Subgroups — #19138966

Order 5504.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063324	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063302	C₂	1	1	yes	maximal, characteristic
3	#14582841	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582838	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582811	C₂	1	1	yes	maximal, characteristic
6	#14582811	C₂	1	1	yes	maximal, characteristic
7	#14582805	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582803	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582803	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582803	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582792	C₂	1	1	yes	maximal, characteristic
12	#14582757	C₂	1	1	yes	maximal, characteristic
13	C₄×Dic₄₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#388	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
18	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
19	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic
20	#13063324	—	2	4	no	—
21	#13063302	—	2	4	no	—
22	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₄₃⋊#95	C₂²	1	1	yes	characteristic
35	C₄₃⋊#95	C₂²	1	1	yes	characteristic
36	C₄₃⋊#95	C₂²	1	1	yes	characteristic
37	C₄₃⋊#95	C₂²	1	1	yes	characteristic
38	C₄₃⋊#95	C₂²	1	1	yes	characteristic
39	C₄₃⋊#95	C₂²	1	1	yes	characteristic
40	C₄₃⋊#95	C₂²	1	1	yes	characteristic
41	C₄₃⋊#95	C₂²	1	1	yes	characteristic
42	C₄₃⋊#95	C₂²	1	1	yes	characteristic
43	C₄₃⋊#95	C₂²	1	1	yes	characteristic
44	C₄₃⋊#95	C₂²	1	1	yes	characteristic
45	C₄₃⋊#95	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
51	#95×C₄₃	C₂²	1	1	yes	characteristic
52	#95×C₄₃	C₂²	1	1	yes	characteristic
53	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	2	2	no	—
61	C₂²×Dic₄₃	—	2	2	no	—
62	C₂²×Dic₄₃	—	2	2	no	—
63	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
67	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
68	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
70	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	—	2	2	no	—
73	C₂×Dic₈₆	—	2	2	no	—
74	C₂×Dic₈₆	—	2	2	no	—
75	C₂×Dic₈₆	—	2	2	no	—
76	C₂×Dic₈₆	—	2	2	no	—
77	C₂×Dic₈₆	—	2	2	no	—
78	C₂×Dic₈₆	—	4	1	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₄⋊C₄×C₄₃	—	2	2	no	—
81	C₂²⋊C₄×C₄₃	—	2	2	no	—
82	C₂²⋊C₄×C₄₃	—	4	1	no	—
83	C₂²⋊C₄×C₄₃	Q₈	1	4	yes	complemented
84	C₄²×C₄₃	—	4	1	no	—
85	C₂²⋊Dic₄₃	—	2	2	no	—
86	C₂²⋊Dic₄₃	—	2	2	no	—
87	C₂²⋊Dic₄₃	—	4	1	no	—
88	C₄⋊Dic₄₃	—	2	2	no	—
89	C₄⋊Dic₄₃	—	2	2	no	—
90	Dic₄₃⋊C₄	—	2	2	no	—
91	Dic₄₃⋊C₄	—	2	2	no	—
92	Dic₄₃⋊C₄	—	2	2	no	—
93	Dic₄₃⋊C₄	D₄	1	4	yes	—
94	C₄×Dic₄₃	—	2	2	no	—
95	C₄×Dic₄₃	—	2	2	no	—
96	C₄×Dic₄₃	—	4	1	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	—	2	1	no	—
99	C₂³×C₄₃	—	2	1	no	—
100	C₂³×C₄₃	—	2	1	no	—
101	C₂³×C₄₃	—	2	1	no	—
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
104	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	C₂×Dic₄₃	—	2	1	no	—
