Subgroups — #24995661

Order 11008.

602 conjugacy classes of subgroups.

The subgroup lattice has 364 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	C₄⋊C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582648	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×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063322	C₂²	1	1	yes	characteristic
9	C₂×#13063314	C₄	1	4	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₃	—	2	1	no	—
13	#14582800	—	2	1	no	—
14	#14582800	—	2	1	no	—
15	#14582800	—	2	2	no	—
16	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582653	—	2	1	no	—
18	#14582653	—	2	2	no	—
19	#14582648	—	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×#333	—	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	8	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
38	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
46	C₈×Dic₄₃	—	4	2	no	—
47	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₃	—	2	1	no	—
49	C₂²×C₈×C₄₃	—	2	1	no	—
50	C₄⋊Q₈×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	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₄⋊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	—	2	1	no	—
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
61	C₂³×D₄₃	—	2	2	no	—
62	C₂×Q₈×C₄₃	—	2	2	no	—
63	C₂×Q₈×C₄₃	—	2	2	no	—
64	C₂×Q₈×C₄₃	—	4	4	no	—
65	C₂²×C₄×C₄₃	—	2	2	no	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
71	C₂×D₁₇₂	—	2	2	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	—	4	1	no	—
74	C₂×D₁₇₂	—	4	1	no	—
75	C₂×D₁₇₂	—	4	1	no	—
76	C₂×D₁₇₂	—	4	1	no	—
77	C₂×D₁₇₂	—	4	1	no	—
78	C₂×D₁₇₂	—	4	2	no	—
79	C₂×D₁₇₂	—	4	1	no	—
80	C₂×D₁₇₂	—	4	2	no	—
81	C₂×D₁₇₂	—	4	1	no	—
82	C₂×D₁₇₂	—	4	1	no	—
83	C₂×D₁₇₂	—	4	4	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	4	1	no	—
86	C₂×C₈×C₄₃	—	4	1	no	—
87	C₂×C₈×C₄₃	—	4	1	no	—
88	C₂×C₈×C₄₃	—	4	2	no	—
89	C₄⋊C₄×C₄₃	—	4	1	no	—
90	C₄⋊C₄×C₄₃	—	4	1	no	—
91	C₄⋊C₄×C₄₃	—	4	1	no	—
92	C₄⋊C₄×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₃	#41	1	1	yes	characteristic
96	C₄²×C₄₃	#41	1	1	yes	characteristic
97	(C₂×D₄₃)⋊C₄	—	4	8	no	—
98	(C₂×D₄₃)⋊C₄	—	4	8	no	—
99	C₄×Dic₄₃	—	2	2	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	2	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	—	2	4	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	2	no	—
111	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×D₄₃	—	4	2	no	—
113	C₂²×D₄₃	—	4	4	no	—
114	C₂²×D₄₃	—	4	2	no	—
115	C₂²×D₄₃	—	4	4	no	—
116	C₂²×D₄₃	—	4	2	no	—
117	Q₈×C₄₃	—	8	2	no	—
118	Q₈×C₄₃	—	8	2	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	4	no	—
122	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
129	C₂×C₄×C₄₃	#135	1	2	yes	—
130	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#135	1	2	yes	—
132	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
134	C₂×Dic₄₃	—	2	2	no	—
135	C₂×Dic₄₃	—	2	2	no	—
136	C₂×Dic₄₃	—	2	2	no	—
137	C₂×Dic₄₃	—	2	2	no	—
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	#100	1	4	yes	—
140	D₁₇₂	—	8	2	no	—
141	D₁₇₂	—	8	2	no	—
