Subgroups — #25014859

Order 11008.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583284	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582715	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582713	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582711	C₂	1	1	yes	maximal, characteristic
5	M₃₂×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134765	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	—	2	1	no	—
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063314	—	2	1	no	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	#14583284	—	2	1	no	—
13	#14583284	—	2	1	no	—
14	#14583284	C₄	1	2	yes	—
15	#14583284	C₄	1	2	yes	—
16	C₂×#13063261	C₂²	1	1	yes	characteristic
17	C₂×#13063260	—	2	1	no	—
18	#14582922	—	2	1	no	—
19	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	#14582715	—	2	2	no	—
21	#14582713	—	2	2	no	—
22	#14582711	—	2	2	no	—
23	#14582674	—	2	1	no	—
24	C₁₆×Dic₄₃	—	2	1	no	—
25	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#362	—	2	2	no	—
29	C₂²×D₁₇₂	—	2	1	no	—
30	C₂×#10621884	—	2	2	no	—
31	C₂×#10621884	—	2	2	no	—
32	C₂×#10621884	—	4	2	no	—
33	C₂×#10621884	—	4	2	no	—
34	C₂×D₃₄₄	—	2	1	no	—
35	C₂×D₃₄₄	—	2	1	no	—
36	C₂×D₃₄₄	—	2	1	no	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
40	C₂×#10621877	—	2	1	no	—
41	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
42	#13063314	—	2	2	no	—
43	#13063314	—	2	2	no	—
44	#13063314	—	4	2	no	—
45	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
46	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
47	#13063265	—	2	1	no	—
48	#13063265	D₄	1	2	yes	—
49	C₂×C₄₃⋊C₁₆	—	2	1	no	—
50	#13063261	—	2	1	no	—
51	#13063261	—	2	1	no	—
52	#13063260	—	4	2	no	—
53	Dic₄₃⋊C₈	—	4	2	no	—
54	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₄₃	D₄	1	1	yes	characteristic
57	C₈×Dic₄₃	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₄₃	—	2	1	no	—
59	C₂×Q₁₆×C₄₃	—	2	1	no	—
60	C₂×Q₁₆×C₄₃	—	2	1	no	—
61	C₂×Q₁₆×C₄₃	—	2	1	no	—
62	C₂×Q₁₆×C₄₃	—	2	1	no	—
63	C₂×Q₁₆×C₄₃	—	2	1	no	—
64	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₄₃	—	2	1	no	—
66	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	M₃₂×C₄₃	—	2	1	no	—
68	M₃₂×C₄₃	D₄	1	2	yes	—
69	C₂×C₁₆×C₄₃	—	2	1	no	—
70	#108×C₄₃	—	2	1	no	—
71	#108×C₄₃	—	2	1	no	—
72	C₂³×D₄₃	—	4	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×Dic₄₃	SD₁₆	1	2	yes	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	4	1	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×D₁₇₂	—	4	1	no	—
87	C₂×D₁₇₂	—	4	1	no	—
88	Q₁₆×C₄₃	—	2	2	no	—
89	Q₁₆×C₄₃	—	2	2	no	—
90	Q₁₆×C₄₃	—	4	1	no	—
91	Q₁₆×C₄₃	—	4	1	no	—
92	M₁₆×C₄₃	—	4	1	no	—
93	M₁₆×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	—	4	1	no	—
95	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
101	#10621884	—	4	4	no	—
102	#10621884	—	4	4	no	—
103	(C₂×D₄₃)⋊C₄	—	4	4	no	—
104	(C₂×D₄₃)⋊C₄	—	4	4	no	—
105	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×Dic₄₃	D₈	1	2	yes	—
108	#10621877	—	4	1	no	—
109	#10621877	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	1	no	—
111	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄₃⋊C₈	D₈	1	2	yes	—
114	C₂×C₄₃⋊C₈	SD₁₆	1	2	yes	—
115	D₃₄₄	—	2	2	no	—
116	D₃₄₄	—	2	2	no	—
117	D₃₄₄	—	4	1	no	—
118	D₃₄₄	—	4	1	no	—
119	C₁₆×C₄₃	—	2	2	no	—
120	C₄₃⋊C₁₆	—	2	2	no	—
