Subgroups — #19149201

Order 5504.

912 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	C₄₃⋊#564	C₂	1	2	yes	maximal, complemented
2	#14583653	C₂	1	2	yes	maximal, complemented
3	#14583626	C₂	1	2	yes	maximal, complemented
4	#14583604	C₂	1	2	yes	maximal, complemented
5	#14583587	C₂	1	2	yes	maximal, complemented
6	#14583584	C₂	1	4	yes	maximal, complemented
7	#14583584	C₂	1	2	yes	maximal, complemented
8	#14583530	C₂	1	2	yes	maximal, complemented
9	#14583529	C₂	1	2	yes	maximal, complemented
10	#14583512	C₂	1	2	yes	maximal, complemented
11	#14583508	C₂	1	4	yes	maximal, complemented
12	#14583496	C₂	1	2	yes	maximal
13	#14583456	C₂	1	1	yes	maximal, characteristic, complemented
14	#14583433	C₂	1	1	yes	maximal, characteristic
15	C₂×#125×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#13063325	C₂²	1	1	yes	characteristic
17	#13063325	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	2	yes	—
19	#13063324	C₂²	1	4	yes	—
20	#13063324	C₂²	1	4	yes	—
21	#13063320	C₂²	1	4	yes	—
22	#13063320	C₂²	1	4	yes	—
23	#13063320	C₂²	1	4	yes	—
24	#13063320	C₂²	1	4	yes	—
25	#13063319	C₂²	1	2	yes	—
26	#13063317	C₂²	1	2	yes	—
27	#13063317	C₂²	1	1	yes	characteristic
28	#13063316	C₂²	1	2	yes	—
29	#13063316	C₂²	1	4	yes	—
30	#13063316	C₂²	1	4	yes	—
31	#13063316	C₂²	1	4	yes	—
32	#13063315	C₂²	1	4	yes	—
33	#13063315	C₂²	1	4	yes	—
34	#13063315	C₂²	1	4	yes	—
35	#13063315	C₂²	1	4	yes	—
36	#13063315	C₂²	1	4	yes	—
37	#13063313	C₂²	1	1	yes	characteristic
38	#13063310	C₂²	1	4	yes	—
39	#13063308	C₂²	1	2	yes	—
40	#13063308	C₂²	1	4	yes	—
41	#13063305	C₂²	1	2	yes	—
42	#13063305	C₂²	1	4	yes	—
43	#13063305	C₂²	1	2	yes	—
44	#13063304	C₂²	1	2	yes	—
45	#13063304	C₂²	1	4	yes	—
46	#13063304	C₂²	1	1	yes	characteristic
47	#13063304	C₂²	1	2	yes	—
48	#13063303	C₂²	1	4	yes	—
49	#13063303	C₂²	1	4	yes	—
50	#13063299	C₂²	1	1	yes	characteristic
51	C₄×Dic₈₆	C₂²	1	2	yes	—
52	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
53	C₄×Dic₈₆	C₂²	1	2	yes	—
54	#13063290	C₂²	1	4	yes	—
55	#13063290	C₂²	1	2	yes	—
56	#13063290	C₂²	1	2	yes	—
57	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
58	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
59	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
60	#13063284	C₂²	1	2	yes	—
61	#13063284	C₂²	1	4	yes	—
62	#13063283	C₂²	1	4	yes	—
63	#13063283	C₂²	1	2	yes	—
64	#13063282	C₂²	1	2	yes	—
65	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
66	#125×C₄₃	C₂²	1	2	yes	—
67	#125×C₄₃	C₂²	1	2	yes	—
68	#125×C₄₃	C₂²	1	2	yes	—
69	#125×C₄₃	C₂²	1	2	yes	—
70	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
72	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
74	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
80	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
81	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
82	C₂×C₄×D₄₃	C₂³	1	2	yes	—
83	C₂×C₄×D₄₃	C₂³	1	4	yes	—
