Subgroups — #24995712

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142953	C₂	1	1	yes	maximal, characteristic
2	#19134711	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582645	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582642	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582642	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063322	C₂²	1	1	yes	characteristic
9	C₂×#13063314	—	2	2	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13063235	—	2	1	no	—
13	#14582800	—	2	1	no	—
14	#14582800	—	2	1	no	—
15	#14582800	C₄	1	4	yes	—
16	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582653	—	2	1	no	—
18	#14582653	—	2	2	no	—
19	#14582645	—	2	2	no	—
20	#14582642	—	2	2	no	—
21	#14582642	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#332	—	2	2	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂²×C₄₃⋊C₈	—	2	1	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	#13063322	—	2	1	no	—
32	#13063322	—	2	1	no	—
33	#13063322	—	2	1	no	—
34	#13063322	—	2	1	no	—
35	#13063314	—	4	4	no	—
36	#13063314	—	4	4	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
39	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
47	#13063235	—	4	2	no	—
48	C₂²×Q₈×C₄₃	—	2	1	no	—
49	C₂²×C₈×C₄₃	—	2	1	no	—
50	C₂²×C₈×C₄₃	—	2	1	no	—
51	C₄⋊Q₈×C₄₃	—	2	1	no	—
52	C₄⋊Q₈×C₄₃	—	2	1	no	—
53	C₄⋊Q₈×C₄₃	—	2	1	no	—
54	C₄⋊Q₈×C₄₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
62	C₂³×D₄₃	—	2	2	no	—
63	C₂×Q₈×C₄₃	—	4	1	no	—
64	C₂×Q₈×C₄₃	—	4	1	no	—
65	C₂×Q₈×C₄₃	—	4	2	no	—
66	C₂×Q₈×C₄₃	—	4	2	no	—
67	C₂²×C₄×C₄₃	—	2	2	no	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
73	C₂×D₁₇₂	—	2	2	no	—
74	C₂×D₁₇₂	—	2	2	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	1	no	—
77	C₂×D₁₇₂	—	4	1	no	—
78	C₂×D₁₇₂	—	4	1	no	—
79	C₂×D₁₇₂	—	4	1	no	—
80	C₂×D₁₇₂	—	4	1	no	—
81	C₂×D₁₇₂	—	4	2	no	—
82	C₂×D₁₇₂	—	4	2	no	—
83	C₂×D₁₇₂	—	4	1	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	4	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	4	1	no	—
88	C₂×C₈×C₄₃	—	4	1	no	—
89	C₂×C₈×C₄₃	—	4	1	no	—
90	C₂×C₈×C₄₃	—	4	2	no	—
91	C₄⋊C₄×C₄₃	—	2	2	no	—
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	4	1	no	—
94	C₄⋊C₄×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	#41	1	1	yes	characteristic
98	C₄²×C₄₃	#41	1	1	yes	characteristic
99	(C₂×D₄₃)⋊C₄	—	4	8	no	—
100	(C₂×D₄₃)⋊C₄	—	4	8	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	C₄×Dic₄₃	—	2	2	no	—
105	C₄×Dic₄₃	—	2	2	no	—
106	C₄×Dic₄₃	—	2	1	no	—
107	C₄×Dic₄₃	—	4	2	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	1	no	—
111	C₂×C₄₃⋊C₈	—	4	1	no	—
112	C₂×C₄₃⋊C₈	—	4	2	no	—
113	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×D₄₃	—	4	2	no	—
115	C₂²×D₄₃	—	4	4	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	C₂²×D₄₃	—	4	2	no	—
118	C₂²×D₄₃	—	4	4	no	—
119	Q₈×C₄₃	—	8	2	no	—
120	Q₈×C₄₃	—	8	1	no	—
121	Q₈×C₄₃	—	8	1	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	4	no	—
125	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
130	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
133	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
137	C₂×Dic₄₃	—	2	2	no	—
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	—	2	2	no	—