121	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₄₃	—	4	2	no	—
123	C₂²×D₄₃	—	8	4	no	—
124	C₂²×D₄₃	—	8	1	no	—
125	C₂²×D₄₃	—	8	1	no	—
126	Q₈×C₄₃	—	4	2	no	—
127	Q₈×C₄₃	—	4	2	no	—
128	Q₈×C₄₃	—	8	1	no	—
129	Q₈×C₄₃	—	8	1	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	8	1	no	—
132	C₂×C₄×C₄₃	—	8	1	no	—
133	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	#102	1	2	yes	—
141	C₂×Dic₄₃	#102	1	2	yes	—
142	D₁₇₂	—	4	2	no	—
143	D₁₇₂	—	4	2	no	—
144	D₁₇₂	—	8	1	no	—
145	D₁₇₂	—	8	1	no	—
146	C₈×C₄₃	—	8	1	no	—
147	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
148	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
149	C₈×C₄₃	#121	1	1	yes	characteristic
150	C₈×C₄₃	#121	1	1	yes	characteristic
151	C₄₃⋊C₈	—	8	1	no	—
152	C₄₃⋊C₈	#102	1	2	yes	—
153	C₄₃⋊C₈	#102	1	2	yes	—
154	#13214	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
157	C₂²×C₄₃	#486	1	1	yes	characteristic
158	C₂²×C₄₃	#492	1	1	yes	characteristic
159	C₂×D₄₃	—	8	4	no	—
160	C₂×D₄₃	—	8	2	no	—
161	C₂×D₄₃	—	16	2	no	—
162	C₂×D₄₃	—	16	2	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	#390	1	1	yes	characteristic
165	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
166	C₄×C₄₃	#486	1	1	yes	characteristic
167	C₄×C₄₃	#493	1	1	yes	characteristic
168	Dic₄₃	—	2	2	no	—
169	Dic₄₃	—	2	2	no	—
170	C₂×#495	—	43	1	no	—
171	C₂×#362	—	43	1	no	—
172	C₂×#362	D₄₃	1	1	yes	characteristic, complemented
173	C₂×#361	—	43	1	no	—
174	C₂×#361	—	43	1	no	—
175	C₄×M₃₂	—	43	1	no	—
176	#1848	—	43	1	no	—
177	C₂×C₄₃	—	2	1	no	—
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	#1957	1	1	yes	characteristic
180	C₂×C₄₃	#2148	1	2	yes	—
181	D₄₃	—	16	2	no	—
182	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
183	C₂²×SD₁₆	—	86	1	no	—
184	C₂²×D₈	—	43	1	no	—
185	C₂×#124	—	86	1	no	—
186	C₂×M₃₂	—	43	1	no	—
187	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
188	#495	—	43	2	no	—
189	#495	—	43	2	no	—
190	#495	—	86	1	no	—
191	#495	—	86	1	no	—
192	C₂×#108	—	43	1	no	—
193	C₂×#108	—	86	1	no	—
194	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
195	#423	—	86	1	no	—
196	C₂×C₄×C₈	—	43	1	no	—
197	#362	—	2	2	no	—
198	#362	—	86	2	no	—
199	#361	—	86	2	no	—
200	#361	—	86	2	no	—
201	#346	—	86	1	no	—
202	C₄×C₁₆	—	86	1	no	—
203	C₄₃	#13214	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×D₄	—	86	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₄×D₄₃	1	2	yes	—
212	C₂×SD₁₆	—	86	2	no	—
213	C₂×SD₁₆	—	86	2	no	—
214	C₂×SD₁₆	—	172	2	no	—
215	C₂×SD₁₆	—	172	2	no	—
216	C₂×D₈	—	43	2	no	—
217	C₂×D₈	—	86	1	no	—
218	C₂×D₈	—	86	1	no	—
219	C₂×D₈	—	86	1	no	—
220	C₂×D₈	—	86	1	no	—
221	C₂×D₈	—	86	1	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	—	86	1	no	—
224	C₂²×C₈	—	43	1	no	—
225	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
226	#124	—	86	2	no	—
227	#124	—	86	2	no	—
228	#124	—	172	2	no	—
229	C₂×C₂²⋊C₄	—	172	2	no	—
230	C₂×C₄²	—	43	1	no	—
231	M₃₂	—	2	1	no	—
232	M₃₂	—	43	2	no	—
233	M₃₂	—	86	1	no	—
234	M₃₂	C₂²⋊D₄₃	1	2	yes	—
235	C₂×C₁₆	—	2	1	no	—
236	C₂×C₁₆	—	86	1	no	—
237	#108	—	2	1	no	—
238	#108	—	2	1	no	—
239	#108	—	86	1	no	—
240	#108	—	86	1	no	—
241	#108	—	172	2	no	—
242	C₄⋊C₈	—	172	2	no	—
243	C₄×C₈	—	43	1	no	—
244	C₄×C₈	—	43	1	no	—
245	C₄×C₈	—	43	1	no	—
246	C₄×C₈	—	43	1	no	—
