Subgroups — #25011810

Order 11008.

526 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582981	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582701	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582699	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582674	C₂	1	1	yes	maximal, characteristic, complemented
5	#107×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132686	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
9	C₂×#13063287	—	2	1	no	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14582981	—	2	2	no	—
12	#14582981	C₄	1	4	yes	complemented
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063245	—	2	1	no	—
15	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₄₃	—	2	1	no	—
17	#14582701	C₄	1	4	yes	complemented
18	#14582699	C₄	1	4	yes	complemented
19	#14582674	—	2	2	no	—
20	#14582661	—	2	1	no	—
21	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×D₁₇₂	—	2	1	no	—
25	C₂²×C₄×D₄₃	—	4	1	no	—
26	C₂×D₃₄₄	—	2	1	no	—
27	C₂×D₃₄₄	—	2	1	no	—
28	C₂×D₃₄₄	—	2	1	no	—
29	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
30	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
31	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	#13063287	—	2	4	no	—
34	#13063287	—	4	2	no	—
35	C₂×C₄⋊Dic₄₃	—	2	1	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	4	1	no	—
37	C₂×C₄×Dic₄₃	—	4	1	no	—
38	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄₃⋊C₁₆	—	2	1	no	—
40	C₂×C₄₃⋊C₁₆	—	2	1	no	—
41	C₂×C₄₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
42	#13063259	—	2	1	no	—
43	#13063259	C₂×C₄	1	2	yes	complemented
44	#13063245	—	2	4	no	—
45	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₄₃	D₄	1	2	yes	—
47	C₄₃⋊#95	—	4	1	no	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
51	C₂×C₁₆×C₄₃	—	2	1	no	—
52	C₂×C₁₆×C₄₃	—	2	1	no	—
53	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	complemented
54	#107×C₄₃	—	2	1	no	—
55	#107×C₄₃	—	2	1	no	—
56	#107×C₄₃	C₂×C₄	1	2	yes	—
57	#107×C₄₃	C₂×C₄	1	2	yes	complemented
58	C₂³×D₄₃	—	4	1	no	—
59	C₂²×C₄×C₄₃	—	4	1	no	—
60	C₂²×C₄×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	4	1	no	—
63	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
64	C₂²×Dic₄₃	D₈	1	1	yes	characteristic
65	C₂×D₁₇₂	—	2	2	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×D₁₇₂	—	2	2	no	—
68	C₂×D₁₇₂	—	4	1	no	—
69	C₂×D₁₇₂	—	4	1	no	—
70	C₂×D₁₇₂	—	4	1	no	—
71	C₂×C₄×D₄₃	—	4	4	no	—
72	C₂×C₄×D₄₃	—	8	2	no	—
73	C₂×C₄×D₄₃	—	8	2	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
79	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
80	C₄⋊C₄×C₄₃	—	2	2	no	—
81	C₄⋊C₄×C₄₃	—	2	2	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	(C₂×D₄₃)⋊C₄	—	4	4	no	—
85	C₄⋊Dic₄₃	—	2	2	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	C₄×Dic₄₃	—	8	2	no	—
88	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₄₃	SD₁₆	1	2	yes	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	2	1	no	—
92	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
93	D₃₄₄	—	2	2	no	—
94	D₃₄₄	—	2	2	no	—
95	D₃₄₄	—	2	2	no	—
96	D₃₄₄	C₄²	1	4	yes	complemented
97	C₁₆×C₄₃	—	2	2	no	—
98	C₄₃⋊C₁₆	—	2	2	no	—
99	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₄₃	—	4	2	no	—
101	C₂²×D₄₃	—	4	2	no	—
102	C₂²×D₄₃	—	4	2	no	—
103	C₂²×D₄₃	—	8	1	no	—
104	C₂²×D₄₃	—	8	1	no	—
105	C₂²×D₄₃	—	8	1	no	—
106	C₂²×D₄₃	—	8	1	no	—
107	C₂×C₄×C₄₃	—	4	2	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	—	8	1	no	—
110	C₂×C₄×C₄₃	—	8	1	no	—
111	C₂×C₄×C₄₃	—	8	1	no	—
112	C₂×C₄×C₄₃	—	8	1	no	—