140	C₂×Dic₄₃	—	2	2	no	—
141	C₂×Dic₄₃	—	2	2	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	D₁₇₂	—	8	2	no	—
144	D₁₇₂	—	8	2	no	—
145	D₁₇₂	—	8	1	no	—
146	D₁₇₂	—	8	1	no	—
147	D₁₇₂	—	8	1	no	—
148	D₁₇₂	—	8	1	no	—
149	D₁₇₂	—	8	2	no	—
150	D₁₇₂	—	8	1	no	—
151	D₁₇₂	—	8	1	no	—
152	C₈×C₄₃	—	4	2	no	—
153	C₈×C₄₃	—	8	1	no	—
154	C₄₃⋊C₈	—	4	2	no	—
155	C₄₃⋊C₈	—	8	1	no	—
156	#9710	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	#390	1	1	yes	characteristic
158	C₂²×C₄₃	#413	1	2	yes	—
159	C₂²×C₄₃	#416	1	1	yes	characteristic
160	C₂²×C₄₃	#417	1	1	yes	characteristic
161	C₂²×C₄₃	#418	1	2	yes	—
162	C₂×D₄₃	—	8	4	no	—
163	C₂×D₄₃	—	8	4	no	—
164	C₂×D₄₃	—	8	2	no	—
165	C₂×D₄₃	—	8	2	no	—
166	C₂×D₄₃	—	8	2	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	8	2	no	—
170	C₄×C₄₃	#463	1	1	yes	characteristic
171	C₄×C₄₃	#463	1	1	yes	characteristic
172	C₄×C₄₃	#464	1	1	yes	characteristic
173	C₄×C₄₃	#464	1	1	yes	characteristic
174	C₄×C₄₃	#490	1	2	yes	—
175	C₄×C₄₃	#496	1	1	yes	characteristic
176	C₄×C₄₃	#496	1	1	yes	characteristic
177	Dic₄₃	—	4	2	no	—
178	Dic₄₃	—	4	2	no	—
179	#2858	—	43	1	no	—
180	#1831	—	43	1	no	—
181	#1767	—	43	1	no	—
182	C₂×#332	—	43	1	no	—
183	C₂×#332	D₄₃	1	1	yes	characteristic, complemented
184	C₂×#331	—	43	1	no	—
185	C₂×#331	—	43	1	no	—
186	C₂×C₄₃	#1553	1	2	yes	—
187	C₂×C₄₃	#1557	1	2	yes	—
188	C₂×C₄₃	#1961	1	1	yes	characteristic
189	C₂×C₄₃	#1967	1	1	yes	characteristic
190	C₂×C₄₃	#1983	1	1	yes	characteristic
191	D₄₃	—	16	2	no	—
192	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
193	C₂×#127	—	43	1	no	—
194	C₂×#124	—	86	2	no	—
195	C₂×C₄⋊C₈	—	43	1	no	—
196	C₂×C₄⋊C₈	—	43	1	no	—
197	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
198	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×#97	—	86	1	no	—
200	#390	—	43	4	no	—
201	#390	—	86	1	no	—
202	#390	—	86	1	no	—
203	C₄³	—	43	1	no	—
204	#336	—	86	2	no	—
205	#336	—	86	1	no	—
206	#332	—	2	2	no	—
207	#332	—	86	2	no	—
208	#331	—	86	2	no	—
209	#331	—	86	2	no	—
210	C₄₃	#9710	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	—	43	2	no	—
213	C₂²×D₄	—	86	1	no	—
214	C₂²×D₄	—	86	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	86	1	no	—
218	C₂²×C₈	—	86	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	#127	—	86	1	no	—
224	#127	—	86	1	no	—
225	#127	—	86	1	no	—
226	#127	—	86	1	no	—
227	#124	—	172	4	no	—
228	#124	—	172	4	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
231	C₂×C₂²⋊C₄	—	86	4	no	—
232	C₂×C₂²⋊C₄	—	86	4	no	—
233	C₂×C₄²	—	43	2	no	—
234	C₂×C₄²	—	43	1	no	—
235	C₂×C₄²	—	43	1	no	—
236	C₂×C₄²	—	43	2	no	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	43	2	no	—
242	C₄⋊C₈	—	86	1	no	—
243	C₄⋊C₈	—	86	1	no	—
244	C₄⋊C₈	—	86	1	no	—
245	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
246	#97	—	172	2	no	—
247	C₂⁴	—	86	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×D₄	—	86	2	no	—
253	C₂×D₄	—	86	2	no	—
254	C₂×D₄	—	172	4	no	—
255	C₂×D₄	—	172	1	no	—
256	C₂×D₄	—	172	1	no	—
257	C₂×D₄	—	172	1	no	—
258	C₂×D₄	—	172	1	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂×D₄	—	172	1	no	—
261	C₂×D₄	—	172	2	no	—
262	C₂×D₄	—	172	2	no	—
