Subgroups — #24987223

Order 11008.

578 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063290	C₂	1	1	yes	maximal, characteristic
2	#19134450	C₂	1	1	yes	maximal, characteristic
3	#19134340	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582631	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582629	C₂	1	1	yes	maximal, characteristic
6	#19132436	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
9	C₂×#13063290	C₄	1	2	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×#13063235	—	2	1	no	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×Dic₄₃	C₄	1	2	yes	—
14	C₄×Dic₄₃⋊C₄	—	2	1	no	—
15	#14582750	—	2	2	no	—
16	C₄²×Dic₄₃	—	2	1	no	—
17	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582653	—	2	2	no	—
19	#14582653	—	2	1	no	—
20	#14582631	C₄	1	4	yes	complemented
21	#14582629	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	1	no	—
28	C₄×Dic₈₆	—	2	1	no	—
29	C₄×Dic₈₆	—	2	1	no	—
30	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
31	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
32	#13063290	—	2	4	no	—
33	#13063290	—	2	4	no	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₄₃⋊C₄	—	2	1	no	—
36	C₂×Dic₄₃⋊C₄	—	2	2	no	—
37	C₂×C₄×Dic₄₃	—	2	1	no	—
38	C₂×C₄×Dic₄₃	—	2	2	no	—
39	C₂×C₄×Dic₄₃	—	2	2	no	—
40	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
43	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
46	#13063235	—	4	2	no	—
47	C₄×C₄₃⋊C₈	—	2	1	no	—
48	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₄₃⋊#95	—	2	2	no	—
50	C₂²×C₈×C₄₃	—	2	1	no	—
51	C₂²×C₈×C₄₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₄₃	—	2	1	no	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	—	2	1	no	—
56	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₄₃	—	2	1	no	—
58	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
59	#95×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	—	2	2	no	—
70	C₂×Dic₈₆	C₄²	1	2	yes	—
71	C₂×Dic₈₆	M₁₆	1	4	yes	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	4	2	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₄⋊C₄×C₄₃	—	2	2	no	—
78	C₄⋊C₄×C₄₃	M₁₆	1	4	yes	—
79	C₄²×C₄₃	—	4	1	no	—
80	C₄²×C₄₃	—	4	1	no	—
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₃	#41	1	1	yes	characteristic
84	C₄²×C₄₃	#41	1	1	yes	characteristic
85	C₄⋊Dic₄₃	—	2	1	no	—
86	C₄⋊Dic₄₃	C₄²	1	2	yes	—
87	Dic₄₃⋊C₄	—	2	8	no	—
88	Dic₄₃⋊C₄	—	2	2	no	—
89	Dic₄₃⋊C₄	—	2	2	no	—
90	C₄×Dic₄₃	—	2	8	no	—
91	C₄×Dic₄₃	—	2	2	no	—
92	C₄×Dic₄₃	—	2	2	no	—
93	C₄×Dic₄₃	—	2	2	no	—
94	C₄×Dic₄₃	—	2	2	no	—
95	C₄×Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	4	2	no	—
98	C₄×Dic₄₃	—	4	4	no	—
99	C₄×Dic₄₃	SD₁₆	1	2	yes	—
100	C₄×Dic₄₃	Q₁₆	1	2	yes	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	—	4	2	no	—
104	C₂×C₄₃⋊C₈	—	4	1	no	—
105	C₂×C₄₃⋊C₈	—	4	1	no	—
106	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	—	4	2	no	—
110	C₂×C₄×C₄₃	—	4	2	no	—
111	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	2	no	—
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	2	4	no	—
126	C₂×Dic₄₃	—	2	2	no	—
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	2	2	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	#103	1	4	yes	—
133	Dic₈₆	—	2	2	no	—
134	Dic₈₆	—	2	4	no	—
135	Dic₈₆	#97	1	4	yes	complemented
136	C₈×C₄₃	—	4	2	no	—
137	C₈×C₄₃	—	4	2	no	—
138	C₄₃⋊C₈	—	4	2	no	—
139	C₄₃⋊C₈	—	4	2	no	—
140	#8175	—	43	1	no	maximal, Sylow
141	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₃	#407	1	2	yes	—
143	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
144	C₂²×C₄₃	#416	1	1	yes	characteristic
145	C₂²×C₄₃	#421	1	2	yes	—
146	C₄×C₄₃	—	2	1	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	4	4	no	—
149	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
150	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
151	C₄×C₄₃	#432	1	2	yes	—
152	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
153	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
154	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
155	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
156	Dic₄₃	—	2	4	no	—
157	Dic₄₃	—	2	4	no	—
158	Dic₄₃	—	4	2	no	—
159	Dic₄₃	—	4	2	no	—
160	C₄²×Q₈	—	43	1	no	—
161	#1767	—	43	1	no	—
162	#1744	—	43	1	no	—
163	C₂×Q₈⋊C₈	—	43	1	no	—
164	C₂×Q₈⋊C₈	—	43	1	no	—
165	#1279	—	43	1	no	—
166	#1279	D₄₃	1	1	yes	characteristic
167	C₂×C₄₃	#1487	1	2	yes	—
168	C₂×C₄₃	#1488	1	2	yes	—
169	C₂×C₄₃	#1749	1	1	yes	characteristic
170	C₂×C₄₃	#1759	1	1	yes	characteristic
171	C₂×C₄₃	C₄×#103	1	1	yes	characteristic
172	C₂×C₄×Q₈	—	43	2	no	—
173	C₂×C₄×Q₈	—	43	1	no	—
174	C₂×C₄⋊C₈	—	43	1	no	—
