Subgroups — #24999174

Order 11008.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583295	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583190	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583101	C₂	1	1	yes	maximal, characteristic
4	#19135388	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582647	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132429	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	—	2	1	no	—
9	C₂×#13063325	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
11	#14583295	—	2	2	no	—
12	#14583295	—	2	2	no	—
13	#14583190	—	2	2	no	—
14	#14583190	—	2	2	no	—
15	#14583101	—	2	2	no	—
16	#14583101	—	2	2	no	—
17	C₂×#13063257	—	2	1	no	—
18	C₂×#13063256	—	2	2	no	—
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13063249	—	2	1	no	—
21	C₂×#13063247	—	2	1	no	—
22	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	#14582647	—	2	2	no	—
25	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#332	—	2	2	no	—
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	—	2	1	no	—
32	C₂×#10621884	—	2	4	no	—
33	C₂×#10621884	—	4	2	no	—
34	C₂×#10621884	—	4	2	no	—
35	C₂²×C₄₃⋊C₈	—	2	1	no	—
36	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
37	#13063325	—	2	1	no	—
38	#13063325	—	2	1	no	—
39	#13063325	—	2	1	no	—
40	#13063325	D₄	1	2	yes	—
41	C₄×D₁₇₂	—	2	1	no	—
42	C₄×D₁₇₂	—	2	1	no	—
43	C₄×D₁₇₂	—	2	1	no	—
44	C₄×D₁₇₂	—	2	1	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	—	2	1	no	—
47	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
49	#13063257	—	2	4	no	—
50	#13063256	—	2	8	no	—
51	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
52	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
53	#13063249	—	2	4	no	—
54	#13063247	—	2	4	no	—
55	(C₂×D₄₃)⋊C₈	—	4	4	no	—
56	C₄×C₄₃⋊C₈	—	2	1	no	—
57	C₄×C₄₃⋊C₈	—	2	1	no	—
58	C₂²×Q₈×C₄₃	—	2	1	no	—
59	C₂²×C₈×C₄₃	—	2	1	no	—
60	C₂²×C₈×C₄₃	—	2	1	no	—
61	C₄⋊Q₈×C₄₃	—	2	1	no	—
62	C₄⋊Q₈×C₄₃	—	2	1	no	—
63	C₄⋊Q₈×C₄₃	—	2	1	no	—
64	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₄₃	—	2	1	no	—
69	C₄⋊C₈×C₄₃	—	2	1	no	—
70	C₄⋊C₈×C₄₃	—	2	1	no	—
71	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
72	C₂³×D₄₃	—	2	1	no	—
73	C₂×Q₈×C₄₃	—	2	2	no	—
74	C₂×Q₈×C₄₃	—	4	1	no	—
75	C₂×Q₈×C₄₃	—	4	1	no	—
76	C₂×Q₈×C₄₃	—	4	1	no	—
77	C₂×Q₈×C₄₃	—	4	1	no	—
78	C₂×Q₈×C₄₃	—	4	1	no	—
79	C₂²×C₄×C₄₃	—	4	1	no	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
83	C₂²×Dic₄₃	—	2	1	no	—
84	C₂²×Dic₄₃	—	4	1	no	—
85	C₂×D₁₇₂	—	2	1	no	—
86	C₂×D₁₇₂	—	2	1	no	—
87	C₂×D₁₇₂	—	2	1	no	—
88	C₂×D₁₇₂	—	2	1	no	—
89	C₂×D₁₇₂	—	2	1	no	—
90	C₂×D₁₇₂	SD₁₆	1	2	yes	—
91	C₂×C₄×D₄₃	—	4	1	no	—
92	C₂×C₄×D₄₃	—	4	1	no	—
93	C₂×C₄×D₄₃	—	4	2	no	—
94	C₂×C₄×D₄₃	—	4	2	no	—
95	C₂×Dic₈₆	—	2	2	no	—
96	C₂×Dic₈₆	—	4	1	no	—
