Subgroups — #24997866

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583304	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583180	C₂	1	1	yes	maximal, characteristic
3	#19139544	C₂	1	1	yes	maximal, characteristic
4	#19135518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582644	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582628	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063325	C₂²	1	1	yes	characteristic
9	C₂×#13063325	C₂²	1	1	yes	characteristic
10	C₂×#13063305	—	2	2	no	—
11	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	#14583304	—	2	2	no	—
13	#14583304	—	2	2	no	—
14	#14583180	—	2	2	no	—
15	#14583180	—	2	2	no	—
16	C₂×#13063259	—	2	2	no	—
17	C₂×#13063259	—	2	1	no	—
18	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×#13063248	—	2	2	no	—
20	C₂×#13063247	—	2	1	no	—
21	#14582836	—	2	2	no	—
22	#14582801	—	2	1	no	—
23	#14582769	—	2	1	no	—
24	#14582644	—	2	2	no	—
25	#14582628	—	2	2	no	—
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×Dic₈₆	—	2	1	no	—
31	C₂²×D₁₇₂	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₄₃	—	2	1	no	—
33	C₂²×C₄₃⋊C₈	—	2	1	no	—
34	#13063325	—	2	1	no	—
35	#13063325	—	2	1	no	—
36	#13063325	—	2	1	no	—
37	#13063325	—	2	1	no	—
38	#13063325	—	2	1	no	—
39	#13063325	—	2	1	no	—
40	#13063325	—	2	1	no	—
41	#13063325	D₄	1	2	yes	—
42	#13063305	—	4	4	no	—
43	#13063305	—	4	4	no	—
44	C₄×D₁₇₂	—	2	1	no	—
45	C₄×D₁₇₂	—	2	1	no	—
46	C₄×D₁₇₂	—	2	1	no	—
47	C₄×D₁₇₂	—	2	1	no	—
48	C₂×C₄⋊Dic₄₃	—	2	1	no	—
49	C₂×C₄⋊Dic₄₃	—	2	1	no	—
50	C₂×C₄⋊Dic₄₃	—	2	4	no	—
51	C₂×C₄⋊Dic₄₃	—	2	1	no	—
52	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
53	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
55	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
56	#13063259	—	2	8	no	—
57	#13063259	—	2	4	no	—
58	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
59	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
60	#13063248	—	4	4	no	—
61	#13063247	—	2	4	no	—
62	C₂²×C₈×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
65	C₂×C₄²×C₄₃	—	2	1	no	—
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₃	—	2	1	no	—
68	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	C₄×C₈×C₄₃	D₄	1	2	yes	—
71	#95×C₄₃	—	2	1	no	—
72	C₂³×D₄₃	—	2	1	no	—
73	C₂²×C₄×C₄₃	—	2	2	no	—
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₄₃	—	2	2	no	—
78	C₂²×Dic₄₃	—	2	1	no	—
79	C₂²×Dic₄₃	—	4	1	no	—
80	C₂×D₁₇₂	—	2	1	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×D₁₇₂	—	2	1	no	—
83	C₂×D₁₇₂	—	2	1	no	—
84	C₂×D₁₇₂	—	2	1	no	—
85	C₂×D₁₇₂	Q₁₆	1	2	yes	—
86	C₂×C₄×D₄₃	—	4	1	no	—
87	C₂×C₄×D₄₃	—	4	1	no	—
88	C₂×C₄×D₄₃	—	4	2	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×Dic₈₆	—	2	2	no	—
91	C₂×Dic₈₆	—	4	2	no	—
92	C₂×Dic₈₆	—	4	2	no	—
93	C₂×Dic₈₆	—	4	1	no	—
94	C₂×Dic₈₆	—	4	1	no	—
95	C₂×Dic₈₆	—	4	1	no	—
96	C₂×Dic₈₆	—	4	2	no	—