142	D₁₇₂	—	8	1	no	—
143	D₁₇₂	—	8	1	no	—
144	D₁₇₂	—	8	1	no	—
145	D₁₇₂	—	8	1	no	—
146	D₁₇₂	—	8	2	no	—
147	D₁₇₂	—	8	1	no	—
148	D₁₇₂	—	8	1	no	—
149	C₈×C₄₃	—	4	2	no	—
150	C₈×C₄₃	—	8	1	no	—
151	C₄₃⋊C₈	—	4	2	no	—
152	C₄₃⋊C₈	—	8	1	no	—
153	#9699	—	43	1	no	maximal, Sylow
154	C₂²×C₄₃	#390	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#100	1	2	yes	—
156	C₂²×C₄₃	#416	1	1	yes	characteristic
157	C₂²×C₄₃	#416	1	2	yes	—
158	C₂²×C₄₃	#417	1	1	yes	characteristic
159	C₂×D₄₃	—	8	4	no	—
160	C₂×D₄₃	—	8	4	no	—
161	C₂×D₄₃	—	8	2	no	—
162	C₂×D₄₃	—	8	2	no	—
163	C₂×D₄₃	—	8	2	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	8	2	no	—
167	C₄×C₄₃	#463	1	1	yes	characteristic
168	C₄×C₄₃	#463	1	1	yes	characteristic
169	C₄×C₄₃	#464	1	1	yes	characteristic
170	C₄×C₄₃	#464	1	1	yes	characteristic
171	C₄×C₄₃	#488	1	2	yes	—
172	C₄×C₄₃	#495	1	1	yes	characteristic
173	C₄×C₄₃	#495	1	1	yes	characteristic
174	Dic₄₃	—	2	4	no	—
175	Dic₄₃	—	4	2	no	—
176	#2858	—	43	1	no	—
177	#1831	—	43	1	no	—
178	C₄×C₄⋊C₈	—	43	1	no	—
179	C₂×#333	—	43	1	no	—
180	C₂×#333	D₄₃	1	1	yes	characteristic, complemented
181	C₂×#331	—	43	1	no	—
182	C₂×#331	—	43	1	no	—
183	C₂×C₄₃	#1543	1	2	yes	—
184	C₂×C₄₃	#1551	1	2	yes	—
185	C₂×C₄₃	#1958	1	1	yes	characteristic
186	C₂×C₄₃	#1965	1	1	yes	characteristic
187	C₂×C₄₃	#1983	1	1	yes	characteristic
188	D₄₃	—	16	2	no	—
189	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#127	—	43	1	no	—
191	C₂×#124	—	43	4	no	—
192	C₂×C₄⋊C₈	—	43	1	no	—
193	C₂×C₄⋊C₈	—	43	1	no	—
194	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
195	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
196	C₂×C₄×C₈	—	86	1	no	—
197	#390	—	86	2	no	—
198	#390	—	86	1	no	—
199	#390	—	86	1	no	—
200	C₄³	—	43	1	no	—
201	#336	—	86	2	no	—
202	#336	—	86	1	no	—
203	#333	—	2	2	no	—
204	#333	—	86	2	no	—
205	#331	—	86	2	no	—
206	#331	—	86	2	no	—
207	C₄₃	#9699	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	C₄×D₄₃	1	2	yes	—
209	C₂²×D₄	—	43	2	no	—
210	C₂²×D₄	—	86	1	no	—
211	C₂²×D₄	—	86	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	86	1	no	—
215	C₂²×C₈	—	86	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	86	1	no	—
221	#127	—	86	1	no	—
222	#127	—	86	1	no	—
223	#127	—	86	1	no	—
224	#124	—	172	8	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	86	4	no	—
228	C₂×C₂²⋊C₄	—	86	4	no	—
229	C₂×C₄²	—	43	2	no	—
230	C₂×C₄²	—	43	2	no	—
231	C₂×C₄²	—	43	1	no	—
232	C₂×C₄²	—	43	1	no	—
233	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	43	2	no	—
238	C₄⋊C₈	—	86	1	no	—
239	C₄⋊C₈	—	86	1	no	—
240	C₄⋊C₈	—	86	1	no	—
241	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
242	C₄×C₈	—	172	2	no	—
243	C₂⁴	—	86	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×D₄	—	86	2	no	—
248	C₂×D₄	—	86	2	no	—
249	C₂×D₄	—	172	4	no	—
250	C₂×D₄	—	172	1	no	—
251	C₂×D₄	—	172	1	no	—
252	C₂×D₄	—	172	1	no	—
253	C₂×D₄	—	172	1	no	—
254	C₂×D₄	—	172	1	no	—
255	C₂×D₄	—	172	2	no	—
256	C₂×D₄	—	172	1	no	—
257	C₂×D₄	—	172	2	no	—
258	C₂×D₄	—	172	1	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	43	2	no	—
