Subgroups — #24999168

Order 11008.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583250	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583180	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583105	C₂	1	1	yes	maximal, characteristic
4	#19135389	C₂	1	1	yes	maximal, characteristic
5	C₂×#14582646	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582644	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621873	—	2	1	no	—
9	C₂×#13063325	C₂²	1	1	yes	characteristic
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	#14583250	—	2	2	no	—
13	#14583250	—	2	2	no	—
14	#14583180	—	2	2	no	—
15	#14583180	—	2	2	no	—
16	#14583105	—	2	2	no	—
17	#14583105	—	2	2	no	—
18	C₂×#13063259	—	2	1	no	—
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#13063250	—	2	1	no	—
21	C₂×#13063250	—	2	2	no	—
22	C₂×#13063247	—	2	1	no	—
23	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
24	#14582646	—	2	2	no	—
25	#14582644	—	2	2	no	—
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×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	C₂×#10621873	—	2	4	no	—
34	C₂×#10621873	—	4	2	no	—
35	C₂×#10621873	—	4	2	no	—
36	C₂²×C₄₃⋊C₈	—	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	—	2	1	no	—
42	#13063325	—	2	1	no	—
43	#13063325	D₄	1	2	yes	—
44	#13063325	D₄	1	2	yes	—
45	C₄×D₁₇₂	—	2	1	no	—
46	C₄×D₁₇₂	—	2	1	no	—
47	C₄×D₁₇₂	—	2	1	no	—
48	C₄×D₁₇₂	—	2	1	no	—
49	C₂×C₄⋊Dic₄₃	—	2	1	no	—
50	C₂×C₄⋊Dic₄₃	—	2	1	no	—
51	C₂×C₄⋊Dic₄₃	—	2	1	no	—
52	C₂×C₄⋊Dic₄₃	—	2	1	no	—
53	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
55	#13063259	—	2	4	no	—
56	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
57	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
58	#13063250	—	2	4	no	—
59	#13063250	—	2	8	no	—
60	#13063247	—	2	4	no	—
61	(C₂×D₄₃)⋊C₈	—	4	4	no	—
62	C₂²×C₈×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	—	2	1	no	—
64	C₂²×C₈×C₄₃	D₄	1	2	yes	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	—	2	1	no	—
67	C₄⋊C₈×C₄₃	—	2	1	no	—
68	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
69	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
70	C₄×C₈×C₄₃	—	2	1	no	—
71	C₄×C₈×C₄₃	—	2	1	no	—
72	C₂³×D₄₃	—	2	1	no	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
76	C₂²×Dic₄₃	—	2	1	no	—
77	C₂²×Dic₄₃	—	4	1	no	—
78	C₂²×Dic₄₃	—	4	1	no	—
79	C₂×D₁₇₂	—	2	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	SD₁₆	1	2	yes	—
85	C₂×C₄×D₄₃	—	4	1	no	—
86	C₂×C₄×D₄₃	—	4	1	no	—
87	C₂×C₄×D₄₃	—	4	2	no	—
88	C₂×C₄×D₄₃	—	4	2	no	—
89	C₂×Dic₈₆	—	2	2	no	—
90	C₂×Dic₈₆	—	2	2	no	—
91	C₂×Dic₈₆	—	4	1	no	—
92	C₂×Dic₈₆	—	4	1	no	—
93	C₂×Dic₈₆	—	4	2	no	—
94	C₂×Dic₈₆	—	4	1	no	—
95	C₂×Dic₈₆	—	4	2	no	—
96	C₂×Dic₈₆	—	4	1	no	—
97	C₂×Dic₈₆	—	4	1	no	—
98	C₂×Dic₈₆	—	4	1	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	—	4	1	no	—
105	C₂×C₈×C₄₃	—	4	1	no	—
106	C₂×C₈×C₄₃	—	4	1	no	—
107	C₂×C₈×C₄₃	—	4	1	no	—
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	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
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	1	no	—
118	C₄⋊Dic₄₃	—	2	2	no	—
119	C₄⋊Dic₄₃	—	2	2	no	—
120	C₄⋊Dic₄₃	—	4	1	no	—
121	C₄⋊Dic₄₃	—	4	1	no	—
122	C₄⋊Dic₄₃	—	4	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₂×C₄₃⋊C₈	—	4	1	no	—
125	C₂×C₄₃⋊C₈	—	4	2	no	—
126	#10621873	—	4	8	no	—
127	C₂³×C₄₃	#121	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	#120	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#135	1	2	yes	—