97	C₂×Dic₈₆	—	4	2	no	—
98	C₂×C₈×C₄₃	—	2	4	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	4	1	no	—
102	C₂×C₈×C₄₃	—	4	1	no	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	D₈	1	4	yes	—
105	C₄²×C₄₃	—	4	1	no	—
106	C₄²×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	#41	1	1	yes	characteristic
111	(C₂×D₄₃)⋊C₄	—	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	2	no	—
116	C₄⋊Dic₄₃	—	2	1	no	—
117	C₄⋊Dic₄₃	—	2	2	no	—
118	C₄⋊Dic₄₃	—	2	1	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	—	4	8	no	—
121	C₄⋊Dic₄₃	—	4	1	no	—
122	C₄⋊Dic₄₃	—	4	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₄⋊Dic₄₃	—	4	1	no	—
125	C₂×C₄₃⋊C₈	—	4	1	no	—
126	C₂×C₄₃⋊C₈	—	4	2	no	—
127	C₂³×C₄₃	#120	1	1	yes	characteristic
128	C₂²×D₄₃	—	2	2	no	—
129	C₂²×D₄₃	—	4	1	no	—
130	C₂²×D₄₃	—	4	2	no	—
131	C₂²×D₄₃	—	4	1	no	—
132	C₂²×D₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	2	2	no	—
134	C₂×C₄×C₄₃	—	2	4	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
146	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	4	1	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	—	4	4	no	—
154	C₂×Dic₄₃	—	8	2	no	—
155	D₁₇₂	—	2	2	no	—
156	D₁₇₂	—	2	2	no	—
157	D₁₇₂	—	4	1	no	—
158	D₁₇₂	—	4	1	no	—
159	C₄×D₄₃	—	8	2	no	—
160	C₄×D₄₃	—	8	1	no	—
161	C₄×D₄₃	—	8	1	no	—
162	Dic₈₆	—	4	4	no	—
163	Dic₈₆	—	8	2	no	—
164	Dic₈₆	—	8	1	no	—
165	Dic₈₆	—	8	1	no	—
166	Dic₈₆	—	8	1	no	—
167	Dic₈₆	—	8	1	no	—
168	C₈×C₄₃	—	2	4	no	—
169	C₈×C₄₃	—	4	2	no	—
170	C₄₃⋊C₈	—	8	1	no	—
171	#10076	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	#392	1	1	yes	characteristic
173	C₂²×C₄₃	#447	1	2	yes	—
174	C₂²×C₄₃	#451	1	1	yes	characteristic
175	C₂²×C₄₃	#452	1	1	yes	characteristic
176	C₂²×C₄₃	#456	1	2	yes	—
177	C₂×D₄₃	—	4	4	no	—
178	C₂×D₄₃	—	4	2	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₂×D₄₃	—	8	1	no	—
181	C₂×D₄₃	—	8	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	1	no	—
184	C₄×C₄₃	—	4	4	no	—
185	C₄×C₄₃	#451	1	1	yes	characteristic
186	C₄×C₄₃	#452	1	1	yes	characteristic
187	C₄×C₄₃	#459	1	2	yes	—
188	C₄×C₄₃	#467	1	1	yes	characteristic
189	C₄×C₄₃	#470	1	1	yes	characteristic
190	C₄×C₄₃	#493	1	1	yes	characteristic
191	C₄×C₄₃	#496	1	1	yes	characteristic
192	Dic₄₃	—	8	2	no	—
193	Dic₄₃	—	8	2	no	—
194	Dic₄₃	—	8	1	no	—
195	C₂×#500	—	43	1	no	—
196	C₂×#474	—	43	1	no	—
197	#2542	—	43	1	no	—
198	#1981	—	43	1	no	—
199	C₂×#332	—	43	1	no	—
200	C₂×D₄⋊C₈	—	43	1	no	—
201	#1279	D₄₃	1	1	yes	characteristic, complemented
202	C₂×C₄₃	#1647	1	2	yes	—
203	C₂×C₄₃	#1649	1	2	yes	—
204	C₂×C₄₃	#2001	1	1	yes	characteristic
205	C₂×C₄₃	#2005	1	1	yes	characteristic
206	C₂×C₄₃	#2011	1	1	yes	characteristic
207	D₄₃	—	8	2	no	—
208	C₂×C₄⋊Q₈	—	43	1	no	—
209	C₂×C₄⋊Q₈	—	43	1	no	—
210	C₂×C₂²⋊Q₈	—	86	2	no	—
211	C₂×C₄×D₄	—	43	1	no	—
212	#500	—	86	2	no	—
213	#500	—	86	2	no	—
214	#474	—	86	2	no	—
215	#474	—	86	2	no	—
