Subgroups — #19137149

Order 5504.

516 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063318	C₂	1	1	yes	maximal, characteristic
2	#14582793	C₂	1	1	yes	maximal, characteristic
3	#14582787	C₂	1	1	yes	maximal, characteristic
4	#14582783	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582782	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582780	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582766	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582750	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582750	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂²⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
14	C₄²×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#388	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic
17	#13063318	C₄	1	8	yes	complemented
18	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×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	C₂×C₄×Dic₄₃	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	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊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×C₄²×C₄₃	C₂²	1	1	yes	characteristic
50	#95×C₄₃	C₂²	1	1	yes	characteristic
51	#95×C₄₃	C₂²	1	1	yes	characteristic
52	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×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×Dic₄₃	—	2	2	no	—
60	C₂²×Dic₄₃	—	2	2	no	—
61	C₂²×Dic₄₃	—	2	2	no	—
62	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₄₃	—	2	2	no	—
71	C₂²⋊C₄×C₄₃	—	2	2	no	—
72	C₂²⋊C₄×C₄₃	—	2	2	no	—
73	C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
74	C₄²×C₄₃	—	2	2	no	—
75	C₂²⋊Dic₄₃	—	2	2	no	—
76	C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
77	C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
78	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	complemented
79	Dic₄₃⋊C₄	—	2	2	no	—
80	Dic₄₃⋊C₄	—	2	2	no	—
81	Dic₄₃⋊C₄	C₂×C₄	1	4	yes	complemented
82	Dic₄₃⋊C₄	C₂×C₄	1	4	yes	complemented
83	C₄×Dic₄₃	—	2	2	no	—
84	C₄×Dic₄₃	—	2	2	no	—
85	C₄×Dic₄₃	—	2	2	no	—
86	C₄×Dic₄₃	—	2	2	no	—
87	C₄×Dic₄₃	—	2	2	no	—
88	C₄×Dic₄₃	—	2	2	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	2	no	—
91	C₄×Dic₄₃	C₂×C₄	1	4	yes	complemented
92	C₂³×C₄₃	—	2	1	no	—
93	C₂³×C₄₃	—	2	1	no	—
94	C₂³×C₄₃	—	2	1	no	—
95	C₂³×C₄₃	—	2	1	no	—
96	C₂³×C₄₃	—	2	1	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
99	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₄₃	—	2	1	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
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	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₄₃	#41	1	2	yes	—
117	C₂×C₄×C₄₃	#41	1	2	yes	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
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	—	4	4	no	—
135	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
136	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
137	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
138	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
139	C₂×Dic₄₃	#41	1	2	yes	—
140	C₂×Dic₄₃	#41	1	2	yes	—
141	C₂×Dic₄₃	#41	1	2	yes	—
142	C₂×Dic₄₃	#41	1	2	yes	—
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
156	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	4	1	no	—
162	C₄×C₄₃	—	4	1	no	—
163	Dic₄₃	—	2	2	no	—
164	Dic₄₃	—	2	2	no	—
165	Dic₄₃	—	2	2	no	—
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	2	2	no	—
168	Dic₄₃	—	4	1	no	—
169	Dic₄₃	—	4	1	no	—
170	Dic₄₃	#126	1	4	yes	complemented
171	C₄×#126	—	43	1	no	maximal, Sylow
172	C₂×C₄₃	—	4	2	no	—
173	C₂×C₄₃	C₄×#41	1	1	yes	characteristic
174	C₂×C₄₃	C₄×#41	1	1	yes	characteristic
175	C₂×C₄₃	C₄×#41	1	1	yes	characteristic
176	C₂×C₄₃	C₂×#126	1	1	yes	characteristic
177	C₂×C₄₃	#529	1	1	yes	characteristic
178	C₂×C₄₃	#529	1	1	yes	characteristic
179	C₂×C₄₃	#529	1	1	yes	characteristic
180	C₂×#126	—	43	1	no	—
181	#388	—	43	1	no	—
182	#388	—	43	1	no	—
183	#388	D₄₃	1	1	yes	characteristic
184	#387	—	43	1	no	—
185	#387	—	43	1	no	—
186	#387	—	43	1	no	—
187	#383	—	43	1	no	—
