Subgroups — #24987236

Order 11008.

618 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137060	C₂	1	1	yes	maximal, characteristic
2	#19134450	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#19132435	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132434	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063284	—	2	1	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#13063235	—	2	1	no	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₃	—	2	2	no	—
13	#14582802	C₄	1	2	yes	—
14	#14582798	C₄	1	2	yes	—
15	#14582792	—	2	2	no	—
16	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊Dic₄₃	—	2	1	no	—
18	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582653	—	2	1	no	—
20	D₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
21	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×D₄⋊C₈	C₄	1	4	yes	complemented
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	#13063284	—	4	2	no	—
28	#13063284	—	4	2	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄×Dic₄₃	—	2	1	no	—
34	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
36	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
38	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
40	#13063235	—	4	2	no	—
41	C₄×C₄₃⋊C₈	—	2	1	no	—
42	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₈×Dic₄₃	—	2	8	no	—
44	C₄₃⋊#95	—	2	1	no	—
45	C₄₃⋊#95	—	2	2	no	—
46	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	—	2	1	no	—
48	C₂²×C₈×C₄₃	—	2	1	no	—
49	C₂²×C₈×C₄₃	—	2	1	no	—
50	C₄×D₄×C₄₃	—	2	1	no	—
51	C₄×D₄×C₄₃	—	2	1	no	—
52	C₄×D₄×C₄₃	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₄₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	1	no	—
58	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₃	—	2	1	no	—
60	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
61	C₂⁴×C₄₃	—	2	1	no	—
62	C₂×D₄×C₄₃	—	2	1	no	—
63	C₂×D₄×C₄₃	—	2	2	no	—
64	C₂×D₄×C₄₃	—	2	2	no	—
65	C₂×D₄×C₄₃	C₄²	1	2	yes	—
66	C₂²×C₄×C₄₃	—	2	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	—	4	1	no	—
70	C₂²×C₄×C₄₃	—	4	1	no	—
71	C₂²×C₄×C₄₃	—	4	1	no	—
72	C₂²×C₄×C₄₃	—	4	1	no	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₃	—	2	2	no	—
77	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
86	C₂²⋊C₄×C₄₃	—	2	2	no	—
87	C₂²⋊C₄×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₃	#41	1	1	yes	characteristic
91	C₄²×C₄₃	#41	1	1	yes	characteristic
92	C₂²⋊Dic₄₃	—	4	4	no	—
93	C₂²⋊Dic₄₃	—	4	4	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	4	4	no	—
97	C₄×Dic₄₃	—	2	2	no	—
98	C₄×Dic₄₃	—	2	2	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	4	no	—
102	C₄×Dic₄₃	—	4	1	no	—
103	C₄×Dic₄₃	—	4	1	no	—
104	C₄×Dic₄₃	D₈	1	2	yes	—
105	C₄×Dic₄₃	SD₁₆	1	2	yes	—
106	C₂×C₄₃⋊C₈	—	2	4	no	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	1	no	—
111	C₂³×C₄₃	—	2	2	no	—
112	C₂³×C₄₃	—	4	2	no	—
113	C₂³×C₄₃	—	4	1	no	—
114	C₂³×C₄₃	—	4	1	no	—
115	C₂³×C₄₃	—	4	2	no	—
116	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	D₄×C₄₃	—	2	2	no	—
118	D₄×C₄₃	—	2	2	no	—
119	D₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	4	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	4	2	no	—
126	C₂×C₄×C₄₃	—	4	2	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#131	1	2	yes	—
135	C₂×C₄×C₄₃	#131	1	2	yes	—
136	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	4	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	#102	1	4	yes	—
148	C₂×Dic₄₃	#104	1	4	yes	—
149	C₈×C₄₃	—	2	4	no	—
150	C₈×C₄₃	—	4	2	no	—
151	C₄₃⋊C₈	—	2	4	no	—
152	C₄₃⋊C₈	—	4	2	no	—
153	#8177	—	43	1	no	maximal, Sylow
154	C₂²×C₄₃	—	4	2	no	—
155	C₂²×C₄₃	—	4	2	no	—
156	C₂²×C₄₃	—	4	2	no	—
157	C₂²×C₄₃	—	8	2	no	—
158	C₂²×C₄₃	—	8	2	no	—
159	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄₃	#408	1	2	yes	—
161	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
162	C₂²×C₄₃	#416	1	1	yes	characteristic
163	C₂²×C₄₃	C₂×#104	1	2	yes	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	4	2	no	—
167	C₄×C₄₃	#405	1	1	yes	characteristic
168	C₄×C₄₃	#405	1	1	yes	characteristic
169	C₄×C₄₃	#433	1	2	yes	—
170	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
171	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
172	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
173	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
174	Dic₄₃	—	2	4	no	—
175	Dic₄₃	—	2	4	no	—
176	Dic₄₃	—	8	2	no	—
177	#2293	—	43	1	no	—
178	#1767	—	43	1	no	—
179	C₄²×C₈	—	43	1	no	—
180	C₂×D₄⋊C₈	—	43	1	no	—
181	C₂×D₄⋊C₈	D₄₃	1	1	yes	characteristic
182	#1279	—	43	1	no	—
183	#1279	—	43	1	no	—
184	C₂×C₄₃	—	8	2	no	—
185	C₂×C₄₃	#1492	1	2	yes	—
186	C₂×C₄₃	#1494	1	2	yes	—
187	C₂×C₄₃	#1750	1	1	yes	characteristic
188	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
189	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
