Subgroups — #19140908

Order 5504.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063324	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063305	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063303	C₂	1	1	yes	maximal, characteristic
5	#14582836	C₂	1	1	yes	maximal, characteristic
6	#14582831	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582829	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582820	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582820	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582812	C₂	1	1	yes	maximal, characteristic
11	#14582791	C₂	1	1	yes	maximal, characteristic
12	#14582790	C₂	1	1	yes	maximal, characteristic
13	#14582784	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582773	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#387	C₂	1	1	yes	maximal, characteristic, complemented
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	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
20	#13063325	—	2	4	no	—
21	#13063324	—	2	4	no	—
22	#13063305	—	2	4	no	—
23	#13063303	—	2	4	no	—
24	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊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×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
37	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
40	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
41	C₂×C₄×Dic₄₃	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	C₄₃⋊#95	C₂²	1	1	yes	characteristic
47	C₄₃⋊#95	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊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	#95×C₄₃	C₂²	1	1	yes	characteristic
54	#95×C₄₃	C₂²	1	1	yes	characteristic
55	C₂³×D₄₃	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×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	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	C₂×C₄×D₄₃	—	2	2	no	—
71	C₂×C₄×D₄₃	—	2	2	no	—
72	C₂×C₄×D₄₃	—	4	1	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	—	2	2	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₂×Dic₈₆	—	4	1	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	2	2	no	—
83	C₄⋊C₄×C₄₃	D₄	1	4	yes	—
84	C₄²×C₄₃	—	2	2	no	—
85	(C₂×D₄₃)⋊C₄	—	2	2	no	—
86	(C₂×D₄₃)⋊C₄	—	2	2	no	—
87	(C₂×D₄₃)⋊C₄	—	2	2	no	—
88	(C₂×D₄₃)⋊C₄	—	2	2	no	—
89	(C₂×D₄₃)⋊C₄	—	4	1	no	—
90	(C₂×D₄₃)⋊C₄	Q₈	1	4	yes	complemented
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	C₄⋊Dic₄₃	—	2	2	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	4	1	no	—
97	C₄⋊Dic₄₃	D₄	1	4	yes	—
98	Dic₄₃⋊C₄	—	2	2	no	—
99	Dic₄₃⋊C₄	—	2	2	no	—
100	Dic₄₃⋊C₄	—	2	2	no	—
101	Dic₄₃⋊C₄	—	2	2	no	—
102	C₄×Dic₄₃	—	2	2	no	—
103	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂²×D₄₃	—	2	1	no	—
105	C₂²×D₄₃	—	2	1	no	—
106	C₂²×D₄₃	—	2	1	no	—
107	C₂²×D₄₃	—	2	1	no	—
108	C₂²×D₄₃	—	2	1	no	—
109	C₂²×D₄₃	—	2	1	no	—
110	C₂²×D₄₃	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
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	—	2	1	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	C₂×Dic₄₃	—	2	1	no	—
145	C₂×Dic₄₃	—	2	1	no	—
146	C₂×Dic₄₃	—	2	1	no	—
147	C₂×Dic₄₃	—	2	1	no	—
148	C₂×Dic₄₃	—	2	1	no	—
149	C₂×Dic₄₃	—	2	1	no	—
