Subgroups — #25097049

Order 11008.

1070 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582942	C₂	1	2	yes	maximal
2	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582854	C₂	1	1	yes	maximal, characteristic
4	#19137215	C₂	1	2	yes	maximal
5	#19134760	C₂	1	4	yes	maximal, complemented
6	#19134758	C₂	1	2	yes	maximal, complemented
7	#19134753	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	1	yes	characteristic
10	C₂×#13063284	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	#14582942	—	2	8	no	—
13	C₂×#13063259	—	2	4	no	—
14	C₂×#13063259	C₂²	1	2	yes	—
15	C₂×#13063259	C₂²	1	2	yes	—
16	C₂×#13063258	C₂²	1	4	yes	—
17	C₂×#13063256	C₂²	1	4	yes	—
18	#14582854	C₄	1	8	yes	complemented
19	C₂×#13063235	C₂²	1	2	yes	—
20	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
21	#14582797	—	2	4	no	—
22	#14582771	—	2	4	no	—
23	#14582768	C₂²	1	4	yes	—
24	#14582764	C₂²	1	4	yes	—
25	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#427	—	2	4	no	—
28	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#106×C₄₃	C₂²	1	2	yes	—
31	C₂³×Dic₄₃	—	2	2	no	—
32	C₂×#10621877	C₂×C₄	1	4	yes	—
33	C₂×#10621877	D₄	1	4	yes	—
34	C₂×#10621877	Q₈	1	4	yes	—
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	#13063284	C₂×C₄	1	4	yes	—
37	#13063284	Q₈	1	4	yes	—
38	C₂×C₄⋊Dic₄₃	—	2	4	no	—
39	C₂×C₄⋊Dic₄₃	—	2	4	no	—
40	C₂×C₄⋊Dic₄₃	—	2	2	no	—
41	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
43	C₂×Dic₄₃⋊C₄	—	2	8	no	—
44	C₂×C₂²⋊Dic₄₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
46	#13063259	—	2	4	no	—
47	#13063259	—	2	4	no	—
48	#13063258	—	2	8	no	—
49	#13063256	—	2	8	no	—
50	#13063235	C₂×C₄	1	8	yes	complemented
51	C₈×Dic₄₃	C₂×C₄	1	8	yes	complemented
52	C₄₃⋊#95	—	2	8	no	—
53	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₄₃	D₄	1	4	yes	—
56	C₂²×C₈×C₄₃	Q₈	1	4	yes	—
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	#117×C₄₃	—	2	4	no	—
60	#117×C₄₃	—	2	4	no	—
61	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
64	C₂×C₄²×C₄₃	—	2	2	no	—
65	#107×C₄₃	—	2	2	no	—
66	#107×C₄₃	—	2	2	no	—
67	#106×C₄₃	—	2	4	no	—
68	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	2	2	no	—
71	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	2	4	no	—
80	C₂²×Dic₄₃	—	4	4	no	—
81	C₂²×Dic₄₃	—	8	4	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	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	2	no	—
94	C₂²⋊C₄×C₄₃	—	4	4	no	—
95	C₄²×C₄₃	—	4	4	no	—
96	C₂²⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
97	C₄⋊Dic₄₃	—	2	4	no	—
98	C₄⋊Dic₄₃	—	2	4	no	—
99	C₄⋊Dic₄₃	—	4	4	no	—
100	C₄⋊Dic₄₃	C₄⋊C₄	1	8	yes	—
101	Dic₄₃⋊C₄	—	8	8	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	2	no	—
109	C₂³×C₄₃	—	2	2	no	—
110	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂³×C₄₃	#127	1	2	yes	—
112	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
113	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	4	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₄₃	#127	1	2	yes	—
122	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
124	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	4	no	—
132	C₂×Dic₄₃	—	2	4	no	—
133	C₂×Dic₄₃	—	4	4	no	—
134	C₂×Dic₄₃	—	4	8	no	—
135	C₂×Dic₄₃	—	8	8	no	—
136	C₂×Dic₄₃	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
139	C₄₃⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
140	#20803	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	2	4	no	—
144	C₂²×C₄₃	—	2	4	no	—
145	C₂²×C₄₃	#384	1	4	yes	—
146	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄₃	#576	1	2	yes	—
150	C₂²×C₄₃	#578	1	2	yes	—
151	C₄×C₄₃	—	8	2	no	—
152	C₄×C₄₃	—	8	2	no	—
153	C₄×C₄₃	#384	1	4	yes	—
154	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	Dic₄₃	—	2	8	no	—
159	Dic₄₃	—	8	4	no	—
160	C₂×#428	—	43	2	no	—
161	C₂×#427	D₄₃	1	1	yes	characteristic
162	C₂²×#107	—	43	1	no	—
163	C₂×#405	—	43	1	no	—
164	#2317	—	43	2	no	—
165	#1847	—	43	4	no	—
166	#1846	—	43	2	no	—
167	#1845	—	43	2	no	—
168	C₂×C₄₃	—	2	4	no	—
169	C₂×C₄₃	C₂×#384	1	1	yes	characteristic
170	C₂×C₄₃	#2913	1	2	yes	—
171	C₂×C₄₃	#3155	1	1	yes	characteristic
172	C₂×C₄₃	#3156	1	1	yes	characteristic
173	C₂×C₄₃	#3164	1	2	yes	—
174	C₂²×M₁₆	—	43	1	no	—
175	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×#117	—	43	1	no	—
177	C₂×#117	C₂×D₄₃	1	2	yes	—
178	C₂²×C₄⋊C₄	—	43	2	no	—
179	#428	—	86	8	no	—
180	#427	—	2	4	no	—
181	C₂×#107	—	43	2	no	—
182	C₂×#107	—	43	2	no	—
183	C₂×#107	—	43	4	no	—
184	C₂×#107	—	86	4	no	—