97	C₂×Dic₈₆	—	8	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	—	4	1	no	—
100	C₂×C₈×C₄₃	—	4	1	no	—
101	C₂×C₈×C₄₃	—	4	1	no	—
102	C₂×C₈×C₄₃	—	4	2	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄⋊C₄×C₄₃	—	4	1	no	—
106	C₄⋊C₄×C₄₃	—	4	1	no	—
107	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
111	#10621884	—	4	8	no	—
112	(C₂×D₄₃)⋊C₄	—	4	1	no	—
113	(C₂×D₄₃)⋊C₄	—	4	1	no	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	—	2	1	no	—
116	C₄⋊Dic₄₃	—	2	2	no	—
117	C₄⋊Dic₄₃	—	2	1	no	—
118	C₄⋊Dic₄₃	—	4	1	no	—
119	C₄⋊Dic₄₃	—	4	1	no	—
120	C₂×C₄₃⋊C₈	—	2	2	no	—
121	C₂×C₄₃⋊C₈	—	2	2	no	—
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	4	1	no	—
124	C₂×C₄₃⋊C₈	—	4	1	no	—
125	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	complemented
126	C₂³×C₄₃	#121	1	1	yes	characteristic
127	C₂²×D₄₃	—	2	2	no	—
128	C₂²×D₄₃	—	4	1	no	—
129	C₂²×D₄₃	—	4	2	no	—
130	C₂²×D₄₃	—	4	1	no	—
131	C₂²×D₄₃	—	4	2	no	—
132	Q₈×C₄₃	—	4	2	no	—
133	Q₈×C₄₃	—	4	2	no	—
134	Q₈×C₄₃	—	8	1	no	—
135	Q₈×C₄₃	—	8	1	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	8	1	no	—
139	C₂×C₄×C₄₃	—	8	1	no	—
140	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
149	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
151	C₂×C₄×C₄₃	#137	1	2	yes	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	8	2	no	—
156	D₁₇₂	—	2	2	no	—
157	D₁₇₂	—	2	2	no	—
158	D₁₇₂	—	4	1	no	—
159	D₁₇₂	—	4	1	no	—
160	C₄×D₄₃	—	8	2	no	—
161	C₄×D₄₃	—	8	1	no	—
162	C₄×D₄₃	—	8	1	no	—
163	Dic₈₆	—	8	2	no	—
164	Dic₈₆	—	8	2	no	—
165	C₈×C₄₃	—	4	2	no	—
166	C₈×C₄₃	—	8	1	no	—
167	C₄₃⋊C₈	—	2	4	no	—
168	C₄₃⋊C₈	—	4	2	no	—
169	#10271	—	43	1	no	maximal, Sylow
170	C₂²×C₄₃	#392	1	1	yes	characteristic
171	C₂²×C₄₃	#461	1	2	yes	—
172	C₂²×C₄₃	#465	1	1	yes	characteristic
173	C₂²×C₄₃	#468	1	1	yes	characteristic
174	C₂²×C₄₃	#471	1	2	yes	—
175	C₂×D₄₃	—	4	4	no	—
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	1	no	—
179	C₂×D₄₃	—	8	1	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	8	2	no	—
183	C₄×C₄₃	#448	1	1	yes	characteristic
184	C₄×C₄₃	#448	1	2	yes	—
185	C₄×C₄₃	#449	1	1	yes	characteristic
186	C₄×C₄₃	#450	1	1	yes	characteristic
187	C₄×C₄₃	#452	1	1	yes	characteristic
188	C₄×C₄₃	#492	1	1	yes	characteristic
189	C₄×C₄₃	#497	1	1	yes	characteristic
190	Dic₄₃	—	8	1	no	—
191	Dic₄₃	—	8	2	no	—
192	C₂×#498	—	43	1	no	—
193	C₂×#476	—	43	1	no	—
194	C₂×#461	—	43	1	no	—
195	#1954	—	43	1	no	—
196	C₂×#333	—	43	1	no	—
197	C₂×#332	D₄₃	1	1	yes	characteristic, complemented
198	#1277	—	43	1	no	—
199	C₂×C₄₃	#1671	1	2	yes	—
200	C₂×C₄₃	#1691	1	2	yes	—
201	C₂×C₄₃	#2000	1	1	yes	characteristic
202	C₂×C₄₃	#2013	1	1	yes	characteristic
203	C₂×C₄₃	#2019	1	1	yes	characteristic
204	D₄₃	—	8	2	no	—
205	C₂²×SD₁₆	—	86	1	no	—
206	C₂×C₄⋊Q₈	—	43	1	no	—
