Subgroups — #24995684

Order 11008.

610 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142953	C₂	1	1	yes	maximal, characteristic
2	#19134746	C₂	1	1	yes	maximal, characteristic, complemented
3	#19134450	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582645	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582642	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582642	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063322	C₂²	1	1	yes	characteristic
9	C₂×#13063314	—	2	1	no	—
10	C₂×#13063314	—	2	1	no	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₈	—	2	2	no	—
14	C₂×#13063235	—	2	1	no	—
15	#14582800	—	2	2	no	—
16	#14582800	C₄	1	4	yes	—
17	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582653	—	2	1	no	—
19	#14582645	—	2	2	no	—
20	#14582642	—	2	2	no	—
21	#14582642	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#332	—	2	2	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
29	C₂²×C₄₃⋊C₈	—	2	1	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	#13063322	—	2	1	no	—
32	#13063322	—	2	1	no	—
33	#13063322	—	2	1	no	—
34	#13063322	—	2	1	no	—
35	#13063314	—	4	2	no	—
36	#13063314	—	4	2	no	—
37	#13063314	—	4	2	no	—
38	#13063314	—	4	2	no	—
39	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
41	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
43	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
49	Dic₄₃⋊C₈	—	4	4	no	—
50	#13063235	—	4	2	no	—
51	C₂²×Q₈×C₄₃	—	2	1	no	—
52	C₂²×C₈×C₄₃	—	2	1	no	—
53	C₂²×C₈×C₄₃	—	2	1	no	—
54	C₄⋊Q₈×C₄₃	—	2	1	no	—
55	C₄⋊Q₈×C₄₃	—	2	1	no	—
56	C₄⋊Q₈×C₄₃	—	2	1	no	—
57	C₄⋊Q₈×C₄₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₃	—	2	1	no	—
62	C₄⋊C₈×C₄₃	—	2	1	no	—
63	C₄⋊C₈×C₄₃	—	2	1	no	—
64	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
65	C₂³×D₄₃	—	2	2	no	—
66	C₂×Q₈×C₄₃	—	4	1	no	—
67	C₂×Q₈×C₄₃	—	4	1	no	—
68	C₂×Q₈×C₄₃	—	4	1	no	—
69	C₂×Q₈×C₄₃	—	4	1	no	—
70	C₂×Q₈×C₄₃	—	4	1	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂²×C₄×C₄₃	—	2	2	no	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
77	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
78	C₂×D₁₇₂	—	2	2	no	—
79	C₂×D₁₇₂	—	2	2	no	—
80	C₂×D₁₇₂	—	4	1	no	—
81	C₂×D₁₇₂	—	4	1	no	—
82	C₂×D₁₇₂	—	4	1	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	C₂×D₁₇₂	—	4	2	no	—
89	C₂×D₁₇₂	—	4	2	no	—
90	C₂×D₁₇₂	—	4	4	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	4	1	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	—	4	1	no	—
95	C₂×C₈×C₄₃	—	4	2	no	—
96	C₄⋊C₄×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	4	1	no	—
99	C₄⋊C₄×C₄₃	—	4	1	no	—
100	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₄₃	#41	1	1	yes	characteristic
103	C₄²×C₄₃	#41	1	1	yes	characteristic
104	(C₂×D₄₃)⋊C₄	—	4	8	no	—
105	(C₂×D₄₃)⋊C₄	—	4	8	no	—
106	C₄×Dic₄₃	—	2	2	no	—
107	C₄×Dic₄₃	—	2	1	no	—
108	C₄×Dic₄₃	—	2	1	no	—
109	C₄×Dic₄₃	—	2	2	no	—
110	C₄×Dic₄₃	—	4	2	no	—
111	C₄×Dic₄₃	D₈	1	2	yes	—
112	C₄×Dic₄₃	SD₁₆	1	2	yes	—
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	—	4	1	no	—
115	C₂×C₄₃⋊C₈	—	4	1	no	—
116	C₂×C₄₃⋊C₈	—	4	1	no	—
117	C₂×C₄₃⋊C₈	—	4	2	no	—
118	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×D₄₃	—	4	2	no	—
120	C₂²×D₄₃	—	4	4	no	—
121	C₂²×D₄₃	—	4	2	no	—
122	C₂²×D₄₃	—	4	4	no	—
123	C₂²×D₄₃	—	4	2	no	—
124	Q₈×C₄₃	—	8	1	no	—
125	Q₈×C₄₃	—	8	1	no	—
126	Q₈×C₄₃	—	8	1	no	—
