Subgroups — #19142011

Order 5504.

424 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14582841	C₂	1	1	yes	maximal, characteristic, complemented
2	#14582838	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582820	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582820	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	#14582792	C₂	1	1	yes	maximal, characteristic
10	#14582774	C₂	1	1	yes	maximal, characteristic
11	#14582774	C₂	1	1	yes	maximal, characteristic
12	#14582771	C₂	1	1	yes	maximal, characteristic
13	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582764	C₂	1	1	yes	maximal, characteristic, complemented
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	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
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×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₄₃⋊#95	C₂²	1	1	yes	characteristic
31	C₄₃⋊#95	C₂²	1	1	yes	characteristic
32	C₄₃⋊#95	C₂²	1	1	yes	characteristic
33	C₄₃⋊#95	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	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊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×C₄²×C₄₃	C₂²	1	1	yes	characteristic
49	#95×C₄₃	C₂²	1	1	yes	characteristic
50	#95×C₄₃	C₂²	1	1	yes	characteristic
51	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×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×Dic₄₃	—	4	1	no	—
59	C₂²×Dic₄₃	—	4	1	no	—
60	C₂²×Dic₄₃	—	4	1	no	—
61	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
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	C₂×Dic₈₆	—	2	2	no	—
70	C₂×Dic₈₆	—	2	2	no	—
71	C₂×Dic₈₆	—	4	1	no	—
72	C₂×Dic₈₆	—	4	1	no	—
73	C₂×Dic₈₆	—	4	1	no	—
74	C₂×Dic₈₆	—	4	1	no	—
75	C₄⋊C₄×C₄₃	—	4	1	no	—
76	C₂²⋊C₄×C₄₃	—	2	2	no	—
77	C₂²⋊C₄×C₄₃	—	4	1	no	—
78	C₂²⋊C₄×C₄₃	Q₈	1	4	yes	complemented
79	C₄²×C₄₃	—	2	2	no	—
80	C₂²⋊Dic₄₃	—	2	2	no	—
81	C₂²⋊Dic₄₃	—	4	1	no	—
82	C₂²⋊Dic₄₃	—	4	1	no	—
83	C₄⋊Dic₄₃	—	4	1	no	—
84	C₄⋊Dic₄₃	—	4	1	no	—
85	C₄⋊Dic₄₃	—	4	1	no	—
86	C₄⋊Dic₄₃	—	4	1	no	—
87	C₄⋊Dic₄₃	—	4	1	no	—
88	Dic₄₃⋊C₄	—	4	1	no	—
89	Dic₄₃⋊C₄	—	4	1	no	—
90	C₄×Dic₄₃	—	2	2	no	—
91	C₂³×C₄₃	—	2	1	no	—
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	C₂×Q₈	1	2	yes	—
98	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₄₃	—	2	1	no	—
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	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
116	C₂×C₄×C₄₃	#41	1	2	yes	—
117	C₂×Dic₄₃	—	2	1	no	—
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	—	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	—	4	4	no	—
141	C₂×Dic₄₃	#41	1	2	yes	—
142	Dic₈₆	—	4	4	no	—
143	Dic₈₆	—	4	4	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	C₂²×Q₈	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	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
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	C₄×C₄₃	—	4	1	no	—
164	Dic₄₃	—	2	2	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	—	4	1	no	—
169	Dic₄₃	—	4	1	no	—
170	Dic₄₃	—	4	1	no	—
171	Dic₄₃	—	4	1	no	—
172	#2763	—	43	1	no	maximal, Sylow
173	C₂×C₄₃	—	4	2	no	—
174	C₂×C₄₃	#542	1	1	yes	characteristic
175	C₂×C₄₃	#542	1	1	yes	characteristic
176	C₂×C₄₃	#544	1	1	yes	characteristic
177	C₂×C₄₃	#551	1	1	yes	characteristic
178	C₂×C₄₃	#554	1	1	yes	characteristic
179	C₂×C₄₃	#554	1	1	yes	characteristic
180	C₂×C₄₃	H(3,𝔽₄)	1	1	yes	characteristic
181	#400	—	43	1	no	—
182	#400	—	43	1	no	—
183	#395	—	43	1	no	—
184	#395	—	43	1	no	—
185	#393	—	43	1	no	—
186	#393	—	43	1	no	—