207	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₄×D₄	—	43	1	no	—
209	#498	—	86	2	no	—
210	#498	—	86	2	no	—
211	#476	—	86	2	no	—
212	#476	—	86	2	no	—
213	#461	—	86	2	no	—
214	#461	—	86	2	no	—
215	C₂×#106	—	86	2	no	—
216	C₂×#106	—	86	1	no	—
217	C₂×C₄⋊C₈	—	43	1	no	—
218	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
220	C₂×#103	—	86	1	no	—
221	C₂×#102	—	86	1	no	—
222	C₂×C₂²⋊C₈	—	86	2	no	—
223	C₂×C₄×C₈	—	43	1	no	—
224	#333	—	86	2	no	—
225	#332	—	2	2	no	—
226	C₄₃	#10271	1	1	yes	characteristic, complemented, Sylow
227	C₂²×Q₈	—	2	1	no	—
228	C₂²×Q₈	—	86	1	no	—
229	C₂²×D₄	—	43	1	no	—
230	C₂³×C₄	—	86	1	no	—
231	C₂×SD₁₆	—	86	4	no	—
232	C₂×SD₁₆	—	172	2	no	—
233	C₂×SD₁₆	—	172	2	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	43	2	no	—
237	C₂²×C₈	—	86	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	43	2	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	C₄⋊Q₈	—	86	1	no	—
244	C₄⋊Q₈	—	86	1	no	—
245	C₄⋊Q₈	D₁₇₂	1	2	yes	—
246	C₄×D₄	—	86	1	no	—
247	C₄×D₄	—	86	1	no	—
248	C₄×D₄	—	86	1	no	—
249	C₄×D₄	—	86	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	43	1	no	—
253	C₂×C₄⋊C₄	—	86	1	no	—
254	C₂×C₄⋊C₄	—	86	1	no	—
255	C₂×C₂²⋊C₄	—	86	1	no	—
256	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
257	#106	—	86	8	no	—
258	#106	—	86	4	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	43	2	no	—
263	C₄⋊C₈	—	86	1	no	—
264	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
265	#103	—	86	4	no	—
266	#102	—	86	4	no	—
267	C₂²⋊C₈	—	172	4	no	—
268	C₄×C₈	—	86	1	no	—
269	C₄×C₈	—	86	1	no	—
270	C₂⁴	—	86	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	1	no	—
275	C₂×Q₈	—	4	1	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	86	2	no	—
278	C₂×Q₈	—	344	2	no	—
279	C₂×Q₈	—	172	1	no	—
280	C₂×D₄	—	43	2	no	—
281	C₂×D₄	—	86	1	no	—
282	C₂×D₄	—	86	1	no	—
283	C₂×D₄	—	86	1	no	—
284	C₂×D₄	—	86	1	no	—
285	C₂×D₄	—	86	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	86	1	no	—
288	C₂²×C₄	—	172	1	no	—
289	C₂²×C₄	—	172	1	no	—
290	C₂²×C₄	—	172	1	no	—
291	C₂²×C₄	—	172	2	no	—
292	C₂²×C₄	—	172	2	no	—
293	C₂²×C₄	#10621898	1	1	yes	characteristic
294	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
296	SD₁₆	—	172	8	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	4	1	no	—
300	C₂×C₈	—	4	1	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	43	4	no	—
303	C₂×C₈	—	86	2	no	—
304	C₂×C₈	—	86	2	no	—
305	C₂×C₈	—	86	2	no	—
306	C₂×C₈	—	172	1	no	—
307	C₂×C₈	—	172	1	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	86	2	no	—
313	C₄⋊C₄	—	86	1	no	—
314	C₄⋊C₄	—	86	1	no	—
315	C₄⋊C₄	—	86	2	no	—
316	C₄⋊C₄	—	172	1	no	—
317	C₄⋊C₄	—	172	1	no	—
318	C₂²⋊C₄	—	172	1	no	—
319	C₂²⋊C₄	—	172	1	no	—
320	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