247	C₂⁴	—	172	1	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×D₄	—	86	2	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	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	43	2	no	—
262	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	4	1	no	—
266	Q₁₆	—	4	1	no	—
267	SD₁₆	—	172	4	no	—
268	SD₁₆	—	172	4	no	—
269	D₈	—	86	2	no	—
270	D₈	—	86	2	no	—
271	D₈	—	172	1	no	—
272	D₈	—	172	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	172	1	no	—
276	M₁₆	—	172	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	43	1	no	—
279	C₂×C₈	—	43	1	no	—
280	C₂×C₈	—	43	2	no	—
281	C₂×C₈	—	43	2	no	—
282	C₂×C₈	—	172	1	no	—
283	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
284	C₂×C₈	#10621898	1	1	yes	characteristic
285	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
286	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
287	C₂×C₈	#10621900	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
289	C₂²⋊C₄	—	172	4	no	—
290	C₂²⋊C₄	—	172	4	no	—
291	C₄²	—	43	1	no	—
292	C₄²	—	43	1	no	—
293	C₄²	—	43	2	no	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	86	2	no	—
296	C₂³	—	344	4	no	—
297	C₂³	—	344	1	no	—
298	C₂³	—	344	1	no	—
299	C₂³	—	172	2	no	—
300	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	1	no	—
304	Q₈	—	8	1	no	—
305	D₄	—	344	1	no	—
306	D₄	—	344	1	no	—
307	D₄	—	172	2	no	—
308	D₄	—	172	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	43	2	no	—
313	C₂×C₄	—	43	2	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
316	C₂×C₄	#13063311	1	1	yes	characteristic
317	C₂×C₄	#13063321	1	1	yes	characteristic
318	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
319	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
320	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
321	C₈	—	8	1	no	—
322	C₈	—	43	2	no	—
323	C₈	—	43	2	no	—
324	C₈	—	344	1	no	—
325	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
326	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
327	C₈	#13063297	1	1	yes	characteristic
328	C₈	#13063298	1	1	yes	characteristic
329	C₂²	—	2	2	no	—
330	C₂²	—	344	4	no	—
331	C₂²	—	344	2	no	—
332	C₂²	—	688	2	no	—
333	C₂²	—	688	2	no	—
334	C₂²	#102×D₄₃	1	1	yes	characteristic
335	C₂²	#14583244	1	1	yes	characteristic
336	C₂²	#14583270	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	86	2	no	—
339	C₄	—	86	2	no	—
340	C₄	#14582799	1	1	yes	characteristic
341	C₄	#102×D₄₃	1	1	yes	characteristic
342	C₄	#14583244	1	1	yes	characteristic
343	C₄	#14583274	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	688	2	no	—
347	C₂	#19135399	1	1	yes	characteristic
348	C₂	#19136343	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₄₃
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#362	C₂
socle	C₂²×C₄₃	C₂×#102
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#362	D₄₃
hypocenter	C₄₃	#13214
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#362

Complements

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

N	H ≅ G/N	type
2. C₂×#14582715	C₂	semidirect
5. M₃₂×Dic₄₃	C₂	semidirect
6. #19134765	C₂	semidirect
7. C₂×C₄₃×#362	C₂	semidirect
172. C₂×#362	D₄₃	semidirect
203. C₄₃	#13214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583284	2
C₂×#14582715	2
C₂×#14582713	2
C₂×#14582711	2
M₃₂×Dic₄₃	2
#19134765	2
C₂×C₄₃×#362	2
#13214	43