Subgroups — #24991239

Order 11008.

586 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143618	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063314	C₂	1	1	yes	maximal, characteristic
3	#19132990	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132915	C₂	1	1	yes	maximal, characteristic
5	#19132427	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063314	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	C₈×D₁₇₂	—	2	2	no	—
11	#14582964	—	2	2	no	—
12	#14582952	C₂²	1	1	yes	characteristic
13	#14582952	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×D₄₃	—	2	2	no	—
15	#14582806	C₄	1	1	yes	characteristic
16	#14582800	C₄	1	1	yes	characteristic
17	#14582789	—	2	2	no	—
18	#14582764	C₄	1	1	yes	characteristic
19	C₄×(C₂×D₄₃)⋊C₄	—	2	2	no	—
20	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
21	Q₈⋊(C₄₃⋊C₈)	—	2	2	no	—
22	#14582622	—	2	2	no	—
23	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₄₃×#431	C₂²	1	1	yes	characteristic
26	C₄₃×Q₈⋊C₈	—	2	2	no	—
27	C₂²×D₁₇₂	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₃	D₄	1	2	yes	—
29	C₂×C₈×D₄₃	—	2	4	no	—
30	#13063314	—	2	2	no	—
31	#13063314	C₈	1	4	yes	complemented
32	C₄×D₁₇₂	—	2	1	no	—
33	C₄×D₁₇₂	—	2	1	no	—
34	C₄×D₁₇₂	—	2	1	no	—
35	C₄×D₁₇₂	—	2	1	no	—
36	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	C₂×(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
40	C₂×C₄×Dic₄₃	—	2	1	no	—
41	C₂×C₄×Dic₄₃	—	2	1	no	—
42	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
46	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
48	(C₂×D₄₃)⋊C₈	—	2	4	no	—
49	C₄×C₄₃⋊C₈	—	2	1	no	—
50	C₄×C₄₃⋊C₈	—	2	1	no	—
51	C₄₃⋊#95	—	2	2	no	—
52	C₄₃⋊#95	—	2	2	no	—
53	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₄₃	—	2	1	no	—
55	C₄×Q₈×C₄₃	—	2	1	no	—
56	C₄×Q₈×C₄₃	—	2	1	no	—
57	C₄×Q₈×C₄₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	—	2	1	no	—
61	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₄₃	—	2	1	no	—
63	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
64	C₂²⋊C₈×C₄₃	—	2	1	no	—
65	C₂²⋊C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₄×C₈×C₄₃	—	2	1	no	—
68	C₂³×D₄₃	C₂²⋊C₄	1	2	yes	—
69	C₂×Q₈×C₄₃	—	2	1	no	—
70	C₂×Q₈×C₄₃	—	2	1	no	—
71	C₂×Q₈×C₄₃	—	2	1	no	—
72	C₂×Q₈×C₄₃	—	2	1	no	—
73	C₂×Q₈×C₄₃	—	2	1	no	—
74	C₂×Q₈×C₄₃	C₂×C₈	1	2	yes	complemented
75	C₂²×C₄×C₄₃	—	2	1	no	—
76	C₂²×C₄×C₄₃	—	2	1	no	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₄₃	—	2	1	no	—
81	C₂²×Dic₄₃	—	2	1	no	—
82	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
83	C₂×D₁₇₂	—	2	1	no	—
84	C₂×D₁₇₂	—	2	1	no	—
85	C₂×D₁₇₂	—	2	1	no	—
86	C₂×D₁₇₂	—	2	1	no	—
87	C₂×D₁₇₂	C₂×C₈	1	2	yes	—
88	C₂×D₁₇₂	M₁₆	1	2	yes	—
89	C₂×C₄×D₄₃	—	2	2	no	—
90	C₂×C₄×D₄₃	—	2	2	no	—
91	C₂×C₄×D₄₃	—	4	4	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₄⋊C₄×C₄₃	—	2	1	no	—
95	C₄⋊C₄×C₄₃	—	2	1	no	—
96	C₄⋊C₄×C₄₃	—	4	1	no	—
97	C₄⋊C₄×C₄₃	—	4	1	no	—
98	C₄²×C₄₃	—	4	1	no	—
99	C₄²×C₄₃	—	4	1	no	—
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	#41	1	1	yes	characteristic
103	C₄²×C₄₃	#41	1	1	yes	characteristic
104	(C₂×D₄₃)⋊C₄	—	2	2	no	—
105	(C₂×D₄₃)⋊C₄	—	2	2	no	—
106	(C₂×D₄₃)⋊C₄	—	2	8	no	—
107	(C₂×D₄₃)⋊C₄	—	4	4	no	—
108	C₄⋊Dic₄₃	—	2	1	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	C₄×Dic₄₃	—	2	1	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	C₄×Dic₄₃	—	4	2	no	—
114	C₄×Dic₄₃	—	4	2	no	—
115	C₄×Dic₄₃	C₂×C₈	1	2	yes	complemented
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	4	1	no	—
118	C₈×D₄₃	—	8	4	no	—