84	C₂×Dic₈₆	C₂³	1	2	yes	—
85	C₂×Dic₈₆	C₂³	1	4	yes	—
86	C₂×Dic₈₆	C₂³	1	2	yes	—
87	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
89	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
90	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
91	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
92	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
93	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
94	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
95	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
96	C₄²×C₄₃	C₂³	1	1	yes	characteristic
97	C₄²×C₄₃	C₂³	1	1	yes	characteristic
98	C₄²×C₄₃	C₂³	1	1	yes	characteristic
99	C₄²×C₄₃	C₂³	1	1	yes	characteristic
100	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
101	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
102	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
103	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
104	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
105	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
106	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
107	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
108	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
109	C₄⋊Dic₄₃	C₂³	1	2	yes	—
110	C₄⋊Dic₄₃	C₂³	1	4	yes	—
111	C₄⋊Dic₄₃	C₂³	1	2	yes	—
112	C₄⋊Dic₄₃	C₂³	1	4	yes	—
113	C₄⋊Dic₄₃	C₂³	1	4	yes	—
114	C₄⋊Dic₄₃	C₂³	1	2	yes	—
115	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
116	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
117	C₄⋊Dic₄₃	C₂³	1	2	yes	—
118	Dic₄₃⋊C₄	C₂³	1	4	yes	—
119	Dic₄₃⋊C₄	C₂³	1	2	yes	—
120	Dic₄₃⋊C₄	C₂³	1	2	yes	—
121	Dic₄₃⋊C₄	C₂³	1	4	yes	—
122	Dic₄₃⋊C₄	C₂³	1	2	yes	—
123	Dic₄₃⋊C₄	C₂³	1	2	yes	—
124	Dic₄₃⋊C₄	C₂³	1	4	yes	—
125	Dic₄₃⋊C₄	C₂³	1	4	yes	—
126	Dic₄₃⋊C₄	C₂³	1	4	yes	—
127	Dic₄₃⋊C₄	C₂³	1	4	yes	—
128	Dic₄₃⋊C₄	C₂³	1	4	yes	—
129	C₄×Dic₄₃	C₂³	1	2	yes	—
130	C₄×Dic₄₃	C₂³	1	4	yes	—
131	C₄×Dic₄₃	C₂³	1	4	yes	—
132	C₄×Dic₄₃	C₂³	1	4	yes	—
133	C₄×Dic₄₃	C₂³	1	4	yes	—
134	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
135	C₂²×D₄₃	C₂⁴	1	2	yes	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	2	4	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
143	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
144	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
149	C₂²⋊D₄₃	—	2	4	no	—
150	C₂×Dic₄₃	C₂⁴	1	2	yes	—
151	C₂×Dic₄₃	C₂⁴	1	4	yes	—
152	C₂×Dic₄₃	C₂⁴	1	2	yes	—
153	C₂×Dic₄₃	C₂⁴	1	2	yes	—
154	C₂×Dic₄₃	C₂⁴	1	4	yes	—
155	D₁₇₂	—	4	1	no	—
156	C₄×D₄₃	—	2	4	no	—
157	C₄×D₄₃	—	4	4	no	—
158	Dic₈₆	—	2	4	no	—
159	Dic₈₆	—	4	1	no	—
160	Dic₈₆	—	4	4	no	—
161	Dic₈₆	—	4	2	no	—
162	C₂²×C₄₃	—	2	1	no	—
163	C₂²×C₄₃	—	2	1	no	—
164	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
165	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	2	2	no	—
168	C₂×D₄₃	—	2	2	no	—
169	C₄×C₄₃	—	2	1	no	—
170	C₄×C₄₃	—	2	4	no	—