175	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂×#97	—	86	1	no	—
177	C₂×C₄×C₈	—	43	1	no	—
178	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	43	2	no	—
180	C₄×C₄⋊C₄	—	86	2	no	—
181	C₄×C₄⋊C₄	—	86	1	no	—
182	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
183	C₄³	—	43	1	no	—
184	C₄³	—	86	1	no	—
185	#336	—	86	2	no	—
186	#336	—	86	1	no	—
187	Q₈⋊C₈	—	43	4	no	—
188	Q₈⋊C₈	—	43	4	no	—
189	C₄₃	#8175	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	43	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	86	1	no	—
194	C₂²×C₈	—	86	1	no	—
195	C₄×Q₈	—	43	2	no	—
196	C₄×Q₈	—	43	2	no	—
197	C₄×Q₈	—	86	4	no	—
198	C₄×Q₈	—	86	4	no	—
199	C₄×Q₈	—	86	1	no	—
200	C₄×Q₈	—	86	1	no	—
201	C₂×C₄⋊C₄	—	43	1	no	—
202	C₂×C₄⋊C₄	—	86	2	no	—
203	C₂×C₄⋊C₄	—	86	1	no	—
204	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	43	1	no	—
207	C₂×C₄²	—	43	2	no	—
208	C₂×C₄²	—	43	1	no	—
209	C₂×C₄²	—	43	2	no	—
210	C₂×C₄²	—	86	2	no	—
211	C₂×C₄²	—	86	1	no	—
212	C₂×C₄²	—	86	2	no	—
213	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	43	2	no	—
216	C₄⋊C₈	—	86	1	no	—
217	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
218	#97	—	172	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	43	2	no	—
221	C₄×C₈	—	86	1	no	—
222	C₄×C₈	C₄×D₄₃	1	2	yes	—
223	#95	—	2	1	no	—
224	#95	—	86	2	no	—
225	C₂×Q₈	—	43	2	no	—
226	C₂×Q₈	—	43	4	no	—
227	C₂×Q₈	—	86	1	no	—
228	C₂×Q₈	—	86	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	43	2	no	—
231	C₂²×C₄	—	43	2	no	—
232	C₂²×C₄	—	86	1	no	—
233	C₂²×C₄	—	86	1	no	—
234	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
236	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	86	2	no	—
243	C₂×C₈	—	86	2	no	—
244	C₂×C₈	—	172	2	no	—
245	C₂×C₈	—	172	1	no	—
246	C₂×C₈	—	172	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	43	2	no	—
249	C₄⋊C₄	—	86	8	no	—
250	C₄⋊C₄	—	86	1	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	86	2	no	—
253	C₄⋊C₄	#10621872	1	4	yes	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	43	2	no	—
257	C₄²	—	43	2	no	—
258	C₄²	—	86	8	no	—
259	C₄²	—	86	2	no	—
260	C₄²	—	86	2	no	—
261	C₄²	—	86	2	no	—
262	C₄²	—	86	2	no	—
263	C₄²	—	86	1	no	—
264	C₄²	—	86	1	no	—
265	C₄²	—	172	2	no	—
266	C₄²	—	172	4	no	—
267	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
268	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₄²	#10621900	1	1	yes	characteristic
270	C₄²	#10621902	1	1	yes	characteristic
271	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
272	Q₈	—	43	4	no	—
273	Q₈	—	86	2	no	—
274	Q₈	—	86	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	43	4	no	—
280	C₂×C₄	—	86	2	no	—
281	C₂×C₄	—	86	4	no	—
282	C₂×C₄	—	86	2	no	—
283	C₂×C₄	—	86	2	no	—
284	C₂×C₄	—	86	2	no	—
285	C₂×C₄	—	86	2	no	—
286	C₂×C₄	—	86	2	no	—
287	C₂×C₄	—	86	2	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
290	C₂×C₄	#13063283	1	1	yes	characteristic
291	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
293	C₂×C₄	#13063287	1	1	yes	characteristic
294	C₂×C₄	#13063287	1	1	yes	characteristic
295	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
296	C₂×C₄	C₂×#10621872	1	2	yes	—
297	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
298	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
299	C₂×C₄	#13063344	1	1	yes	characteristic
300	C₂×C₄	#13063346	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	172	2	no	—
304	C₈	—	172	2	no	—
305	C₂²	#14582746	1	1	yes	characteristic
306	C₂²	#14582861	1	2	yes	—
307	C₂²	#103×D₄₃	1	1	yes	characteristic
308	C₂²	#14582892	1	1	yes	characteristic
309	C₂²	#14582914	1	2	yes	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	4	no	—
313	C₄	—	86	4	no	—
314	C₄	—	86	4	no	—
315	C₄	—	172	2	no	—
316	C₄	—	172	2	no	—
317	C₄	#97×D₄₃	1	1	yes	characteristic
318	C₄	C₄×#10621872	1	1	yes	characteristic
319	C₄	#14582955	1	2	yes	—
320	C₄	#14582986	1	1	yes	characteristic
321	C₄	#14582990	1	1	yes	characteristic
322	C₄	#14582991	1	1	yes	characteristic
323	C₄	#14582993	1	1	yes	characteristic
324	C₂	#19133022	1	2	yes	—
325	C₂	#19133026	1	2	yes	—
326	C₂	#19134356	1	1	yes	characteristic
327	C₂	#19134404	1	1	yes	characteristic
328	C₂	#19134414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1279	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1279	D₄₃
hypocenter	C₄₃	#8175
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
20. #14582631	C₄	semidirect
21. #14582629	C₄	semidirect
135. Dic₈₆	#97	semidirect
189. C₄₃	#8175	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063290	2
#19134450	2
#19134340	2
C₂×#14582631	2
C₂×#14582629	2
#19132436	2
C₄₃×#1279	2
#8175	43