187	#391	—	43	1	no	—
188	#391	—	43	1	no	—
189	#388	—	43	1	no	—
190	#388	D₄₃	1	1	yes	characteristic
191	#385	—	43	1	no	—
192	#385	—	43	1	no	—
193	#385	—	43	1	no	—
194	#382	—	43	1	no	—
195	#382	—	43	1	no	—
196	C₄₃	#2763	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	43	1	no	—
198	C₂²×Q₈	—	43	1	no	—
199	C₂³×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	—	43	1	no	—
204	C₂×C₄⋊C₄	—	43	1	no	—
205	C₂×C₄⋊C₄	—	43	1	no	—
206	C₂×C₄⋊C₄	—	43	1	no	—
207	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	43	1	no	—
209	C₂×C₂²⋊C₄	—	43	1	no	—
210	C₂×C₂²⋊C₄	—	43	1	no	—
211	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
214	C₂×C₄²	—	43	1	no	—
215	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
216	#95	—	43	1	no	—
217	#95	—	43	1	no	—
218	#95	—	43	1	no	—
219	#95	—	43	1	no	—
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	C₂×Q₈	—	86	2	no	—
234	C₂×Q₈	—	86	2	no	—
235	C₂×Q₈	—	172	1	no	—
236	C₂×Q₈	—	172	1	no	—
237	C₂×Q₈	—	172	1	no	—
238	C₂×Q₈	—	172	1	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	43	1	no	—
241	C₂²×C₄	—	43	1	no	—
242	C₂²×C₄	—	43	1	no	—
243	C₂²×C₄	—	43	1	no	—
244	C₂²×C₄	—	43	1	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	43	1	no	—
247	C₂²×C₄	—	172	1	no	—
248	C₂²×C₄	—	172	1	no	—
249	C₂²×C₄	—	172	1	no	—
250	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	172	1	no	—
258	C₄⋊C₄	—	172	1	no	—
259	C₄⋊C₄	—	172	1	no	—
260	C₄⋊C₄	—	172	1	no	—
261	C₄⋊C₄	—	172	1	no	—
262	C₄⋊C₄	—	172	1	no	—
263	C₄⋊C₄	—	172	1	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	86	2	no	—
267	C₂²⋊C₄	—	172	1	no	—
268	C₂²⋊C₄	—	172	1	no	—
269	C₂²⋊C₄	Dic₈₆	1	4	yes	complemented
270	C₄²	—	2	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×Dic₈₆	1	2	yes	—
279	C₂³	C₂³×D₄₃	1	1	yes	characteristic
280	Q₈	—	172	4	no	—
281	Q₈	—	172	4	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	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	43	2	no	—
298	C₂×C₄	—	86	1	no	—
299	C₂×C₄	—	86	1	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	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	—	86	1	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	172	4	no	—
322	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
323	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
324	C₂×C₄	#10621900	1	2	yes	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
333	C₂²	#41×D₄₃	1	1	yes	characteristic
334	C₂²	C₂×#10621900	1	1	yes	characteristic
335	C₂²	#13063370	1	1	yes	characteristic
336	C₂²	#13063371	1	1	yes	characteristic
337	C₂²	#13063371	1	1	yes	characteristic
338	C₂²	#13063372	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	86	2	no	—
346	C₄	—	172	1	no	—
347	C₄	—	172	1	no	—
348	C₄	—	172	1	no	—
349	C₄	—	172	1	no	—
350	C₄	—	172	1	no	—
351	C₄	—	172	1	no	—
352	C₄	—	172	1	no	—
353	C₂	—	4	2	no	—
354	C₂	#14583515	1	1	yes	characteristic
355	C₂	#14583516	1	1	yes	characteristic
356	C₂	#14583527	1	1	yes	characteristic
357	C₂	#14583572	1	1	yes	characteristic
358	C₂	#14583606	1	1	yes	characteristic
359	C₂	#14583608	1	1	yes	characteristic
360	C₂	#14583641	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	#2763
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. #14582841	C₂	semidirect
2. #14582838	C₂	semidirect
3. #14582820	C₂	semidirect
4. #14582820	C₂	semidirect
7. #14582805	C₂	semidirect
8. #14582803	C₂	semidirect
13. #14582765	C₂	semidirect
14. #14582764	C₂	semidirect
78. C₂²⋊C₄×C₄₃	Q₈	semidirect
196. C₄₃	#2763	semidirect
269. C₂²⋊C₄	Dic₈₆	semidirect