119	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×D₄₃	—	2	2	no	—
121	C₂²×D₄₃	—	2	2	no	—
122	C₂²×D₄₃	—	4	2	no	—
123	C₂²×D₄₃	—	4	2	no	—
124	C₂²×D₄₃	C₂²⋊C₈	1	4	yes	—
125	Q₈×C₄₃	—	2	2	no	—
126	Q₈×C₄₃	—	2	2	no	—
127	Q₈×C₄₃	—	4	1	no	—
128	Q₈×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	2	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	4	1	no	—
150	C₂×Dic₄₃	—	4	1	no	—
151	C₂×Dic₄₃	—	4	1	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	2	2	no	—
156	D₁₇₂	—	2	2	no	—
157	D₁₇₂	—	2	2	no	—
158	C₄×D₄₃	—	4	4	no	—
159	C₄×D₄₃	—	8	2	no	—
160	C₈×C₄₃	—	4	2	no	—
161	C₈×C₄₃	—	8	1	no	—
162	C₄₃⋊C₈	—	4	2	no	—
163	C₄₃⋊C₈	—	8	1	no	—
164	#8957	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	#386	1	1	yes	characteristic
168	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₄₃	#408	1	1	yes	characteristic
170	C₂×D₄₃	—	2	4	no	—
171	C₂×D₄₃	—	4	2	no	—
172	C₂×D₄₃	—	4	2	no	—
173	C₂×D₄₃	—	4	4	no	—
174	C₂×D₄₃	—	8	2	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	4	2	no	—
179	C₄×C₄₃	—	8	1	no	—
180	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
181	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
182	C₄×C₄₃	#435	1	1	yes	characteristic
183	C₄×C₄₃	#435	1	1	yes	characteristic
184	C₄×C₄₃	#455	1	2	yes	—
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	4	2	no	—
187	Dic₄₃	—	4	2	no	—
188	#2972	—	43	1	no	—
189	C₄×#124	—	43	1	no	—
190	#1479	—	43	1	no	—
191	#1479	D₄₃	1	1	yes	characteristic, complemented
192	#1460	—	43	1	no	—
193	#1276	—	43	1	no	—
194	#1276	—	43	1	no	—
195	C₂×C₄₃	—	2	1	no	—
196	C₂×C₄₃	—	2	1	no	—
197	C₂×C₄₃	#1869	1	1	yes	characteristic
198	C₂×C₄₃	#1884	1	1	yes	characteristic
199	C₂×C₄₃	#1885	1	1	yes	characteristic
200	D₄₃	—	4	4	no	—
201	C₂×#124	—	43	1	no	—
202	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
203	C₂×C₄×D₄	—	43	1	no	—
204	C₈×D₄	—	86	2	no	—
205	C₈×D₄	—	86	2	no	—
206	#431	—	43	1	no	—
207	#431	—	43	1	no	—
208	#431	C₂×D₄₃	1	1	yes	characteristic
209	#431	C₂×D₄₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₈	—	86	2	no	—
211	#391	—	43	1	no	—
212	#390	—	43	1	no	—
213	#388	—	86	2	no	—
214	#382	—	43	1	no	—
215	C₄×C₂²⋊C₄	—	86	2	no	—
216	C₄³	—	43	1	no	—
217	Q₈⋊C₈	—	2	2	no	—
218	Q₈⋊C₈	—	86	2	no	—
219	#323	—	86	2	no	—
220	C₄₃	#8957	1	1	yes	characteristic, complemented, Sylow
221	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
222	C₂²×D₄	—	43	1	no	—
223	C₂³×C₄	—	43	2	no	—
224	C₂²×C₈	—	86	4	no	—
225	#124	—	43	4	no	—
226	#124	—	86	2	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×D₄	—	86	1	no	—
232	C₄×D₄	—	86	1	no	—
233	C₄×D₄	—	86	1	no	—
234	C₄×D₄	—	86	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	43	1	no	—
237	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	—	43	2	no	—
239	C₂×C₂²⋊C₄	—	86	2	no	—
240	C₂×C₂²⋊C₄	—	86	2	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	—	43	1	no	—
243	C₂×C₄²	—	43	1	no	—
244	C₂×C₄²	—	86	1	no	—
245	C₂×C₄²	—	86	1	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	43	2	no	—
249	C₄⋊C₈	—	86	1	no	—
250	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
251	C₂²⋊C₈	—	2	1	no	—
252	C₂²⋊C₈	—	2	1	no	—
253	C₂²⋊C₈	—	86	4	no	—
254	C₂²⋊C₈	—	86	1	no	—
255	C₂²⋊C₈	—	86	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	86	1	no	—
259	C₄×C₈	—	86	1	no	—
260	#95	—	86	2	no	—
261	#95	—	86	2	no	—
262	C₂⁴	—	43	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	C₈×D₄₃	1	2	yes	complemented
269	C₂×D₄	—	43	2	no	—
270	C₂×D₄	—	43	2	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	86	1	no	—