113	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	8	1	no	—
125	C₂×Dic₄₃	—	8	1	no	—
126	C₂×Dic₄₃	#102	1	2	yes	—
127	C₂×Dic₄₃	#102	1	2	yes	—
128	D₁₇₂	—	2	4	no	—
129	D₁₇₂	—	4	2	no	—
130	D₁₇₂	—	4	2	no	—
131	D₁₇₂	—	4	2	no	—
132	C₄×D₄₃	—	8	8	no	—
133	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
134	C₈×C₄₃	#117	1	1	yes	characteristic
135	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
137	C₄₃⋊C₈	—	2	2	no	—
138	#12637	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
140	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
141	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
142	C₂²×C₄₃	#509	1	1	yes	characteristic
143	C₂²×C₄₃	#509	1	1	yes	characteristic
144	C₂²×C₄₃	#509	1	1	yes	characteristic
145	C₂²×C₄₃	#510	1	1	yes	characteristic
146	C₂×D₄₃	—	4	4	no	—
147	C₂×D₄₃	—	8	2	no	—
148	C₂×D₄₃	—	8	2	no	—
149	C₂×D₄₃	—	8	2	no	—
150	C₂×D₄₃	—	8	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₄×C₄₃	—	8	2	no	—
154	C₄×C₄₃	—	8	2	no	—
155	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
156	C₄×C₄₃	#416	1	1	yes	characteristic
157	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
158	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
159	Dic₄₃	—	2	2	no	—
160	Dic₄₃	—	2	2	no	—
161	Dic₄₃	—	8	2	no	—
162	C₂×C₄×D₈	—	43	1	no	—
163	C₂×#357	—	43	1	no	—
164	C₂×#357	—	43	1	no	—
165	C₂×#346	—	43	1	no	—
166	C₄×#107	—	43	1	no	—
167	#1380	—	43	1	no	—
168	#1380	D₄₃	1	1	yes	characteristic, complemented
169	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
170	C₂×C₄₃	#2139	1	1	yes	characteristic
171	C₂×C₄₃	#2140	1	1	yes	characteristic
172	C₂×C₄₃	#2175	1	1	yes	characteristic
173	C₂×C₄₃	#2175	1	1	yes	characteristic
174	C₂×C₄₃	#2177	1	1	yes	characteristic
175	C₂×C₄₃	#2177	1	1	yes	characteristic
176	D₄₃	—	8	4	no	—
177	C₂²×D₈	—	43	1	no	—
178	C₂×C₄×D₄	—	86	1	no	—
179	C₂²×C₁₆	—	43	1	no	—
180	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
181	C₄×D₈	—	43	4	no	—
182	C₄×D₈	—	86	2	no	—
183	C₂×#107	—	43	1	no	—
184	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
185	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
186	C₂×#102	—	86	1	no	—
187	C₂×C₄×C₈	—	43	1	no	—
188	C₄×C₄⋊C₄	—	86	1	no	—
189	#357	—	43	4	no	—
190	#357	—	43	4	no	—
191	#346	—	86	2	no	—
192	#340	—	86	1	no	—
193	C₄₃	#12637	1	1	yes	characteristic, complemented, Sylow
194	C₂²×D₄	—	86	1	no	—
195	C₂³×C₄	—	172	1	no	—
196	C₂×D₈	—	43	2	no	—
197	C₂×D₈	—	43	2	no	—
198	C₂×D₈	—	43	2	no	—
199	C₂×D₈	—	86	1	no	—
200	C₂×D₈	—	86	1	no	—
201	C₂×D₈	—	86	1	no	—
202	C₂²×C₈	—	43	1	no	—
203	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	C₄×D₄	—	86	4	no	—
205	C₄×D₄	—	172	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	86	1	no	—
209	C₂×C₂²⋊C₄	—	172	1	no	—
210	C₂×C₄²	—	43	1	no	—
211	C₂×C₄²	—	172	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	43	2	no	—
215	C₂×C₁₆	—	86	1	no	—
216	C₂×C₁₆	—	86	1	no	—
217	C₂×C₁₆	C₄×D₄₃	1	2	yes	complemented
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	43	2	no	—
221	#107	—	86	1	no	—
222	#107	C₄×D₄₃	1	2	yes	—
223	#107	C₄×D₄₃	1	2	yes	complemented
224	#102	—	86	4	no	—
225	C₄×C₈	—	43	2	no	—
226	C₄×C₈	—	43	2	no	—
227	#95	—	172	1	no	—
228	C₂⁴	—	172	1	no	—
229	C₂×D₄	—	86	2	no	—
230	C₂×D₄	—	86	2	no	—
231	C₂×D₄	—	86	2	no	—
232	C₂×D₄	—	172	1	no	—
233	C₂×D₄	—	172	1	no	—
234	C₂×D₄	—	172	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	43	1	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	344	2	no	—
240	C₂²×C₄	—	344	2	no	—
241	C₂²×C₄	—	172	4	no	—
242	C₂²×C₄	—	172	1	no	—
