Subgroups — #25068341

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143629	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19133510	C₂	1	2	yes	maximal, complemented
4	#19133426	C₂	1	4	yes	maximal, complemented
5	#19133420	C₂	1	2	yes	maximal, complemented
6	#19133358	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133039	C₂	1	2	yes	maximal, complemented
8	C₄₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₄×D₃₄₄	C₄	1	2	yes	—
12	C₄×D₃₄₄	C₄	1	1	yes	characteristic
13	C₄×D₃₄₄	C₄	1	1	yes	characteristic
14	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
15	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
16	C₄×D₃₄₄	C₂²	1	2	yes	—
17	#14582978	C₂²	1	4	yes	complemented
18	#14582969	—	2	4	no	—
19	C₈×D₁₇₂	C₂²	1	4	yes	complemented
20	#14582956	C₂²	1	2	yes	—
21	C₂×#13063259	C₄	1	1	yes	characteristic
22	C₂×#13063247	C₄	1	2	yes	—
23	#14582861	—	2	2	no	—
24	C₂²⋊C₈×D₄₃	—	2	2	no	—
25	#14582652	C₂²	1	2	yes	complemented
26	#14582650	C₂²	1	2	yes	complemented
27	#14582628	C₂²	1	4	yes	complemented
28	#14582628	C₂²	1	4	yes	complemented
29	C₄₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#322	C₂²	1	2	yes	complemented
33	C₄₃×#322	C₂²	1	2	yes	complemented
34	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₄₃	—	2	2	no	—
36	C₂×D₃₄₄	—	2	4	no	—
37	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
38	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
39	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
40	C₂×#10621872	—	2	4	no	—
41	C₂×C₈×D₄₃	—	2	4	no	—
42	C₄×D₁₇₂	—	2	2	no	—
43	C₄×D₁₇₂	—	2	2	no	—
44	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
45	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
46	C₄×D₁₇₂	C₂³	1	2	yes	—
47	C₄×D₁₇₂	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
49	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
50	#13063259	C₂×C₄	1	2	yes	—
51	#13063259	C₂×C₄	1	1	yes	characteristic
52	#13063259	C₂×C₄	1	1	yes	characteristic
53	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
54	#13063247	C₂×C₄	1	2	yes	—
55	#13063247	C₂×C₄	1	2	yes	—
56	#13063247	C₂×C₄	1	2	yes	—
57	#13063247	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
59	(C₂×D₄₃)⋊C₈	—	2	4	no	—
60	(C₂×D₄₃)⋊C₈	—	2	4	no	—
61	#13063236	—	2	2	no	—
62	C₂×M₁₆×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
67	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
68	#97×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂³	1	2	yes	—
72	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₃	C₂³	1	2	yes	—
75	C₂³×D₄₃	—	2	2	no	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	2	2	no	—
80	C₂×D₁₇₂	—	2	2	no	—
81	C₂×D₁₇₂	—	2	4	no	—
82	C₂×D₁₇₂	—	4	2	no	—
83	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
84	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₄₃	—	2	2	no	—
86	C₂×C₄×D₄₃	—	2	2	no	—
87	C₂×C₄×D₄₃	—	2	2	no	—
88	C₂×C₄×D₄₃	—	2	2	no	—
89	C₂×C₄×D₄₃	—	8	4	no	—
90	M₁₆×C₄₃	—	4	2	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
99	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	#41	1	1	yes	characteristic
102	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
103	(C₂×D₄₃)⋊C₄	—	2	2	no	—
104	(C₂×D₄₃)⋊C₄	—	2	2	no	—
105	(C₂×D₄₃)⋊C₄	—	2	2	no	—
106	(C₂×D₄₃)⋊C₄	—	2	2	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
109	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
110	C₂×C₄₃⋊C₈	—	2	4	no	—
111	D₃₄₄	—	2	4	no	—
112	D₃₄₄	—	2	2	no	—