171	C₄×C₄₃	—	2	1	no	—
172	C₄×C₄₃	—	2	1	no	—
173	C₄×C₄₃	—	2	1	no	—
174	C₄×C₄₃	—	2	4	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
177	Dic₄₃	—	2	4	no	—
178	Dic₄₃	—	2	2	no	—
179	Dic₄₃	—	2	2	no	—
180	Dic₄₃	—	2	2	no	—
181	Dic₄₃	—	2	4	no	—
182	#3563	—	43	1	no	maximal, Sylow
183	C₂×C₄₃	—	2	2	no	—
184	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
186	C₂×C₄₃	#585	1	1	yes	characteristic
187	D₄₃	—	4	2	no	—
188	#564	—	43	2	no	—
189	#563	—	43	2	no	—
190	#557	—	43	2	no	—
191	#554	—	43	2	no	—
192	#552	—	43	2	no	—
193	#552	—	43	4	no	—
194	#552	—	43	2	no	—
195	#544	—	43	2	no	—
196	#544	—	43	2	no	—
197	#541	—	43	4	no	—
198	#541	—	43	2	no	—
199	#539	—	43	2	no	—
200	#533	—	43	1	no	—
201	#529	—	43	1	no	—
202	C₂×#125	D₄₃	1	1	yes	characteristic, complemented
203	C₄₃	#3563	1	1	yes	characteristic, complemented, Sylow
204	C₄⋊Q₈	—	43	4	no	—
205	C₄⋊Q₈	—	43	2	no	—
206	C₄⋊Q₈	—	43	4	no	—
207	C₄⋊Q₈	—	43	1	no	—
208	C₄⋊Q₈	—	43	1	no	—
209	#126	—	43	4	no	—
210	#126	—	43	4	no	—
211	#126	—	43	4	no	—
212	#126	—	43	4	no	—
213	#126	—	43	2	no	—
214	#125	—	43	4	no	—
215	#125	—	43	4	no	—
216	#125	—	43	4	no	—
217	#125	—	43	4	no	—
218	#125	—	43	4	no	—
219	#125	—	43	4	no	—
220	#125	—	43	2	no	—
221	#125	—	43	4	no	—
222	#125	—	43	4	no	—
223	#125	—	43	1	no	—
224	#125	—	43	2	no	—
225	#125	C₂×D₄₃	1	2	yes	—
226	#125	C₂×D₄₃	1	2	yes	—
227	#125	C₂×D₄₃	1	2	yes	—
228	#125	C₂×D₄₃	1	2	yes	—
229	#124	—	43	1	no	—
230	#123	—	43	2	no	—
231	#123	—	43	4	no	—
232	#123	—	43	4	no	—
233	C₂²⋊Q₈	—	43	1	no	—
234	C₂²⋊Q₈	—	43	2	no	—
235	C₂²⋊Q₈	—	43	2	no	—
236	C₂²⋊Q₈	—	43	4	no	—
237	C₂²⋊Q₈	—	43	4	no	—
238	C₂²⋊Q₈	—	43	2	no	—
239	C₂²⋊Q₈	—	43	2	no	—
240	C₂²⋊Q₈	—	43	4	no	—
241	C₂²⋊Q₈	—	43	4	no	—
242	#121	—	43	1	no	—
243	C₄×Q₈	—	43	4	no	—
244	C₄×Q₈	—	43	2	no	—
245	C₄×Q₈	—	43	2	no	—
246	C₄×Q₈	—	43	1	no	—
247	C₄×Q₈	—	43	2	no	—
248	C₄×Q₈	—	43	2	no	—
249	C₄×D₄	—	43	2	no	—
250	C₄×D₄	—	43	4	no	—
251	C₄×D₄	—	43	1	no	—
252	#117	—	43	2	no	—
253	#117	—	43	4	no	—
254	#117	—	43	4	no	—
255	#117	—	43	2	no	—
256	#117	—	43	2	no	—
257	C₂×C₄⋊C₄	—	43	2	no	—
258	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
261	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
262	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
263	C₂×Q₈	—	43	2	no	—
264	C₂×Q₈	—	43	4	no	—
265	C₂×Q₈	—	43	2	no	—
266	C₂×Q₈	—	43	1	no	—
267	C₂×D₄	—	43	2	no	—
268	C₂×D₄	—	43	1	no	—
269	C₂²×C₄	—	43	2	no	—
270	C₂²×C₄	—	43	4	no	—
271	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
274	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
276	C₄⋊C₄	—	43	2	no	—
277	C₄⋊C₄	—	43	4	no	—
278	C₄⋊C₄	—	43	4	no	—
279	C₄⋊C₄	—	43	2	no	—
280	C₄⋊C₄	—	43	4	no	—
281	C₄⋊C₄	—	43	2	no	—
282	C₄⋊C₄	—	43	4	no	—
283	C₄⋊C₄	—	43	2	no	—
284	C₄⋊C₄	—	43	2	no	—
285	C₄⋊C₄	—	43	1	no	—
286	C₄⋊C₄	—	43	4	no	—
287	C₄⋊C₄	—	43	2	no	—
288	C₄⋊C₄	—	43	2	no	—
289	C₄⋊C₄	—	43	1	no	—
290	C₄⋊C₄	—	43	4	no	—