150	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
151	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
152	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
153	C₄×D₄₃	—	4	4	no	—
154	Dic₈₆	—	4	4	no	—
155	Dic₈₆	—	4	4	no	—
156	Dic₈₆	—	4	4	no	—
157	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
158	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
159	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
160	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
161	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×D₄₃	—	2	2	no	—
165	C₂×D₄₃	—	2	2	no	—
166	C₂×D₄₃	—	2	2	no	—
167	C₂×D₄₃	—	4	1	no	—
168	C₂×D₄₃	—	4	1	no	—
169	C₂×D₄₃	—	4	1	no	—
170	C₂×D₄₃	—	4	1	no	—
171	C₄×C₄₃	—	2	2	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	2	2	no	—
174	C₄×C₄₃	—	2	2	no	—
175	C₄×C₄₃	—	2	2	no	—
176	C₄×C₄₃	—	4	1	no	—
177	C₄×C₄₃	—	4	1	no	—
178	Dic₄₃	—	2	2	no	—
179	Dic₄₃	—	2	2	no	—
180	Dic₄₃	—	2	2	no	—
181	Dic₄₃	—	4	1	no	—
182	Dic₄₃	—	4	1	no	—
183	Dic₄₃	—	4	1	no	—
184	Dic₄₃	—	4	1	no	—
185	#2677	—	43	1	no	maximal, Sylow
186	C₂×C₄₃	#536	1	1	yes	characteristic
187	C₂×C₄₃	#536	1	1	yes	characteristic
188	C₂×C₄₃	#544	1	1	yes	characteristic
189	C₂×C₄₃	D₄²	1	1	yes	characteristic
190	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
191	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
192	C₂×C₄₃	#563	1	1	yes	characteristic
193	D₄₃	—	4	2	no	—
194	C₂×C₄⋊Q₈	—	43	1	no	—
195	C₂×C₄⋊Q₈	—	43	1	no	—
196	C₂×C₂²⋊Q₈	—	43	1	no	—
197	C₂×C₂²⋊Q₈	—	43	1	no	—
198	#399	—	43	1	no	—
199	#398	—	43	1	no	—
200	#398	—	43	1	no	—
201	#395	—	43	1	no	—
202	#395	—	43	1	no	—
203	#393	—	43	1	no	—
204	#388	—	43	1	no	—
205	#388	—	43	1	no	—
206	#387	—	43	1	no	—
207	#387	D₄₃	1	1	yes	characteristic, complemented
208	#385	—	43	1	no	—
209	C₄₃	#2677	1	1	yes	characteristic, complemented, Sylow
210	C₂²×Q₈	—	43	1	no	—
211	C₂²×Q₈	—	43	1	no	—
212	C₂²×Q₈	—	43	1	no	—
213	C₂³×C₄	—	43	1	no	—
214	C₄⋊Q₈	—	86	4	no	—
215	C₄⋊Q₈	—	86	4	no	—
216	C₂²⋊Q₈	—	86	4	no	—
217	C₂²⋊Q₈	—	86	4	no	—
218	C₂×C₄⋊C₄	—	43	1	no	—
219	C₂×C₄⋊C₄	—	43	1	no	—
220	C₂×C₄⋊C₄	—	43	1	no	—
221	C₂×C₄⋊C₄	—	43	1	no	—
222	C₂×C₄⋊C₄	—	43	1	no	—
223	C₂×C₄⋊C₄	—	43	1	no	—
224	C₂×C₄⋊C₄	—	43	1	no	—
225	C₂×C₄⋊C₄	—	43	1	no	—
226	C₂×C₄⋊C₄	—	43	1	no	—
227	C₂×C₄⋊C₄	—	43	1	no	—
228	C₂×C₄⋊C₄	—	43	1	no	—
229	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	—	43	1	no	—
232	C₂×C₂²⋊C₄	—	43	1	no	—
233	C₂×C₂²⋊C₄	—	43	1	no	—
234	C₂×C₂²⋊C₄	—	43	1	no	—
235	C₂×C₂²⋊C₄	—	43	1	no	—
236	C₂×C₂²⋊C₄	—	43	1	no	—
237	C₂×C₄²	—	43	1	no	—
238	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
239	#95	—	43	1	no	—
240	#95	—	43	1	no	—
241	#95	—	43	1	no	—
242	#95	—	43	1	no	—
243	#95	—	43	1	no	—
244	#95	—	43	1	no	—
245	#95	C₂×D₄₃	1	1	yes	characteristic
246	#95	C₂×D₄₃	1	1	yes	characteristic
247	#95	C₂×D₄₃	1	1	yes	characteristic
248	#95	C₂×D₄₃	1	1	yes	characteristic
249	C₂⁴	—	43	1	no	—
250	C₂×Q₈	—	86	2	no	—
251	C₂×Q₈	—	86	2	no	—
252	C₂×Q₈	—	86	2	no	—
253	C₂×Q₈	—	86	2	no	—
254	C₂×Q₈	—	86	2	no	—
255	C₂×Q₈	—	86	2	no	—
256	C₂×Q₈	—	172	1	no	—
257	C₂×Q₈	—	172	1	no	—
258	C₂×Q₈	—	172	1	no	—
259	C₂²×C₄	—	43	1	no	—
260	C₂²×C₄	—	43	1	no	—
261	C₂²×C₄	—	43	1	no	—