263	C₂×D₄	—	172	1	no	—
264	C₂×D₄	—	172	1	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	43	2	no	—
267	C₂²×C₄	—	43	2	no	—
268	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	86	2	no	—
277	C₂×C₈	—	172	1	no	—
278	C₂×C₈	—	172	2	no	—
279	C₂×C₈	—	172	1	no	—
280	C₂×C₈	—	172	1	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	172	8	no	—
286	C₂²⋊C₄	—	172	8	no	—
287	C₄²	—	86	1	no	—
288	C₄²	—	86	1	no	—
289	C₄²	—	86	2	no	—
290	C₄²	—	86	1	no	—
291	C₄²	—	86	2	no	—
292	C₄²	—	86	1	no	—
293	C₄²	—	172	2	no	—
294	C₄²	#10621898	1	1	yes	characteristic
295	C₄²	D₄×D₄₃	1	1	yes	characteristic
296	C₄²	#10621900	1	1	yes	characteristic
297	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
298	C₂³	—	172	2	no	—
299	C₂³	—	172	4	no	—
300	C₂³	—	172	2	no	—
301	C₂³	—	172	2	no	—
302	C₂³	—	172	4	no	—
303	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
304	Q₈	—	8	2	no	—
305	Q₈	—	8	1	no	—
306	Q₈	—	8	1	no	—
307	D₄	—	344	2	no	—
308	D₄	—	344	2	no	—
309	D₄	—	344	1	no	—
310	D₄	—	344	1	no	—
311	D₄	—	344	1	no	—
312	D₄	—	344	1	no	—
313	D₄	—	344	2	no	—
314	D₄	—	344	1	no	—
315	D₄	—	344	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	2	no	—
322	C₂×C₄	—	86	2	no	—
323	C₂×C₄	—	86	2	no	—
324	C₂×C₄	—	86	2	no	—
325	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
327	C₂×C₄	#13063297	1	1	yes	characteristic
328	C₂×C₄	#13063298	1	1	yes	characteristic
329	C₂×C₄	#13063311	1	1	yes	characteristic
330	C₂×C₄	#13063321	1	1	yes	characteristic
331	C₂×C₄	#13063346	1	1	yes	characteristic
332	C₂×C₄	#13063348	1	1	yes	characteristic
333	C₂×C₄	#13063349	1	2	yes	—
334	C₂×C₄	#13063352	1	1	yes	characteristic
335	C₂×C₄	#13063352	1	2	yes	—
336	C₂×C₄	#13063357	1	1	yes	characteristic
337	C₈	—	4	2	no	—
338	C₈	—	8	1	no	—
339	C₈	—	344	1	no	—
340	C₈	—	172	2	no	—
341	C₂²	—	344	2	no	—
342	C₂²	—	344	4	no	—
343	C₂²	—	344	4	no	—
344	C₂²	—	344	2	no	—
345	C₂²	—	344	2	no	—
346	C₂²	#14582799	1	1	yes	characteristic
347	C₂²	#14582882	1	2	yes	—
348	C₂²	#14582894	1	1	yes	characteristic
349	C₂²	#14582899	1	1	yes	characteristic
350	C₂²	#14582902	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	8	2	no	—
354	C₄	—	172	2	no	—
355	C₄	—	172	2	no	—
356	C₄	#14583116	1	1	yes	characteristic
357	C₄	#14583117	1	1	yes	characteristic
358	C₄	#14583123	1	1	yes	characteristic
359	C₄	#14583124	1	1	yes	characteristic
360	C₄	#14583262	1	2	yes	—
361	C₄	#14583285	1	1	yes	characteristic
362	C₄	#14583287	1	1	yes	characteristic
363	C₂	—	688	2	no	—
364	C₂	#19133299	1	2	yes	—
365	C₂	#19133316	1	2	yes	—
366	C₂	#19135417	1	1	yes	characteristic
367	C₂	#19135443	1	1	yes	characteristic
368	C₂	#19135524	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#332	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#332	D₄₃
hypocenter	C₄₃	#9710
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #19134711	C₂	semidirect
3. #19134450	C₂	semidirect
4. C₂×#14582645	C₂	semidirect
7. C₂×C₄₃×#332	C₂	semidirect
183. C₂×#332	D₄₃	semidirect
210. C₄₃	#9710	semidirect

Maximal subgroups

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

subgroup	index
#19142953	2
#19134711	2
#19134450	2
C₂×#14582645	2
C₂×#14582642	2
C₂×C₄₃×#332	2
#9710	43