131	C₂×Dic₄₃	—	2	1	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	2	1	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	4	4	no	—
143	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
144	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
145	C₂×Dic₄₃	#41	1	2	yes	—
146	C₂×Dic₄₃	#41	1	2	yes	—
147	C₂×Dic₄₃	#41	1	2	yes	—
148	Dic₈₆	—	4	4	no	—
149	Dic₈₆	—	4	4	no	—
150	Dic₈₆	—	4	4	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	—	4	1	no	—
155	C₂²×C₄₃	—	4	1	no	—
156	C₂²×C₄₃	—	4	1	no	—
157	C₂²×C₄₃	—	4	1	no	—
158	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
163	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	4	1	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	—	4	1	no	—
171	Dic₄₃	—	2	2	no	—
172	Dic₄₃	—	2	2	no	—
173	Dic₄₃	—	4	1	no	—
174	Dic₄₃	—	4	1	no	—
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	—	4	1	no	—
177	Dic₄₃	—	4	1	no	—
178	Dic₄₃	#124	1	4	yes	—
179	#2492	—	43	1	no	maximal, Sylow
180	C₂×C₄₃	—	4	2	no	—
181	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
182	C₂×C₄₃	#533	1	1	yes	characteristic
183	C₂×C₄₃	#542	1	1	yes	characteristic
184	C₂×C₄₃	#546	1	1	yes	characteristic
185	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
186	C₂×C₄₃	#554	1	1	yes	characteristic
187	C₂×C₄₃	#556	1	1	yes	characteristic
188	C₂×C₄⋊Q₈	—	43	1	no	—
189	C₂×C₂²⋊Q₈	—	43	1	no	—
190	#400	—	43	1	no	—
191	#400	—	43	1	no	—
192	#393	—	43	1	no	—
193	#393	—	43	1	no	—
194	#391	—	43	1	no	—
195	#391	—	43	1	no	—
196	#391	—	43	1	no	—
197	#391	—	43	1	no	—
198	#388	—	43	1	no	—
199	#388	D₄₃	1	1	yes	characteristic
200	#380	—	43	1	no	—
201	C₄×C₄⋊C₄	—	43	1	no	—
202	C₄×C₂²⋊C₄	—	43	1	no	—
203	C₄₃	#2492	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	43	1	no	—
205	C₂²×Q₈	—	43	1	no	—
206	C₂²×Q₈	—	43	1	no	—
207	C₂³×C₄	—	43	1	no	—
208	C₄⋊Q₈	—	86	4	no	—
209	C₂²⋊Q₈	—	86	4	no	—
210	C₂×C₄⋊C₄	—	43	1	no	—
211	C₂×C₄⋊C₄	—	43	1	no	—
212	C₂×C₄⋊C₄	—	43	1	no	—
213	C₂×C₄⋊C₄	—	43	1	no	—
214	C₂×C₄⋊C₄	—	43	1	no	—
215	C₂×C₄⋊C₄	—	43	1	no	—
216	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	—	43	1	no	—
218	C₂×C₂²⋊C₄	—	43	1	no	—
219	C₂×C₂²⋊C₄	—	43	1	no	—
220	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
223	C₂×C₄²	—	43	1	no	—
224	C₂×C₄²	—	43	1	no	—
225	C₂×C₄²	—	43	1	no	—
226	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
227	#95	—	43	1	no	—
228	#95	—	43	1	no	—
229	#95	—	43	1	no	—
230	#95	—	43	1	no	—
231	#95	—	43	1	no	—
232	#95	—	43	1	no	—
233	#95	—	43	1	no	—
234	#95	—	43	1	no	—
235	#95	—	43	1	no	—
236	#95	—	43	1	no	—
237	#95	—	43	1	no	—
238	#95	—	43	1	no	—
239	#95	C₂×D₄₃	1	1	yes	characteristic
240	#95	C₂×D₄₃	1	1	yes	characteristic
241	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
242	C₂×Q₈	—	86	2	no	—
243	C₂×Q₈	—	86	2	no	—
244	C₂×Q₈	—	86	2	no	—
245	C₂×Q₈	—	86	2	no	—
246	C₂×Q₈	—	86	2	no	—
247	C₂×Q₈	—	86	2	no	—
248	C₂×Q₈	—	86	2	no	—