262	C₂²×C₄	—	43	1	no	—
263	C₂²×C₄	—	43	1	no	—
264	C₂²×C₄	—	43	1	no	—
265	C₂²×C₄	—	43	1	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	—	86	2	no	—
268	C₂²×C₄	—	172	1	no	—
269	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
271	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	43	4	no	—
278	C₄⋊C₄	—	86	2	no	—
279	C₄⋊C₄	—	86	2	no	—
280	C₄⋊C₄	—	86	2	no	—
281	C₄⋊C₄	—	86	2	no	—
282	C₄⋊C₄	—	86	2	no	—
283	C₄⋊C₄	—	86	2	no	—
284	C₄⋊C₄	—	86	2	no	—
285	C₄⋊C₄	—	86	2	no	—
286	C₄⋊C₄	—	86	2	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₄⋊C₄	D₁₇₂	1	4	yes	—
289	C₂²⋊C₄	—	43	4	no	—
290	C₂²⋊C₄	—	86	2	no	—
291	C₂²⋊C₄	—	86	2	no	—
292	C₂²⋊C₄	—	86	2	no	—
293	C₂²⋊C₄	—	86	2	no	—
294	C₂²⋊C₄	—	172	1	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	86	2	no	—
297	C₂³	—	43	2	no	—
298	C₂³	—	86	1	no	—
299	C₂³	—	86	1	no	—
300	C₂³	—	86	1	no	—
301	C₂³	—	86	1	no	—
302	C₂³	—	86	1	no	—
303	C₂³	—	86	1	no	—
304	C₂³	C₂³×D₄₃	1	1	yes	characteristic
305	Q₈	—	172	4	no	—
306	Q₈	—	172	4	no	—
307	Q₈	—	172	4	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	43	2	no	—
325	C₂×C₄	—	43	2	no	—
326	C₂×C₄	—	43	2	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	—	86	1	no	—
336	C₂×C₄	—	86	1	no	—
337	C₂×C₄	—	86	1	no	—
338	C₂×C₄	—	86	1	no	—
339	C₂×C₄	—	86	1	no	—
340	C₂×C₄	—	86	1	no	—
341	C₂×C₄	—	86	1	no	—
342	C₂×C₄	—	86	1	no	—
343	C₂×C₄	—	86	1	no	—
344	C₂×C₄	—	86	1	no	—
345	C₂×C₄	—	172	4	no	—
346	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
347	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
348	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
349	C₂×C₄	D₄×D₄₃	1	2	yes	—
350	C₂×C₄	Q₈×D₄₃	1	2	yes	—
351	C₂²	—	86	2	no	—
352	C₂²	—	86	2	no	—
353	C₂²	—	86	2	no	—
354	C₂²	—	172	1	no	—
355	C₂²	—	172	1	no	—
356	C₂²	—	172	1	no	—
357	C₂²	—	172	1	no	—
358	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
359	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
360	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
361	C₂²	#13063371	1	1	yes	characteristic
362	C₂²	#13063372	1	1	yes	characteristic
363	C₂²	#13063372	1	1	yes	characteristic
364	C₂²	#13063372	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	86	2	no	—
373	C₄	—	86	2	no	—
374	C₄	—	86	2	no	—
375	C₄	—	172	1	no	—
376	C₄	—	172	1	no	—
377	C₄	—	172	1	no	—
378	C₄	—	172	1	no	—
379	C₂	—	172	2	no	—
380	C₂	#14583471	1	1	yes	characteristic
381	C₂	#14583475	1	1	yes	characteristic
382	C₂	#14583529	1	1	yes	characteristic
383	C₂	D₄×D₁₇₂	1	1	yes	characteristic
384	C₂	#14583562	1	1	yes	characteristic
385	C₂	Q₈×D₁₇₂	1	1	yes	characteristic
386	C₂	#14583651	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#387	C₂
hypercenter	#387	D₄₃
hypocenter	C₄₃	#2677
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #387

Complements

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

N	H ≅ G/N	type
1. C₂×#13063325	C₂	semidirect
2. C₂×#13063324	C₂	semidirect
6. #14582831	C₂	semidirect
7. #14582829	C₂	semidirect
8. #14582820	C₂	semidirect
9. #14582820	C₂	semidirect
13. #14582784	C₂	semidirect
15. C₄₃×#387	C₂	semidirect
90. (C₂×D₄₃)⋊C₄	Q₈	semidirect
207. #387	D₄₃	semidirect
209. C₄₃	#2677	semidirect