190	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
191	C₂×#117	—	86	1	no	—
192	C₂×C₄⋊C₈	—	43	1	no	—
193	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
194	C₂×#97	—	86	1	no	—
195	C₂×C₄×C₈	—	43	1	no	—
196	C₂×C₄×C₈	—	86	2	no	—
197	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
198	#390	—	43	2	no	—
199	#389	—	43	2	no	—
200	#388	—	86	2	no	—
201	C₄×C₄⋊C₄	—	43	1	no	—
202	C₄×C₂²⋊C₄	—	86	1	no	—
203	C₄³	—	43	1	no	—
204	#336	—	86	1	no	—
205	D₄⋊C₈	—	43	4	no	—
206	D₄⋊C₈	Dic₄₃	1	4	yes	complemented
207	C₄₃	#8177	1	1	yes	characteristic, complemented, Sylow
208	C₂²×D₄	C₄×D₄₃	1	1	yes	characteristic
209	C₂³×C₄	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	86	1	no	—
213	C₂²×C₈	—	86	1	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	C₄×D₄₃	1	2	yes	—
217	C₄×D₄	C₂×Dic₄₃	1	2	yes	—
218	#117	—	172	2	no	—
219	#117	—	172	2	no	—
220	C₂×C₄⋊C₄	—	43	2	no	—
221	C₂×C₄⋊C₄	—	86	2	no	—
222	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	86	2	no	—
225	C₂×C₂²⋊C₄	—	86	2	no	—
226	C₂×C₄²	—	43	1	no	—
227	C₂×C₄²	—	43	2	no	—
228	C₂×C₄²	—	43	2	no	—
229	C₂×C₄²	—	43	1	no	—
230	C₂×C₄²	—	86	1	no	—
231	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	43	2	no	—
234	C₄⋊C₈	—	86	1	no	—
235	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
236	#97	—	172	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	43	2	no	—
239	C₄×C₈	—	86	8	no	—
240	C₄×C₈	—	86	1	no	—
241	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
242	#95	—	86	2	no	—
243	#95	—	86	1	no	—
244	C₂⁴	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₄×Dic₄₃	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	43	2	no	—
257	C₂²×C₄	—	43	2	no	—
258	C₂²×C₄	—	86	2	no	—
259	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
261	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	86	4	no	—
268	C₂×C₈	—	86	2	no	—
269	C₂×C₈	—	86	2	no	—
270	C₂×C₈	—	172	1	no	—
271	C₂×C₈	—	172	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	86	2	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₄⋊C₄	—	172	4	no	—
276	C₄⋊C₄	C₄×Dic₄₃	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	172	4	no	—
280	C₂²⋊C₄	—	172	4	no	—
281	C₄²	—	43	2	no	—
282	C₄²	—	43	2	no	—
283	C₄²	—	86	4	no	—
284	C₄²	—	86	2	no	—
285	C₄²	—	86	2	no	—
286	C₄²	—	86	1	no	—
287	C₄²	—	86	1	no	—
288	C₄²	—	172	1	no	—
289	C₄²	—	172	1	no	—
290	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
291	C₄²	#10621900	1	1	yes	characteristic
292	C₄²	#10621900	1	1	yes	characteristic
293	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	1	no	—
297	C₂³	—	4	1	no	—
298	C₂³	—	4	2	no	—
299	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
300	D₄	—	2	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	43	4	no	—
312	C₂×C₄	—	43	4	no	—
313	C₂×C₄	—	86	2	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
321	C₂×C₄	#13063281	1	1	yes	characteristic
322	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
323	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
324	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
325	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
326	C₂×C₄	#13063331	1	2	yes	—
327	C₂×C₄	#13063333	1	2	yes	—
328	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
330	C₂×C₄	#13063344	1	1	yes	characteristic
331	C₂×C₄	#13063346	1	1	yes	characteristic
332	C₈	—	2	4	no	—
333	C₈	—	4	2	no	—
334	C₈	—	86	4	no	—
335	C₈	—	172	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
342	C₂²	#14582864	1	2	yes	—
343	C₂²	#102×D₄₃	1	1	yes	characteristic
344	C₂²	#14582892	1	1	yes	characteristic
345	C₂²	#104×D₄₃	1	2	yes	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	86	4	no	—
350	C₄	—	86	4	no	—
351	C₄	—	344	2	no	—
352	C₄	#14582854	1	1	yes	characteristic
353	C₄	#14582857	1	1	yes	characteristic
354	C₄	#14582957	1	2	yes	—
355	C₄	#14582979	1	1	yes	characteristic
356	C₄	D₈×Dic₄₃	1	1	yes	characteristic
357	C₄	SD₁₆×Dic₄₃	1	1	yes	characteristic
358	C₄	#14582990	1	1	yes	characteristic
359	C₂	—	8	2	no	—
360	C₂	#19133044	1	2	yes	—
361	C₂	#19133052	1	2	yes	—
362	C₂	#19134363	1	1	yes	characteristic
363	C₂	#104×Dic₄₃	1	1	yes	characteristic
364	C₂	#102×Dic₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×D₄⋊C₈	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×D₄⋊C₈	D₄₃
hypocenter	C₄₃	#8177
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×D₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #19134450	C₂	semidirect
3. C₄×C₈×Dic₄₃	C₂	semidirect
5. #19132435	C₂	semidirect
6. #19132434	C₂	semidirect
20. D₄⋊(C₄₃⋊C₈)	C₄	semidirect
24. C₄₃×D₄⋊C₈	C₄	semidirect
206. D₄⋊C₈	Dic₄₃	semidirect
207. C₄₃	#8177	semidirect

Maximal subgroups

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

subgroup	index
#19137060	2
#19134450	2
C₄×C₈×Dic₄₃	2
C₂×D₄⋊(C₄₃⋊C₈)	2
#19132435	2
#19132434	2
C₂×C₄₃×D₄⋊C₈	2
#8177	43