113	D₃₄₄	M₁₆	1	4	yes	complemented
114	#10621872	—	8	4	no	—
115	C₈×D₄₃	—	8	4	no	—
116	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₃	—	2	2	no	—
118	C₂²×D₄₃	—	2	2	no	—
119	C₂²×D₄₃	—	4	4	no	—
120	C₂²×D₄₃	—	4	4	no	—
121	C₂²×D₄₃	—	8	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₄₃	—	2	2	no	—
133	C₂×Dic₄₃	—	2	2	no	—
134	C₂×Dic₄₃	—	8	2	no	—
135	D₁₇₂	—	2	2	no	—
136	D₁₇₂	—	2	2	no	—
137	D₁₇₂	—	2	2	no	—
138	D₁₇₂	—	4	4	no	—
139	D₁₇₂	C₂×M₁₆	1	4	yes	—
140	C₄×D₄₃	—	4	4	no	—
141	C₄×D₄₃	—	4	4	no	—
142	C₄×D₄₃	—	8	2	no	—
143	C₄×D₄₃	—	8	2	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₈×C₄₃	—	2	1	no	—
146	C₈×C₄₃	—	2	1	no	—
147	C₈×C₄₃	—	2	4	no	—
148	C₈×C₄₃	—	4	2	no	—
149	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
150	C₄₃⋊C₈	—	4	4	no	—
151	#18064	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	#444	1	2	yes	—
155	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₂×D₄₃	—	2	4	no	—
157	C₂×D₄₃	—	4	2	no	—
158	C₂×D₄₃	—	4	2	no	—
159	C₂×D₄₃	—	4	2	no	—
160	C₂×D₄₃	—	4	2	no	—
161	C₂×D₄₃	—	4	2	no	—
162	C₂×D₄₃	—	8	8	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
168	C₄×C₄₃	#568	1	1	yes	characteristic
169	C₄×C₄₃	#576	1	2	yes	—
170	Dic₄₃	—	4	2	no	—
171	Dic₄₃	—	4	2	no	—
172	#2974	—	43	2	no	—
173	C₂×C₄×D₈	—	43	1	no	—
174	#1594	—	43	2	no	—
175	#1581	—	43	4	no	—
176	#1580	—	43	2	no	—
177	#1568	—	43	1	no	—
178	#1490	—	43	2	no	—
179	#1451	D₄₃	1	1	yes	characteristic, complemented
180	C₂×C₄₃	—	2	2	no	—
181	C₂×C₄₃	D₄×M₁₆	1	1	yes	characteristic
182	C₂×C₄₃	#2948	1	1	yes	characteristic
183	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
184	D₄₃	—	4	4	no	—
185	D₄₃	—	8	2	no	—
186	C₂²×D₈	—	43	1	no	—
187	C₂×C₄×D₄	—	43	2	no	—
188	C₄×D₈	—	43	1	no	—
189	C₄×D₈	—	43	2	no	—
190	C₄×D₈	—	43	1	no	—
191	C₄×D₈	—	43	1	no	—
192	C₄×D₈	—	43	2	no	—
193	C₄×D₈	—	43	1	no	—
194	#436	—	43	4	no	—
195	#435	—	86	4	no	—
196	C₈×D₄	—	43	4	no	—
197	#432	—	43	2	no	—
198	#431	C₂×D₄₃	1	1	yes	characteristic
199	C₂×#107	—	43	1	no	—
200	C₂×#102	—	43	2	no	—
201	#407	—	86	2	no	—
202	C₂×C₂²⋊C₈	—	86	2	no	—
203	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
205	#335	—	43	2	no	—
206	#334	—	43	2	no	—
207	D₄⋊C₈	—	43	4	no	—
208	D₄⋊C₈	—	43	4	no	—
209	#322	C₂×D₄₃	1	2	yes	complemented
210	#322	C₂×D₄₃	1	2	yes	complemented
211	C₄₃	#18064	1	1	yes	characteristic, complemented, Sylow
212	C₂²×D₄	—	43	2	no	—
213	C₂³×C₄	—	86	2	no	—
214	C₂×D₈	—	43	1	no	—
215	C₂×D₈	—	43	2	no	—
216	C₂×D₈	—	43	1	no	—
217	C₂×D₈	—	86	4	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	86	4	no	—
220	C₂²×C₈	—	86	4	no	—
221	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
222	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
223	C₄×D₄	—	43	2	no	—
224	C₄×D₄	—	43	2	no	—
225	C₄×D₄	—	43	2	no	—
226	C₄×D₄	—	43	2	no	—
227	C₄×D₄	—	86	2	no	—
228	C₄×D₄	—	86	2	no	—
229	C₂×C₄⋊C₄	—	43	2	no	—
230	C₂×C₂²⋊C₄	—	86	2	no	—
231	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
232	#107	—	43	2	no	—
233	#107	—	43	1	no	—
234	#107	—	43	1	no	—
235	C₄⋊C₈	—	43	4	no	—
236	C₄⋊C₈	D₁₇₂	1	2	yes	—
237	#102	—	43	2	no	—
238	#102	—	43	2	no	—
239	#102	—	43	2	no	—
240	#102	—	43	2	no	—
241	C₂²⋊C₈	—	86	2	no	—
242	C₂²⋊C₈	—	86	4	no	—
243	C₂²⋊C₈	—	86	4	no	—
244	C₂²⋊C₈	D₁₇₂	1	2	yes	—
245	#97	—	2	1	no	—
246	#97	—	86	2	no	—
247	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
248	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₄₃	1	2	yes	—
250	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₄₃	1	2	yes	—