143	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	1	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	8	2	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	C₂×Dic₄₃	—	8	1	no	—
153	D₁₇₂	—	2	2	no	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	4	1	no	—
156	D₁₇₂	—	4	1	no	—
157	C₄×D₄₃	—	8	2	no	—
158	C₄×D₄₃	—	8	1	no	—
159	C₄×D₄₃	—	8	1	no	—
160	Dic₈₆	—	4	4	no	—
161	Dic₈₆	—	4	2	no	—
162	Dic₈₆	—	4	2	no	—
163	Dic₈₆	—	8	1	no	—
164	Dic₈₆	—	8	1	no	—
165	Dic₈₆	—	8	1	no	—
166	Dic₈₆	—	8	1	no	—
167	C₈×C₄₃	—	4	2	no	—
168	C₈×C₄₃	—	4	2	no	—
169	C₈×C₄₃	—	4	2	no	—
170	C₄₃⋊C₈	—	8	1	no	—
171	#10270	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	#392	1	1	yes	characteristic
173	C₂²×C₄₃	#463	1	2	yes	—
174	C₂²×C₄₃	#465	1	1	yes	characteristic
175	C₂²×C₄₃	#468	1	1	yes	characteristic
176	C₂²×C₄₃	#473	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	#448	1	1	yes	characteristic
185	C₄×C₄₃	#449	1	1	yes	characteristic
186	C₄×C₄₃	#449	1	2	yes	—
187	C₄×C₄₃	#450	1	1	yes	characteristic
188	C₄×C₄₃	#452	1	1	yes	characteristic
189	C₄×C₄₃	#495	1	1	yes	characteristic
190	C₄×C₄₃	#496	1	1	yes	characteristic
191	Dic₄₃	—	8	2	no	—
192	Dic₄₃	—	8	2	no	—
193	Dic₄₃	—	8	1	no	—
194	C₂×#487	—	43	1	no	—
195	C₂×#474	—	43	1	no	—
196	C₂×#461	—	43	1	no	—
197	#1954	—	43	1	no	—
198	C₂×#332	—	43	1	no	—
199	C₂×#332	—	43	1	no	—
200	#1277	D₄₃	1	1	yes	characteristic, complemented
201	C₂×C₄₃	#1679	1	2	yes	—
202	C₂×C₄₃	#1699	1	2	yes	—
203	C₂×C₄₃	#2000	1	1	yes	characteristic
204	C₂×C₄₃	#2012	1	1	yes	characteristic
205	C₂×C₄₃	#2020	1	1	yes	characteristic
206	D₄₃	—	8	2	no	—
207	C₂²×SD₁₆	—	86	1	no	—
208	C₂×C₄⋊Q₈	—	43	1	no	—
209	C₂×C₄⋊Q₈	—	43	1	no	—
210	C₂×C₄×D₄	—	43	1	no	—
211	#487	—	86	2	no	—
212	#487	—	86	2	no	—
213	#474	—	86	2	no	—
214	#474	—	86	2	no	—
215	#461	—	86	2	no	—
216	#461	—	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×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
221	C₂×#103	—	86	1	no	—
222	C₂×#103	—	86	2	no	—
223	C₂×#102	—	86	1	no	—
224	C₂×C₂²⋊C₈	—	86	2	no	—
225	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
226	#332	—	86	2	no	—
227	#332	—	86	2	no	—
228	C₄₃	#10270	1	1	yes	characteristic, complemented, Sylow
229	C₂²×Q₈	—	86	1	no	—
230	C₂²×Q₈	—	86	1	no	—
231	C₂²×D₄	—	43	1	no	—
232	C₂³×C₄	—	86	1	no	—
233	C₂×SD₁₆	—	86	4	no	—
234	C₂×SD₁₆	—	172	2	no	—
235	C₂×SD₁₆	—	172	2	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	86	1	no	—
239	C₂²×C₈	D₁₇₂	1	2	yes	—
240	C₄⋊Q₈	—	43	2	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	—	86	1	no	—
246	C₄⋊Q₈	—	86	1	no	—
247	C₄⋊Q₈	—	86	1	no	—
248	C₄×D₄	—	86	1	no	—
249	C₄×D₄	—	86	1	no	—
250	C₄×D₄	—	86	1	no	—
251	C₄×D₄	—	86	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×C₄⋊C₄	—	86	1	no	—
256	C₂×C₄⋊C₄	—	86	1	no	—
257	C₂×C₂²⋊C₄	—	86	1	no	—
258	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
259	#107	—	86	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	86	1	no	—
263	C₄⋊C₈	—	86	1	no	—
264	C₄⋊C₈	D₁₇₂	1	2	yes	—
265	C₄⋊C₈	D₁₇₂	1	2	yes	—
266	#103	—	86	4	no	—
267	#103	—	86	8	no	—
268	#102	—	86	4	no	—
269	C₂²⋊C₈	—	172	4	no	—
270	C₄×C₈	—	2	1	no	—
271	C₄×C₈	—	2	1	no	—
272	C₂⁴	—	86	1	no	—
273	C₂×Q₈	—	86	2	no	—
274	C₂×Q₈	—	86	2	no	—
275	C₂×Q₈	—	172	1	no	—
276	C₂×Q₈	—	172	1	no	—
277	C₂×Q₈	—	172	2	no	—
278	C₂×Q₈	—	172	1	no	—
279	C₂×Q₈	—	172	2	no	—
280	C₂×Q₈	—	172	1	no	—
281	C₂×Q₈	—	172	1	no	—
282	C₂×Q₈	—	172	1	no	—
283	C₂×D₄	—	43	2	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	C₂×D₄	—	86	1	no	—