185	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
186	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#106	—	43	4	no	—
188	C₂×#106	C₂×D₄₃	1	2	yes	—
189	#405	—	43	8	no	—
190	C₂×#97	—	43	2	no	—
191	C₂×C₄×C₈	—	43	2	no	—
192	#389	—	86	4	no	—
193	#385	—	86	4	no	—
194	#384	—	43	4	no	—
195	#382	—	43	4	no	—
196	C₄₃	#20803	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	—	86	2	no	—
198	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
199	C₂×M₁₆	—	43	4	no	—
200	C₂×M₁₆	—	43	4	no	—
201	C₂×M₁₆	—	43	4	no	—
202	C₂²×C₈	—	43	2	no	—
203	C₂²×C₈	Dic₈₆	1	4	yes	—
204	C₂²×C₈	C₄×D₄₃	1	4	yes	—
205	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
206	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
208	#117	—	2	4	no	—
209	#117	—	2	4	no	—
210	#117	—	43	4	no	—
211	#117	—	43	4	no	—
212	C₂×C₄⋊C₄	—	43	2	no	—
213	C₂×C₄⋊C₄	—	43	4	no	—
214	C₂×C₄⋊C₄	—	86	4	no	—
215	C₂×C₄⋊C₄	—	86	4	no	—
216	C₂×C₄⋊C₄	—	86	8	no	—
217	C₂×C₄⋊C₄	—	86	2	no	—
218	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	43	2	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	43	2	no	—
224	#107	—	2	2	no	—
225	#107	—	2	2	no	—
226	#107	—	86	8	no	—
227	#107	—	86	4	no	—
228	#107	—	86	4	no	—
229	#106	—	2	4	no	—
230	#106	—	86	8	no	—
231	#97	—	43	8	no	—
232	C₄×C₈	—	43	8	no	—
233	#95	—	86	8	no	—
234	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	43	4	no	—
238	C₂²×C₄	—	86	4	no	—
239	C₂²×C₄	—	344	4	no	—
240	C₂²×C₄	—	172	4	no	—
241	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
243	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₄₃	1	2	yes	—
245	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
246	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
248	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
249	M₁₆	—	43	16	no	—
250	C₂×C₈	—	43	4	no	—
251	C₂×C₈	—	43	4	no	—
252	C₂×C₈	—	43	4	no	—
253	C₂×C₈	Dic₄₃⋊C₄	1	16	yes	complemented
254	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
255	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
258	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	43	8	no	—
265	C₄⋊C₄	—	86	4	no	—
266	C₄⋊C₄	—	86	4	no	—
267	C₄⋊C₄	—	344	8	no	—
268	C₄⋊C₄	—	172	4	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	43	8	no	—
271	C₄²	—	4	4	no	—
272	C₄²	—	43	4	no	—
273	C₄²	—	43	4	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	C₄⋊C₄×D₄₃	1	2	yes	—
277	C₂³	#13063321	1	2	yes	—
278	C₂³	#13063324	1	2	yes	—
279	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	43	8	no	—
285	C₂×C₄	—	86	4	no	—
286	C₂×C₄	—	86	4	no	—
287	C₂×C₄	—	344	8	no	—
288	C₂×C₄	—	172	4	no	—
289	C₂×C₄	—	172	8	no	—
290	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
291	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
292	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
293	C₂×C₄	#13063321	1	2	yes	—
294	C₂×C₄	#13063323	1	4	yes	—
295	C₂×C₄	#13063324	1	4	yes	—
296	C₂×C₄	#13063324	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
303	C₈	—	43	8	no	—
304	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
305	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	#14582768	1	4	yes	—
311	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
312	C₂²	C₂×#13063321	1	1	yes	characteristic
313	C₂²	C₂×#13063324	1	1	yes	characteristic
314	C₂²	#14583701	1	2	yes	—
315	C₂²	#14583713	1	2	yes	—
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	86	8	no	—
319	C₄	—	344	4	no	—
320	C₄	#14582768	1	4	yes	—
321	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
322	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
323	C₄	C₂×#13063323	1	1	yes	characteristic
324	C₄	C₂×#13063324	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₂×#14582768	1	1	yes	characteristic
327	C₂	#19143192	1	2	yes	—
328	C₂	#19145082	1	1	yes	characteristic
329	C₂	#19145090	1	1	yes	characteristic
330	C₂	#19145135	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#427	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#427	D₄₃
hypocenter	C₄₃	#20803
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
5. #19134760	C₂	semidirect
6. #19134758	C₂	semidirect
7. #19134753	C₂	semidirect
18. #14582854	C₄	semidirect
50. #13063235	C₂×C₄	semidirect
51. C₈×Dic₄₃	C₂×C₄	semidirect
83. C₂×C₈×C₄₃	C₄⋊C₄	semidirect
104. #10621877	C₄⋊C₄	semidirect
196. C₄₃	#20803	semidirect
253. 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×#14582942	2
C₂²×#13063259	2
C₂×#14582854	2
#19137215	2
#19134760	2
#19134758	2
#19134753	2
C₂×C₄₃×#427	2
#20803	43