291	C₄⋊C₄	—	43	4	no	—
292	C₄⋊C₄	—	43	4	no	—
293	C₄⋊C₄	—	43	4	no	—
294	C₄⋊C₄	—	43	2	no	—
295	C₄⋊C₄	—	43	4	no	—
296	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
297	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
298	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
299	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
300	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
301	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
302	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
303	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
304	C₂²⋊C₄	—	43	4	no	—
305	C₂²⋊C₄	—	43	2	no	—
306	C₂²⋊C₄	—	43	2	no	—
307	C₂²⋊C₄	—	43	2	no	—
308	C₂²⋊C₄	—	43	4	no	—
309	C₂²⋊C₄	—	43	2	no	—
310	C₂²⋊C₄	—	43	4	no	—
311	C₂²⋊C₄	—	43	4	no	—
312	C₂²⋊C₄	—	43	2	no	—
313	C₄²	—	43	4	no	—
314	C₄²	—	43	4	no	—
315	C₄²	—	43	2	no	—
316	C₄²	—	43	4	no	—
317	C₄²	—	43	4	no	—
318	C₄²	C₂²×D₄₃	1	1	yes	characteristic
319	C₄²	C₂²×D₄₃	1	1	yes	characteristic
320	C₄²	C₂²×D₄₃	1	1	yes	characteristic
321	C₄²	C₂²×D₄₃	1	1	yes	characteristic
322	C₂³	—	43	2	no	—
323	C₂³	C₂³×D₄₃	1	1	yes	characteristic
324	Q₈	—	86	4	no	—
325	Q₈	—	172	4	no	—
326	Q₈	—	172	1	no	—
327	Q₈	—	172	2	no	—
328	D₄	—	86	4	no	—
329	D₄	—	172	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	43	4	no	—
336	C₂×C₄	—	43	2	no	—
337	C₂×C₄	—	43	2	no	—
338	C₂×C₄	—	43	2	no	—
339	C₂×C₄	—	43	4	no	—
340	C₂×C₄	—	86	4	no	—
341	C₂×C₄	—	172	4	no	—
342	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
343	C₂×C₄	C₂³×D₄₃	1	4	yes	—
344	C₂×C₄	C₂³×D₄₃	1	4	yes	—
345	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
346	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
347	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
349	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	86	2	no	—
353	C₂²	—	86	2	no	—
354	C₂²	—	86	2	no	—
355	C₂²	#13063372	1	2	yes	—
356	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	4	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	4	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	86	2	no	—
365	C₄	—	86	2	no	—
366	C₄	—	86	4	no	—
367	C₄	—	86	4	no	—
368	C₄	—	86	2	no	—
369	C₄	#13063373	1	2	yes	—
370	C₂	—	2	2	no	—
371	C₂	—	172	2	no	—
372	C₂	C₂×#13063372	1	1	yes	characteristic
373	C₂	C₂×#13063373	1	1	yes	characteristic
374	C₂	#14583735	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂×#125×C₄₃	C₂
hypercenter	C₂×#125	D₄₃
hypocenter	C₄₃	#3563
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₄₃⋊#564	C₂	semidirect
2. #14583653	C₂	semidirect
3. #14583626	C₂	semidirect
4. #14583604	C₂	semidirect
5. #14583587	C₂	semidirect
6. #14583584	C₂	semidirect
7. #14583584	C₂	semidirect
8. #14583530	C₂	semidirect
9. #14583529	C₂	semidirect
10. #14583512	C₂	semidirect
11. #14583508	C₂	semidirect
13. #14583456	C₂	semidirect
15. C₂×#125×C₄₃	C₂	semidirect
202. C₂×#125	D₄₃	semidirect
203. C₄₃	#3563	semidirect