127	Q₈×C₄₃	—	8	1	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	4	2	no	—
130	C₂×C₄×C₄₃	—	4	2	no	—
131	C₂×C₄×C₄₃	—	4	2	no	—
132	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₄₃	#137	1	2	yes	—
144	C₂×Dic₄₃	—	2	2	no	—
145	C₂×Dic₄₃	—	2	2	no	—
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	#102	1	4	yes	—
150	D₁₇₂	—	8	2	no	—
151	D₁₇₂	—	8	2	no	—
152	D₁₇₂	—	8	1	no	—
153	D₁₇₂	—	8	1	no	—
154	D₁₇₂	—	8	1	no	—
155	D₁₇₂	—	8	1	no	—
156	D₁₇₂	—	8	2	no	—
157	D₁₇₂	—	8	1	no	—
158	D₁₇₂	—	8	1	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₈×C₄₃	—	8	1	no	—
161	C₄₃⋊C₈	—	4	2	no	—
162	C₄₃⋊C₈	—	8	1	no	—
163	#9704	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	#390	1	1	yes	characteristic
165	C₂²×C₄₃	#413	1	2	yes	—
166	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
167	C₂²×C₄₃	#417	1	2	yes	—
168	C₂²×C₄₃	#418	1	1	yes	characteristic
169	C₂×D₄₃	—	8	4	no	—
170	C₂×D₄₃	—	8	4	no	—
171	C₂×D₄₃	—	8	2	no	—
172	C₂×D₄₃	—	8	2	no	—
173	C₂×D₄₃	—	8	2	no	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	8	2	no	—
177	C₄×C₄₃	#459	1	1	yes	characteristic
178	C₄×C₄₃	#459	1	1	yes	characteristic
179	C₄×C₄₃	#460	1	1	yes	characteristic
180	C₄×C₄₃	#460	1	1	yes	characteristic
181	C₄×C₄₃	#489	1	2	yes	—
182	C₄×C₄₃	#496	1	1	yes	characteristic
183	C₄×C₄₃	#497	1	1	yes	characteristic
184	Dic₄₃	—	2	4	no	—
185	Dic₄₃	—	4	2	no	—
186	#2858	—	43	1	no	—
187	#1842	—	43	1	no	—
188	#1767	—	43	1	no	—
189	C₂×#332	—	43	1	no	—
190	C₂×#332	D₄₃	1	1	yes	characteristic, complemented
191	C₂×#331	—	43	1	no	—
192	C₂×#331	—	43	1	no	—
193	C₂×C₄₃	#1547	1	2	yes	—
194	C₂×C₄₃	#1557	1	2	yes	—
195	C₂×C₄₃	#1959	1	1	yes	characteristic
196	C₂×C₄₃	#1967	1	1	yes	characteristic
197	C₂×C₄₃	#1982	1	1	yes	characteristic
198	D₄₃	—	16	2	no	—
199	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
200	C₂×#127	—	43	1	no	—
201	C₂×#124	—	86	1	no	—
202	C₂×#124	—	86	1	no	—
203	C₂×C₄⋊C₈	—	43	1	no	—
204	C₂×C₄⋊C₈	—	43	1	no	—
205	C₂×C₄⋊C₈	—	86	2	no	—
206	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
207	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
208	C₂×#97	—	86	1	no	—
209	#390	—	43	4	no	—
210	#390	—	86	2	no	—
211	C₄³	—	43	1	no	—
212	#336	—	86	1	no	—
213	#332	—	2	2	no	—
214	#332	—	86	2	no	—
215	#331	—	86	2	no	—
216	#331	—	86	2	no	—
217	C₄₃	#9704	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	2	1	no	—
219	C₂²×D₄	—	43	2	no	—
220	C₂²×D₄	—	86	1	no	—
221	C₂²×D₄	—	86	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	86	1	no	—
225	C₂²×C₈	—	86	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	#127	—	86	1	no	—
231	#127	—	86	1	no	—
232	#127	—	86	1	no	—
233	#127	—	86	1	no	—
234	#124	—	172	2	no	—
235	#124	—	172	2	no	—
236	#124	—	172	2	no	—
237	#124	—	172	2	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
240	C₂×C₂²⋊C₄	—	86	4	no	—
241	C₂×C₂²⋊C₄	—	86	4	no	—
242	C₂×C₄²	—	43	1	no	—
243	C₂×C₄²	—	43	1	no	—
244	C₂×C₄²	—	43	2	no	—
245	C₂×C₄²	—	43	2	no	—
246	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	43	2	no	—
251	C₄⋊C₈	—	86	1	no	—
252	C₄⋊C₈	—	86	1	no	—
253	C₄⋊C₈	—	86	1	no	—
254	C₄⋊C₈	—	172	4	no	—
255	C₄⋊C₈	C₂²⋊D₄₃	1	2	yes	—
256	#97	—	172	2	no	—
257	C₂⁴	—	86	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×D₄	—	86	2	no	—
265	C₂×D₄	—	86	2	no	—
266	C₂×D₄	—	172	4	no	—
267	C₂×D₄	—	172	1	no	—
268	C₂×D₄	—	172	1	no	—
269	C₂×D₄	—	172	1	no	—
270	C₂×D₄	—	172	1	no	—
271	C₂×D₄	—	172	1	no	—
272	C₂×D₄	—	172	1	no	—
