802	1	1	yes	characteristic
298	C₄	#14582806	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#19132824	1	1	yes	characteristic
302	C₂	#19134333	1	2	yes	—
303	C₂	#19135442	1	2	yes	—
304	C₂	#19135587	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator	C₂³×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#423	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#423	D₄₃
hypocenter	C₄₃	#10858
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
8. C₄×#10621877	C₄	semidirect
19. C₄₃×#423	C₄	semidirect
162. #423	Dic₄₃	semidirect
173. C₄₃	#10858	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621877	2
C₂×#14582765	2
C₂×#14582667	2
C₂×C₄₃×#423	2
#10858	43