249	C₂×Q₈	—	172	1	no	—
250	C₂×Q₈	—	172	1	no	—
251	C₂²×C₄	—	43	1	no	—
252	C₂²×C₄	—	43	1	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	43	1	no	—
257	C₂²×C₄	—	43	1	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	86	2	no	—
260	C₂²×C₄	—	86	2	no	—
261	C₂²×C₄	—	86	2	no	—
262	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
265	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
266	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
267	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	43	4	no	—
270	C₄⋊C₄	—	86	2	no	—
271	C₄⋊C₄	—	86	2	no	—
272	C₄⋊C₄	—	86	2	no	—
273	C₄⋊C₄	—	86	2	no	—
274	C₄⋊C₄	—	86	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₂²⋊C₄	—	86	2	no	—
278	C₂²⋊C₄	—	86	2	no	—
279	C₂²⋊C₄	—	172	1	no	—
280	C₂²⋊C₄	Dic₈₆	1	4	yes	complemented
281	C₄²	—	4	1	no	—
282	C₄²	—	86	2	no	—
283	C₄²	—	86	2	no	—
284	C₄²	—	172	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	C₂×Dic₈₆	1	2	yes	—
292	C₂³	C₂³×D₄₃	1	1	yes	characteristic
293	Q₈	—	172	4	no	—
294	Q₈	—	172	4	no	—
295	Q₈	—	172	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	43	2	no	—
312	C₂×C₄	—	43	2	no	—
313	C₂×C₄	—	43	2	no	—
314	C₂×C₄	—	43	2	no	—
315	C₂×C₄	—	43	2	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	86	1	no	—
325	C₂×C₄	—	86	1	no	—
326	C₂×C₄	—	86	1	no	—
327	C₂×C₄	—	86	1	no	—
328	C₂×C₄	—	86	1	no	—
329	C₂×C₄	—	86	1	no	—
330	C₂×C₄	—	86	1	no	—
331	C₂×C₄	—	86	1	no	—
332	C₂×C₄	—	86	1	no	—
333	C₂×C₄	—	86	1	no	—
334	C₂×C₄	—	86	1	no	—
335	C₂×C₄	—	172	4	no	—
336	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
337	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
338	C₂×C₄	D₄×D₄₃	1	2	yes	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
347	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
348	C₂²	#41×D₄₃	1	1	yes	characteristic
349	C₂²	#41×D₄₃	1	1	yes	characteristic
350	C₂²	#41×D₄₃	1	1	yes	characteristic
351	C₂²	#13063370	1	1	yes	characteristic
352	C₂²	#13063372	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	43	4	no	—
360	C₄	—	86	2	no	—
361	C₄	—	86	2	no	—
362	C₄	—	172	1	no	—
363	C₄	—	172	1	no	—
364	C₄	—	172	1	no	—
365	C₄	—	172	1	no	—
366	C₄	—	172	1	no	—
367	C₂	—	4	2	no	—
368	C₂	#124×D₄₃	1	1	yes	characteristic
369	C₂	#14583452	1	1	yes	characteristic
370	C₂	#14583517	1	1	yes	characteristic
371	C₂	#14583538	1	1	yes	characteristic
372	C₂	D₄×Dic₈₆	1	1	yes	characteristic
373	C₂	#14583608	1	1	yes	characteristic
374	C₂	#14583616	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#388	C₂
hypercenter	#388	D₄₃
hypocenter	C₄₃	#2492
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
1. C₂×#13063324	C₂	semidirect
3. #14582841	C₂	semidirect
4. #14582838	C₂	semidirect
7. #14582805	C₂	semidirect
8. #14582803	C₂	semidirect
9. #14582803	C₂	semidirect
10. #14582803	C₂	semidirect
13. C₄×Dic₄₃⋊C₄	C₂	semidirect
83. C₂²⋊C₄×C₄₃	Q₈	semidirect
203. C₄₃	#2492	semidirect
280. C₂²⋊C₄	Dic₈₆	semidirect