273	C₂×D₄	—	172	1	no	—
274	C₂×D₄	—	172	1	no	—
275	C₂×D₄	—	172	2	no	—
276	C₂×D₄	—	172	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	43	2	no	—
279	C₂²×C₄	—	43	2	no	—
280	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	—	86	2	no	—
289	C₂×C₈	—	172	1	no	—
290	C₂×C₈	—	172	2	no	—
291	C₂×C₈	—	172	1	no	—
292	C₂×C₈	—	172	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	172	8	no	—
298	C₂²⋊C₄	—	172	8	no	—
299	C₄²	—	43	2	no	—
300	C₄²	—	43	2	no	—
301	C₄²	—	86	1	no	—
302	C₄²	—	86	2	no	—
303	C₄²	—	86	1	no	—
304	C₄²	—	86	2	no	—
305	C₄²	—	172	2	no	—
306	C₄²	#10621898	1	1	yes	characteristic
307	C₄²	D₄×D₄₃	1	1	yes	characteristic
308	C₄²	#10621900	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
310	C₂³	—	172	2	no	—
311	C₂³	—	172	4	no	—
312	C₂³	—	172	2	no	—
313	C₂³	—	172	4	no	—
314	C₂³	—	172	2	no	—
315	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
316	Q₈	—	8	1	no	—
317	Q₈	—	8	1	no	—
318	Q₈	—	8	1	no	—
319	Q₈	—	8	1	no	—
320	D₄	—	344	2	no	—
321	D₄	—	344	2	no	—
322	D₄	—	344	1	no	—
323	D₄	—	344	1	no	—
324	D₄	—	344	1	no	—
325	D₄	—	344	1	no	—
326	D₄	—	344	2	no	—
327	D₄	—	344	1	no	—
328	D₄	—	344	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	43	4	no	—
334	C₂×C₄	—	86	2	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	86	2	no	—
337	C₂×C₄	—	86	2	no	—
338	C₂×C₄	—	86	2	no	—
339	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
340	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
341	C₂×C₄	#13063297	1	1	yes	characteristic
342	C₂×C₄	#13063298	1	1	yes	characteristic
343	C₂×C₄	#13063311	1	1	yes	characteristic
344	C₂×C₄	#13063321	1	1	yes	characteristic
345	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
346	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
347	C₂×C₄	#13063349	1	1	yes	characteristic
348	C₂×C₄	#13063352	1	1	yes	characteristic
349	C₂×C₄	#13063352	1	2	yes	—
350	C₂×C₄	#13063357	1	2	yes	—
351	C₈	—	4	2	no	—
352	C₈	—	8	1	no	—
353	C₈	—	344	1	no	—
354	C₈	—	172	2	no	—
355	C₂²	—	344	2	no	—
356	C₂²	—	344	4	no	—
357	C₂²	—	344	4	no	—
358	C₂²	—	344	2	no	—
359	C₂²	—	344	2	no	—
360	C₂²	#14582799	1	1	yes	characteristic
361	C₂²	#14582882	1	2	yes	—
362	C₂²	#102×D₄₃	1	1	yes	characteristic
363	C₂²	#14582899	1	2	yes	—
364	C₂²	#14582902	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	86	4	no	—
369	C₄	—	172	2	no	—
370	C₄	#14583092	1	1	yes	characteristic
371	C₄	#14583093	1	1	yes	characteristic
372	C₄	#14583097	1	1	yes	characteristic
373	C₄	#14583098	1	1	yes	characteristic
374	C₄	#14583260	1	2	yes	—
375	C₄	#14583285	1	1	yes	characteristic
376	C₄	#14583292	1	1	yes	characteristic
377	C₂	—	688	2	no	—
378	C₂	#19133271	1	2	yes	—
379	C₂	#19133316	1	2	yes	—
380	C₂	#19135408	1	1	yes	characteristic
381	C₂	#19135443	1	1	yes	characteristic
382	C₂	#19135520	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#332	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#332	D₄₃
hypocenter	C₄₃	#9704
LCS	G ▷ C₂²×C₄×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
2. #19134746	C₂	semidirect
3. #19134450	C₂	semidirect
4. C₂×#14582645	C₂	semidirect
7. C₂×C₄₃×#332	C₂	semidirect
190. C₂×#332	D₄₃	semidirect
217. C₄₃	#9704	semidirect

Maximal subgroups

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

subgroup	index
#19142953	2
#19134746	2
#19134450	2
C₂×#14582645	2
C₂×#14582642	2
C₂×C₄₃×#332	2
#9704	43