216	C₂×#107	—	86	2	no	—
217	C₂×#107	—	86	1	no	—
218	C₂×C₄⋊C₈	—	43	1	no	—
219	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
220	C₂×#103	—	86	2	no	—
221	C₂×#102	—	86	1	no	—
222	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
223	#399	—	86	2	no	—
224	#390	—	86	1	no	—
225	#384	—	86	1	no	—
226	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
227	#332	—	86	2	no	—
228	D₄⋊C₈	—	86	2	no	—
229	C₄₃	#10076	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	86	1	no	—
231	C₂²×Q₈	—	86	1	no	—
232	C₂²×D₄	—	43	1	no	—
233	C₂³×C₄	—	86	1	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	86	1	no	—
237	C₄⋊Q₈	—	43	2	no	—
238	C₄⋊Q₈	—	86	1	no	—
239	C₄⋊Q₈	—	86	1	no	—
240	C₄⋊Q₈	—	86	1	no	—
241	C₄⋊Q₈	—	86	1	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	C₄⋊Q₈	—	86	1	no	—
244	C₄⋊Q₈	—	86	1	no	—
245	C₂²⋊Q₈	—	172	4	no	—
246	C₂²⋊Q₈	—	172	4	no	—
247	C₄×D₄	—	86	1	no	—
248	C₄×D₄	—	86	1	no	—
249	C₄×D₄	—	86	1	no	—
250	C₄×D₄	—	86	1	no	—
251	C₂×C₄⋊C₄	—	43	1	no	—
252	C₂×C₄⋊C₄	—	43	2	no	—
253	C₂×C₄⋊C₄	—	86	1	no	—
254	C₂×C₄⋊C₄	—	86	1	no	—
255	C₂×C₄⋊C₄	—	86	4	no	—
256	C₂×C₄⋊C₄	—	86	1	no	—
257	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
258	C₂×C₂²⋊C₄	—	86	4	no	—
259	C₂×C₂²⋊C₄	—	86	1	no	—
260	C₂×C₄²	—	2	1	no	—
261	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
262	#107	—	86	4	no	—
263	#107	—	86	8	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	86	1	no	—
266	C₄⋊C₈	—	86	1	no	—
267	C₄⋊C₈	D₁₇₂	1	2	yes	—
268	#103	—	172	4	no	—
269	#102	—	86	4	no	—
270	C₄×C₈	—	2	1	no	—
271	C₄×C₈	D₁₇₂	1	2	yes	—
272	#95	—	2	1	no	—
273	C₂⁴	—	86	1	no	—
274	C₂×Q₈	—	86	2	no	—
275	C₂×Q₈	—	172	2	no	—
276	C₂×Q₈	—	172	2	no	—
277	C₂×Q₈	—	172	1	no	—
278	C₂×Q₈	—	172	1	no	—
279	C₂×Q₈	—	172	1	no	—
280	C₂×Q₈	—	172	2	no	—
281	C₂×Q₈	—	172	2	no	—
282	C₂×D₄	—	43	2	no	—
283	C₂×D₄	—	86	1	no	—
284	C₂×D₄	—	86	1	no	—
285	C₂×D₄	—	86	1	no	—
286	C₂×D₄	—	86	1	no	—
287	C₂×D₄	—	86	1	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	86	2	no	—
290	C₂²×C₄	—	86	1	no	—
291	C₂²×C₄	—	172	1	no	—
292	C₂²×C₄	—	172	1	no	—
293	C₂²×C₄	—	172	1	no	—
294	C₂²×C₄	—	172	2	no	—
295	C₂²×C₄	—	172	2	no	—
296	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
297	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
298	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
299	C₂×C₈	—	2	4	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	172	1	no	—
305	C₂×C₈	—	172	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	86	2	no	—
308	C₄⋊C₄	—	86	1	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	86	1	no	—
312	C₄⋊C₄	—	86	2	no	—
313	C₄⋊C₄	—	172	8	no	—
314	C₄⋊C₄	—	172	1	no	—
315	C₄⋊C₄	—	172	1	no	—
316	C₄⋊C₄	—	172	1	no	—
317	C₄⋊C₄	—	172	1	no	—
318	C₄⋊C₄	D₃₄₄	1	4	yes	—
319	C₂²⋊C₄	—	172	8	no	—
320	C₂²⋊C₄	—	172	1	no	—
321	C₂²⋊C₄	—	172	1	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	4	1	no	—