253	C₂⁴	—	86	2	no	—
254	C₂×D₄	—	43	2	no	—
255	C₂×D₄	—	43	2	no	—
256	C₂×D₄	—	86	4	no	—
257	C₂×D₄	—	86	2	no	—
258	C₂×D₄	—	172	2	no	—
259	C₂²×C₄	—	86	2	no	—
260	C₂²×C₄	—	86	2	no	—
261	C₂²×C₄	—	86	2	no	—
262	C₂²×C₄	—	86	2	no	—
263	C₂²×C₄	—	86	2	no	—
264	C₂²×C₄	—	344	4	no	—
265	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
267	C₂²×C₄	#10621898	1	1	yes	characteristic
268	D₈	—	43	4	no	—
269	D₈	—	86	4	no	—
270	D₈	—	86	2	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	—	344	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	86	4	no	—
277	C₂×C₈	—	344	4	no	—
278	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
282	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
283	C₄⋊C₄	—	43	2	no	—
284	C₄⋊C₄	—	43	2	no	—
285	C₄⋊C₄	—	86	2	no	—
286	C₂²⋊C₄	—	86	2	no	—
287	C₂²⋊C₄	—	86	2	no	—
288	C₂²⋊C₄	—	86	2	no	—
289	C₂²⋊C₄	—	86	2	no	—
290	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
291	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
292	C₄²	#10621898	1	1	yes	characteristic
293	C₄²	C₂³×D₄₃	1	1	yes	characteristic
294	C₂³	—	86	2	no	—
295	C₂³	—	86	2	no	—
296	C₂³	—	344	2	no	—
297	C₂³	—	172	4	no	—
298	C₂³	—	172	4	no	—
299	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
300	D₄	—	43	4	no	—
301	D₄	—	86	2	no	—
302	D₄	—	86	2	no	—
303	D₄	—	86	2	no	—
304	D₄	—	172	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	86	2	no	—
310	C₂×C₄	—	86	2	no	—
311	C₂×C₄	—	344	2	no	—
312	C₂×C₄	—	344	2	no	—
313	C₂×C₄	—	344	2	no	—
314	C₂×C₄	—	172	4	no	—
315	C₂×C₄	—	172	4	no	—
316	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
317	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
318	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
321	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	172	4	no	—
328	C₈	M₁₆×D₄₃	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	86	4	no	—
332	C₂²	—	344	8	no	—
333	C₂²	—	172	2	no	—
334	C₂²	—	172	2	no	—
335	C₂²	—	172	2	no	—
336	C₂²	—	172	2	no	—
337	C₂²	—	172	2	no	—
338	C₂²	#14583022	1	2	yes	—
339	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	172	2	no	—
345	C₄	—	172	2	no	—
346	C₄	C₂×M₁₆×D₄₃	1	1	yes	characteristic
347	C₄	#14583665	1	1	yes	characteristic
348	C₄	#14583704	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	344	2	no	—
351	C₂	—	172	4	no	—
352	C₂	M₁₆×D₁₇₂	1	1	yes	characteristic
353	C₂	#19143448	1	1	yes	characteristic
354	C₂	C₂×#14583022	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1451	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂²×D₃₄₄	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1451	D₄₃
hypocenter	C₄₃	#18064
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
1. #19143629	C₂	semidirect
3. #19133510	C₂	semidirect
4. #19133426	C₂	semidirect
5. #19133420	C₂	semidirect
6. #19133358	C₂	semidirect
7. #19133039	C₂	semidirect
8. C₄₃×#1451	C₂	semidirect
17. #14582978	C₂²	semidirect
19. C₈×D₁₇₂	C₂²	semidirect
25. #14582652	C₂²	semidirect
26. #14582650	C₂²	semidirect
27. #14582628	C₂²	semidirect
28. #14582628	C₂²	semidirect
32. C₄₃×#322	C₂²	semidirect
33. C₄₃×#322	C₂²	semidirect
113. D₃₄₄	M₁₆	semidirect
179. #1451	D₄₃	semidirect
209. #322	C₂×D₄₃	semidirect
210. #322	C₂×D₄₃	semidirect
211. C₄₃	#18064	semidirect

Maximal subgroups

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

subgroup	index
#19143629	2
C₂×C₄×D₃₄₄	2
#19133510	2
#19133426	2
#19133420	2
#19133358	2
#19133039	2
C₄₃×#1451	2
#18064	43