273	C₂×D₄	—	86	1	no	—
274	C₂×D₄	—	86	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	1	no	—
277	C₂²×C₄	—	43	2	no	—
278	C₂²×C₄	—	86	1	no	—
279	C₂²×C₄	—	86	1	no	—
280	C₂²×C₄	—	86	2	no	—
281	C₂²×C₄	—	86	2	no	—
282	C₂²×C₄	—	172	4	no	—
283	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	86	2	no	—
289	C₂×C₈	—	344	4	no	—
290	C₂×C₈	—	172	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	86	1	no	—
296	C₄⋊C₄	—	86	1	no	—
297	C₂²⋊C₄	—	86	8	no	—
298	C₂²⋊C₄	—	86	2	no	—
299	C₂²⋊C₄	—	86	2	no	—
300	C₂²⋊C₄	—	172	4	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	43	2	no	—
304	C₄²	—	86	1	no	—
305	C₄²	—	86	1	no	—
306	C₄²	—	86	1	no	—
307	C₄²	—	172	2	no	—
308	C₄²	—	172	2	no	—
309	C₄²	#10621898	1	1	yes	characteristic
310	C₄²	D₄×D₄₃	1	1	yes	characteristic
311	C₄²	#10621900	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₂³	—	43	4	no	—
314	C₂³	—	86	2	no	—
315	C₂³	—	86	2	no	—
316	C₂³	—	172	2	no	—
317	C₂³	—	172	2	no	—
318	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
319	Q₈	—	2	2	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	4	1	no	—
322	Q₈	—	4	1	no	—
323	D₄	—	86	2	no	—
324	D₄	—	86	2	no	—
325	D₄	—	86	2	no	—
326	D₄	—	86	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	86	2	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	86	2	no	—
337	C₂×C₄	—	344	2	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	1	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	172	1	no	—
342	C₂×C₄	—	172	4	no	—
343	C₂×C₄	—	172	2	no	—
344	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
346	C₂×C₄	#13063293	1	1	yes	characteristic
347	C₂×C₄	#13063296	1	1	yes	characteristic
348	C₂×C₄	#13063298	1	1	yes	characteristic
349	C₂×C₄	#13063307	1	1	yes	characteristic
350	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
351	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
352	C₂×C₄	#13063331	1	1	yes	characteristic
353	C₂×C₄	#13063333	1	1	yes	characteristic
354	C₈	—	4	2	no	—
355	C₈	—	8	1	no	—
356	C₈	—	344	1	no	—
357	C₈	—	172	2	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	86	4	no	—
361	C₂²	—	344	2	no	—
362	C₂²	—	172	4	no	—
363	C₂²	—	172	2	no	—
364	C₂²	—	172	2	no	—
365	C₂²	#14582775	1	1	yes	characteristic
366	C₂²	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
367	C₂²	#14582864	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	1	no	—
373	C₄	—	172	2	no	—
374	C₄	—	172	2	no	—
375	C₄	—	172	2	no	—
376	C₄	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
377	C₄	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
378	C₄	#14582968	1	1	yes	characteristic
379	C₄	#14582973	1	1	yes	characteristic
380	C₄	#14583075	1	2	yes	—
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	172	4	no	—
384	C₂	#19134884	1	1	yes	characteristic
385	C₂	#19134975	1	1	yes	characteristic
386	C₂	#19134980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1479	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂²×D₁₇₂	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1479	D₄₃
hypocenter	C₄₃	#8957
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
3. #19132990	C₂	semidirect
5. #19132427	C₂	semidirect
6. #19132427	C₂	semidirect
7. C₄₃×#1479	C₂	semidirect
31. #13063314	C₈	semidirect
74. C₂×Q₈×C₄₃	C₂×C₈	semidirect
115. C₄×Dic₄₃	C₂×C₈	semidirect
191. #1479	D₄₃	semidirect
220. C₄₃	#8957	semidirect
268. C₂×Q₈	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143618	2
C₄×#13063314	2
#19132990	2
#19132915	2
#19132427	2
C₄₃×#1479	2
#8957	43