324	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
325	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
326	C₄²	D₄×D₄₃	1	1	yes	characteristic
327	C₄²	#10621902	1	1	yes	characteristic
328	C₂³	—	86	2	no	—
329	C₂³	—	172	1	no	—
330	C₂³	—	172	1	no	—
331	C₂³	—	172	2	no	—
332	C₂³	—	172	2	no	—
333	C₂³	#13063295	1	1	yes	characteristic
334	Q₈	—	344	2	no	—
335	Q₈	—	344	1	no	—
336	Q₈	—	344	1	no	—
337	Q₈	—	344	1	no	—
338	Q₈	—	344	1	no	—
339	Q₈	—	172	4	no	—
340	D₄	—	86	2	no	—
341	D₄	—	86	2	no	—
342	D₄	—	172	1	no	—
343	D₄	—	172	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	344	2	no	—
349	C₂×C₄	—	344	2	no	—
350	C₂×C₄	—	344	1	no	—
351	C₂×C₄	—	344	1	no	—
352	C₂×C₄	—	172	1	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	—	172	4	no	—
356	C₂×C₄	—	172	2	no	—
357	C₂×C₄	#13063295	1	1	yes	characteristic
358	C₂×C₄	#13063295	1	1	yes	characteristic
359	C₂×C₄	#13063300	1	1	yes	characteristic
360	C₂×C₄	#13063300	1	1	yes	characteristic
361	C₂×C₄	#13063300	1	1	yes	characteristic
362	C₂×C₄	#13063322	1	1	yes	characteristic
363	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
364	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
365	C₂×C₄	#13063344	1	1	yes	characteristic
366	C₂×C₄	#13063353	1	2	yes	—
367	C₂×C₄	#13063354	1	1	yes	characteristic
368	C₂×C₄	#13063354	1	1	yes	characteristic
369	C₈	—	2	4	no	—
370	C₈	—	4	2	no	—
371	C₈	—	344	1	no	—
372	C₂²	—	344	2	no	—
373	C₂²	—	344	1	no	—
374	C₂²	—	344	1	no	—
375	C₂²	—	172	2	no	—
376	C₂²	—	172	4	no	—
377	C₂²	#14582809	1	1	yes	characteristic
378	C₂²	#14583035	1	2	yes	—
379	C₂²	#14583056	1	1	yes	characteristic
380	C₂²	#14583062	1	1	yes	characteristic
381	C₂²	#14583083	1	2	yes	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	4	4	no	—
385	C₄	—	344	1	no	—
386	C₄	—	344	2	no	—
387	C₄	—	344	2	no	—
388	C₄	#14583056	1	1	yes	characteristic
389	C₄	#14583062	1	1	yes	characteristic
390	C₄	#14583095	1	2	yes	—
391	C₄	#14583141	1	1	yes	characteristic
392	C₄	#14583158	1	1	yes	characteristic
393	C₄	#14583276	1	1	yes	characteristic
394	C₄	#14583288	1	1	yes	characteristic
395	C₂	—	344	2	no	—
396	C₂	#19133832	1	2	yes	—
397	C₂	#19133846	1	2	yes	—
398	C₂	#19135606	1	1	yes	characteristic
399	C₂	#19135634	1	1	yes	characteristic
400	C₂	#19135668	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063295
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1279	C₂
socle	C₂³×C₄₃	#120
prime radical	C₂²×D₁₇₂	D₄
hypercenter	#1279	D₄₃
hypocenter	C₄₃	#10076
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
1. C₂×#14583304	C₂	semidirect
4. #19135518	C₂	semidirect
5. C₂×#14582644	C₂	semidirect
7. C₄₃×#1279	C₂	semidirect
201. #1279	D₄₃	semidirect
229. C₄₃	#10076	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583304	2
C₂×#14583180	2
#19139544	2
#19135518	2
C₂×#14582644	2
C₂×#14582628	2
C₄₃×#1279	2
#10076	43