188	C₄×C₄⋊C₄	—	43	1	no	—
189	C₄×C₄⋊C₄	—	43	1	no	—
190	C₄×C₄⋊C₄	—	43	1	no	—
191	C₄×C₂²⋊C₄	—	43	1	no	—
192	C₄×C₂²⋊C₄	—	43	1	no	—
193	C₄×C₂²⋊C₄	—	43	1	no	—
194	C₄³	—	43	1	no	—
195	C₄₃	C₄×#126	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	—	43	1	no	—
197	#126	—	43	8	no	—
198	C₂×C₄⋊C₄	—	43	1	no	—
199	C₂×C₄⋊C₄	—	43	1	no	—
200	C₂×C₄⋊C₄	—	43	1	no	—
201	C₂×C₄⋊C₄	—	43	1	no	—
202	C₂×C₄⋊C₄	—	43	1	no	—
203	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	43	1	no	—
205	C₂×C₂²⋊C₄	—	43	1	no	—
206	C₂×C₂²⋊C₄	—	43	1	no	—
207	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
210	C₂×C₄²	—	43	1	no	—
211	C₂×C₄²	—	43	1	no	—
212	C₂×C₄²	—	43	1	no	—
213	C₂×C₄²	—	43	1	no	—
214	C₂×C₄²	—	43	1	no	—
215	C₂×C₄²	—	43	1	no	—
216	C₂×C₄²	—	43	1	no	—
217	C₂×C₄²	—	43	1	no	—
218	C₂×C₄²	—	43	1	no	—
219	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
220	#95	—	43	1	no	—
221	#95	—	43	1	no	—
222	#95	—	43	1	no	—
223	#95	—	43	1	no	—
224	#95	—	43	1	no	—
225	#95	—	43	1	no	—
226	#95	—	43	1	no	—
227	#95	—	43	1	no	—
228	#95	—	43	1	no	—
229	#95	—	43	1	no	—
230	#95	C₂×D₄₃	1	1	yes	characteristic
231	#95	C₂×D₄₃	1	1	yes	characteristic
232	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	—	43	1	no	—
234	C₂²×C₄	—	43	1	no	—
235	C₂²×C₄	—	43	1	no	—
236	C₂²×C₄	—	43	1	no	—
237	C₂²×C₄	—	43	1	no	—
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	43	1	no	—
241	C₂²×C₄	—	86	2	no	—
242	C₂²×C₄	—	86	2	no	—
243	C₂²×C₄	—	86	2	no	—
244	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	43	4	no	—
252	C₄⋊C₄	—	43	4	no	—
253	C₄⋊C₄	—	43	4	no	—
254	C₄⋊C₄	—	86	2	no	—
255	C₄⋊C₄	—	86	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	43	4	no	—
259	C₂²⋊C₄	—	43	4	no	—
260	C₂²⋊C₄	—	86	2	no	—
261	C₂²⋊C₄	C₄×D₄₃	1	4	yes	—
262	C₄²	—	2	2	no	—
263	C₄²	—	43	4	no	—
264	C₄²	—	86	2	no	—
265	C₄²	—	86	2	no	—
266	C₄²	—	86	2	no	—
267	C₄²	—	86	2	no	—
268	C₄²	—	86	2	no	—
269	C₄²	—	86	2	no	—
270	C₄²	—	86	2	no	—
271	C₄²	—	86	2	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	C₂×C₄×D₄₃	1	2	yes	—
279	C₂³	C₂³×D₄₃	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	43	2	no	—
295	C₂×C₄	—	43	2	no	—
296	C₂×C₄	—	43	2	no	—
297	C₂×C₄	—	43	2	no	—
298	C₂×C₄	—	43	2	no	—
299	C₂×C₄	—	43	2	no	—
300	C₂×C₄	—	43	2	no	—
301	C₂×C₄	—	43	2	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	86	1	no	—
309	C₂×C₄	—	86	1	no	—
310	C₂×C₄	—	86	1	no	—
311	C₂×C₄	—	86	1	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	172	4	no	—
319	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
320	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
321	C₂×C₄	#10621900	1	2	yes	—
322	C₂×C₄	#10621900	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
331	C₂²	#41×D₄₃	1	1	yes	characteristic
332	C₂²	#41×D₄₃	1	1	yes	characteristic
333	C₂²	#41×D₄₃	1	1	yes	characteristic
334	C₂²	#41×D₄₃	1	1	yes	characteristic
335	C₂²	C₂×#10621900	1	1	yes	characteristic
336	C₂²	C₂×#10621900	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	43	4	no	—
344	C₄	—	86	2	no	—
345	C₄	—	86	2	no	—
346	C₄	—	86	2	no	—
347	C₄	—	86	2	no	—
348	C₄	—	86	2	no	—
349	C₄	—	172	1	no	—
350	C₄	—	172	1	no	—
351	C₂	—	4	2	no	—
352	C₂	C₄×#10621900	1	1	yes	characteristic
353	C₂	C₄×#10621900	1	1	yes	characteristic
354	C₂	#14583367	1	1	yes	characteristic
355	C₂	#126×D₄₃	1	1	yes	characteristic
356	C₂	#14583426	1	1	yes	characteristic
357	C₂	#14583428	1	1	yes	characteristic
358	C₂	#14583430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#388	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	#388	D₄₃
hypocenter	C₄₃	C₄×#126
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
4. #14582783	C₂	semidirect
5. #14582782	C₂	semidirect
6. #14582780	C₂	semidirect
7. #14582766	C₂	semidirect
8. C₄⋊C₄×Dic₄₃	C₂	semidirect
9. #14582750	C₂	semidirect
10. #14582750	C₂	semidirect
14. C₄²×Dic₄₃	C₂	semidirect
17. #13063318	C₄	semidirect
78. C₄⋊Dic₄₃	C₂×C₄	semidirect
81. Dic₄₃⋊C₄	C₂×C₄	semidirect
82. Dic₄₃⋊C₄	C₂×C₄	semidirect
91. C₄×Dic₄₃	C₂×C₄	semidirect
170. Dic₄₃	#126	semidirect
195. C₄₃	C₄×#126	semidirect