289	C₂²×C₄	—	86	1	no	—
290	C₂²×C₄	—	172	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	#10621902	1	1	yes	characteristic
299	SD₁₆	—	172	8	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	4	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	—	172	1	no	—
310	C₂×C₈	—	172	2	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	—	86	1	no	—
313	C₄⋊C₄	—	86	2	no	—
314	C₄⋊C₄	—	86	1	no	—
315	C₄⋊C₄	—	86	2	no	—
316	C₄⋊C₄	—	86	2	no	—
317	C₄⋊C₄	—	172	1	no	—
318	C₄⋊C₄	—	172	1	no	—
319	C₄⋊C₄	—	172	1	no	—
320	C₄⋊C₄	—	172	1	no	—
321	C₂²⋊C₄	—	172	1	no	—
322	C₂²⋊C₄	—	172	1	no	—
323	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
324	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
325	C₄²	D₄×D₄₃	1	1	yes	characteristic
326	C₄²	D₄×D₄₃	1	1	yes	characteristic
327	C₂³	—	86	2	no	—
328	C₂³	—	172	1	no	—
329	C₂³	—	172	2	no	—
330	C₂³	—	172	1	no	—
331	C₂³	—	172	2	no	—
332	C₂³	#13063300	1	1	yes	characteristic
333	Q₈	—	344	1	no	—
334	Q₈	—	344	1	no	—
335	Q₈	—	344	1	no	—
336	Q₈	—	344	1	no	—
337	Q₈	—	172	4	no	—
338	Q₈	—	172	2	no	—
339	Q₈	—	172	2	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	—	344	2	no	—
346	C₂×C₄	—	344	1	no	—
347	C₂×C₄	—	344	1	no	—
348	C₂×C₄	—	344	2	no	—
349	C₂×C₄	—	344	1	no	—
350	C₂×C₄	—	344	1	no	—
351	C₂×C₄	—	172	1	no	—
352	C₂×C₄	—	172	2	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	#13063295	1	1	yes	characteristic
356	C₂×C₄	#13063295	1	1	yes	characteristic
357	C₂×C₄	#13063295	1	1	yes	characteristic
358	C₂×C₄	#13063300	1	1	yes	characteristic
359	C₂×C₄	#13063300	1	1	yes	characteristic
360	C₂×C₄	#13063322	1	1	yes	characteristic
361	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
362	C₂×C₄	#13063346	1	1	yes	characteristic
363	C₂×C₄	#13063347	1	2	yes	—
364	C₂×C₄	#13063349	1	1	yes	characteristic
365	C₂×C₄	#13063353	1	2	yes	—
366	C₂×C₄	#13063357	1	1	yes	characteristic
367	C₈	—	4	2	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	344	1	no	—
371	C₂²	—	344	1	no	—
372	C₂²	—	344	2	no	—
373	C₂²	—	344	1	no	—
374	C₂²	—	172	4	no	—
375	C₂²	—	172	2	no	—
376	C₂²	#14582809	1	1	yes	characteristic
377	C₂²	#14583115	1	2	yes	—
378	C₂²	#14583129	1	1	yes	characteristic
379	C₂²	#14583146	1	1	yes	characteristic
380	C₂²	#14583175	1	2	yes	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	344	2	no	—
384	C₄	—	344	1	no	—
385	C₄	—	344	2	no	—
386	C₄	#14583039	1	1	yes	characteristic
387	C₄	#14583045	1	1	yes	characteristic
388	C₄	#14583046	1	2	yes	—
389	C₄	#14583051	1	1	yes	characteristic
390	C₄	#14583064	1	1	yes	characteristic
391	C₄	#14583283	1	1	yes	characteristic
392	C₄	#14583288	1	1	yes	characteristic
393	C₂	—	344	2	no	—
394	C₂	#19134019	1	2	yes	—
395	C₂	#19134143	1	2	yes	—
396	C₂	#19135601	1	1	yes	characteristic
397	C₂	#19135675	1	1	yes	characteristic
398	C₂	#19135717	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063300
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1277	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂²×D₁₇₂	D₄
hypercenter	#1277	D₄₃
hypocenter	C₄₃	#10270
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
1. C₂×#14583250	C₂	semidirect
5. C₂×#14582646	C₂	semidirect
6. C₂×#14582644	C₂	semidirect
7. C₄₃×#1277	C₂	semidirect
200. #1277	D₄₃	semidirect
228. C₄₃	#10270	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583250	2
C₂×#14583180	2
C₂×#14583105	2
#19135389	2
C₂×#14582646	2
C₂×#14582644	2
C₄₃×#1277	2
#10270	43