243	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
244	D₈	—	43	4	no	—
245	D₈	—	86	2	no	—
246	D₈	—	86	2	no	—
247	D₈	—	86	2	no	—
248	C₂×C₈	—	43	2	no	—
249	C₂×C₈	—	86	1	no	—
250	C₂×C₈	—	86	1	no	—
251	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
254	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
255	C₂×C₈	#10621900	1	1	yes	characteristic
256	C₂×C₈	#10621902	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	86	2	no	—
262	C₄⋊C₄	—	172	1	no	—
263	C₂²⋊C₄	—	172	4	no	—
264	C₄²	—	43	2	no	—
265	C₄²	—	43	2	no	—
266	C₄²	—	344	2	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	86	2	no	—
269	C₂³	—	344	1	no	—
270	C₂³	—	344	1	no	—
271	C₂³	—	344	1	no	—
272	C₂³	—	344	1	no	—
273	C₂³	—	172	2	no	—
274	C₂³	—	172	2	no	—
275	C₂³	—	172	2	no	—
276	C₂³	D₈×D₄₃	1	1	yes	characteristic
277	D₄	—	86	4	no	—
278	D₄	—	172	2	no	—
279	D₄	—	172	2	no	—
280	D₄	—	172	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	43	2	no	—
288	C₂×C₄	—	43	2	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	1	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	344	8	no	—
294	C₂×C₄	—	344	1	no	—
295	C₂×C₄	—	344	1	no	—
296	C₂×C₄	—	172	2	no	—
297	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#13063287	1	1	yes	characteristic
300	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
301	C₂×C₄	#13063346	1	1	yes	characteristic
302	C₂×C₄	#13063348	1	1	yes	characteristic
303	C₈	—	86	2	no	—
304	C₈	C₄²×D₄₃	1	1	yes	characteristic
305	C₈	#13063283	1	1	yes	characteristic
306	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
307	C₈	#13063287	1	1	yes	characteristic
308	C₂²	—	344	2	no	—
309	C₂²	—	344	2	no	—
310	C₂²	—	344	2	no	—
311	C₂²	—	344	2	no	—
312	C₂²	—	344	2	no	—
313	C₂²	—	344	2	no	—
314	C₂²	—	172	4	no	—
315	C₂²	#102×D₄₃	1	1	yes	characteristic
316	C₂²	#14582979	1	1	yes	characteristic
317	C₂²	#14582981	1	1	yes	characteristic
318	C₂²	#14583333	1	1	yes	characteristic
319	C₂²	#14583336	1	1	yes	characteristic
320	C₂²	#14583336	1	1	yes	characteristic
321	C₂²	#14583341	1	1	yes	characteristic
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	86	2	no	—
325	C₄	—	86	2	no	—
326	C₄	—	344	2	no	—
327	C₄	#14582746	1	1	yes	characteristic
328	C₄	#14582894	1	1	yes	characteristic
329	C₄	#14582984	1	1	yes	characteristic
330	C₄	#14582986	1	1	yes	characteristic
331	C₂	—	344	4	no	—
332	C₂	#19134409	1	1	yes	characteristic
333	C₂	#19136306	1	1	yes	characteristic
334	C₂	#19136309	1	1	yes	characteristic
335	C₂	#19136458	1	1	yes	characteristic
336	C₂	#19136462	1	1	yes	characteristic
337	C₂	#19136468	1	1	yes	characteristic
338	C₂	#19136470	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₃
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1380	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1380	D₄₃
hypocenter	C₄₃	#12637
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. C₂×#14582674	C₂	semidirect
5. #107×Dic₄₃	C₂	semidirect
6. #19132686	C₂	semidirect
7. C₄₃×#1380	C₂	semidirect
12. #14582981	C₄	semidirect
17. #14582701	C₄	semidirect
18. #14582699	C₄	semidirect
41. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
43. #13063259	C₂×C₄	semidirect
53. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
57. #107×C₄₃	C₂×C₄	semidirect
96. D₃₄₄	C₄²	semidirect
168. #1380	D₄₃	semidirect
193. C₄₃	#12637	semidirect
217. C₂×C₁₆	C₄×D₄₃	semidirect
223. #107	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582981	2
C₂×#14582701	2
C₂×#14582699	2
C₂×#14582674	2
#107×Dic₄₃	2
#19132686	2
C₄₃×#1380	2
#12637	43