321	C₄²	D₄×D₄₃	1	1	yes	characteristic
322	C₄²	D₄×D₄₃	1	1	yes	characteristic
323	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
324	C₂³	—	86	2	no	—
325	C₂³	—	172	1	no	—
326	C₂³	—	172	2	no	—
327	C₂³	—	172	1	no	—
328	C₂³	—	172	2	no	—
329	C₂³	#13063298	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	8	1	no	—
333	Q₈	—	8	1	no	—
334	Q₈	—	344	2	no	—
335	Q₈	—	344	2	no	—
336	D₄	—	86	2	no	—
337	D₄	—	86	2	no	—
338	D₄	—	172	1	no	—
339	D₄	—	172	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	344	2	no	—
345	C₂×C₄	—	344	1	no	—
346	C₂×C₄	—	344	1	no	—
347	C₂×C₄	—	344	2	no	—
348	C₂×C₄	—	172	1	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	#13063293	1	1	yes	characteristic
352	C₂×C₄	#13063295	1	1	yes	characteristic
353	C₂×C₄	#13063295	1	1	yes	characteristic
354	C₂×C₄	#13063298	1	1	yes	characteristic
355	C₂×C₄	#13063299	1	1	yes	characteristic
356	C₂×C₄	#13063321	1	1	yes	characteristic
357	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
358	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
359	C₂×C₄	#13063344	1	1	yes	characteristic
360	C₂×C₄	#13063349	1	1	yes	characteristic
361	C₂×C₄	#13063354	1	1	yes	characteristic
362	C₂×C₄	#13063357	1	2	yes	—
363	C₈	—	4	2	no	—
364	C₈	—	8	1	no	—
365	C₈	—	86	4	no	—
366	C₈	—	172	2	no	—
367	C₂²	—	344	2	no	—
368	C₂²	—	344	1	no	—
369	C₂²	—	344	1	no	—
370	C₂²	—	172	4	no	—
371	C₂²	—	172	2	no	—
372	C₂²	#14582808	1	1	yes	characteristic
373	C₂²	#14583103	1	2	yes	—
374	C₂²	#14583127	1	1	yes	characteristic
375	C₂²	#14583144	1	1	yes	characteristic
376	C₂²	#14583165	1	2	yes	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	8	2	no	—
380	C₄	—	344	1	no	—
381	C₄	—	344	2	no	—
382	C₄	#14583036	1	1	yes	characteristic
383	C₄	#14583039	1	2	yes	—
384	C₄	#14583041	1	1	yes	characteristic
385	C₄	#14583051	1	1	yes	characteristic
386	C₄	#14583062	1	1	yes	characteristic
387	C₄	#14583270	1	1	yes	characteristic
388	C₄	#14583292	1	1	yes	characteristic
389	C₂	—	344	2	no	—
390	C₂	#19133969	1	2	yes	—
391	C₂	#19134093	1	2	yes	—
392	C₂	#19135599	1	1	yes	characteristic
393	C₂	#19135679	1	1	yes	characteristic
394	C₂	#19135708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063298
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#332	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	C₂×#332	D₄₃
hypocenter	C₄₃	#10271
LCS	G ▷ C₂×C₄²×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
1. C₂×#14583295	C₂	semidirect
5. C₂×#14582647	C₂	semidirect
6. #19132429	C₂	semidirect
7. C₂×C₄₃×#332	C₂	semidirect
125. C₂×C₄₃⋊C₈	SD₁₆	semidirect
197. C₂×#332	D₄₃	semidirect
226. C₄₃	#10271	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583295	2
C₂×#14583190	2
C₂×#14583101	2
#19135388	2
C₂×#14582647	2
#19132429	2
C₂×C₄₃×#332	2
#10271	43