262	C₂²×C₄	—	43	2	no	—
263	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
264	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	2	no	—
271	C₂×C₈	—	86	2	no	—
272	C₂×C₈	—	172	1	no	—
273	C₂×C₈	—	172	2	no	—
274	C₂×C₈	—	172	1	no	—
275	C₂×C₈	—	172	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	172	8	no	—
281	C₂²⋊C₄	—	172	8	no	—
282	C₄²	—	86	4	no	—
283	C₄²	—	86	2	no	—
284	C₄²	—	86	1	no	—
285	C₄²	—	86	2	no	—
286	C₄²	—	86	1	no	—
287	C₄²	—	86	1	no	—
288	C₄²	—	86	1	no	—
289	C₄²	#10621898	1	1	yes	characteristic
290	C₄²	D₄×D₄₃	1	1	yes	characteristic
291	C₄²	#10621900	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
293	C₂³	—	172	2	no	—
294	C₂³	—	172	4	no	—
295	C₂³	—	172	2	no	—
296	C₂³	—	172	4	no	—
297	C₂³	—	172	2	no	—
298	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
299	Q₈	—	8	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	344	2	no	—
302	D₄	—	344	2	no	—
303	D₄	—	344	1	no	—
304	D₄	—	344	1	no	—
305	D₄	—	344	1	no	—
306	D₄	—	344	1	no	—
307	D₄	—	344	2	no	—
308	D₄	—	344	1	no	—
309	D₄	—	344	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	43	4	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	2	no	—
317	C₂×C₄	—	86	2	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
321	C₂×C₄	#13063297	1	1	yes	characteristic
322	C₂×C₄	#13063298	1	1	yes	characteristic
323	C₂×C₄	#13063311	1	1	yes	characteristic
324	C₂×C₄	#13063321	1	1	yes	characteristic
325	C₂×C₄	#13063346	1	1	yes	characteristic
326	C₂×C₄	#13063346	1	2	yes	—
327	C₂×C₄	#13063348	1	1	yes	characteristic
328	C₂×C₄	#13063348	1	2	yes	—
329	C₂×C₄	#13063352	1	1	yes	characteristic
330	C₂×C₄	#13063357	1	1	yes	characteristic
331	C₈	—	4	2	no	—
332	C₈	—	8	1	no	—
333	C₈	—	344	1	no	—
334	C₈	—	172	2	no	—
335	C₂²	—	344	2	no	—
336	C₂²	—	344	4	no	—
337	C₂²	—	344	4	no	—
338	C₂²	—	344	2	no	—
339	C₂²	—	344	2	no	—
340	C₂²	#14582799	1	1	yes	characteristic
341	C₂²	#100×D₄₃	1	2	yes	—
342	C₂²	#14582894	1	1	yes	characteristic
343	C₂²	#14582894	1	2	yes	—
344	C₂²	#14582899	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	86	4	no	—
349	C₄	—	172	2	no	—
350	C₄	#14583116	1	1	yes	characteristic
351	C₄	#14583117	1	1	yes	characteristic
352	C₄	#14583123	1	1	yes	characteristic
353	C₄	#14583124	1	1	yes	characteristic
354	C₄	#14583255	1	2	yes	—
355	C₄	#14583280	1	1	yes	characteristic
356	C₄	#14583284	1	1	yes	characteristic
357	C₂	—	688	2	no	—
358	C₂	#19133256	1	2	yes	—
359	C₂	#19133289	1	2	yes	—
360	C₂	#19135403	1	1	yes	characteristic
361	C₂	#19135433	1	1	yes	characteristic
362	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×#333	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#333	D₄₃
hypocenter	C₄₃	#9699
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #19134711	C₂	semidirect
3. C₄⋊C₈×Dic₄₃	C₂	semidirect
4. C₂×#14582648	C₂	semidirect
7. C₂×C₄₃×#333	C₂	semidirect
180. C₂×#333	D₄₃	semidirect
207. C₄₃	#9699	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
C₄⋊C₈×Dic₄₃	2
C₂×#14582648	2
C₂×#14582642	2
C₂×C₄₃×#333	2
#9699	43