Subgroups — #19139603

Order 5504.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063302	C₂	1	1	yes	maximal, characteristic
2	#14582830	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582829	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582819	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582818	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582806	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582804	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582794	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582794	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582772	C₂	1	1	yes	maximal, characteristic
11	#14582771	C₂	1	1	yes	maximal, characteristic
12	#14582757	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
14	C₂×C₄₃⋊#95	C₂	1	1	yes	maximal, characteristic
15	C₂×C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
17	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂³×Dic₄₃	C₂²	1	1	yes	characteristic
19	#13063302	—	2	4	no	—
20	C₂×C₄⋊Dic₄₃	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×Dic₄₃⋊C₄	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×Dic₄₃⋊C₄	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	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	C₄₃⋊#95	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
46	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
47	C₂²⋊Q₈×C₄₃	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
51	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
53	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₄₃	—	2	2	no	—
55	C₂×Q₈×C₄₃	—	2	2	no	—
56	C₂×Q₈×C₄₃	—	4	1	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	4	1	no	—
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×C₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₄₃	—	2	2	no	—
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	4	1	no	—
69	C₂²×Dic₄₃	—	4	1	no	—
70	C₂²×Dic₄₃	—	4	1	no	—
71	C₂²×Dic₄₃	Q₈	1	4	yes	complemented
72	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
73	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
74	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
75	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
76	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
77	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
78	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
79	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
80	C₂×Dic₈₆	—	2	2	no	—
81	C₂×Dic₈₆	—	2	2	no	—
82	C₂×Dic₈₆	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	2	2	no	—
84	C₄⋊C₄×C₄₃	—	2	2	no	—
85	C₄⋊C₄×C₄₃	—	2	2	no	—
86	C₂²⋊C₄×C₄₃	—	2	2	no	—
87	C₂²⋊C₄×C₄₃	D₄	1	4	yes	—
88	C₂²⋊Dic₄₃	—	2	2	no	—
89	C₂²⋊Dic₄₃	Q₈	1	4	yes	complemented
90	C₄⋊Dic₄₃	—	2	2	no	—
91	Dic₄₃⋊C₄	—	2	2	no	—
92	Dic₄₃⋊C₄	—	2	2	no	—
93	Dic₄₃⋊C₄	—	2	2	no	—
94	Dic₄₃⋊C₄	—	2	2	no	—
95	Dic₄₃⋊C₄	—	4	1	no	—
96	Dic₄₃⋊C₄	—	4	1	no	—
97	Dic₄₃⋊C₄	—	4	1	no	—
98	Dic₄₃⋊C₄	—	4	1	no	—
99	C₄×Dic₄₃	—	4	1	no	—
100	C₄×Dic₄₃	—	4	1	no	—
101	C₂³×C₄₃	—	2	1	no	—
102	C₂³×C₄₃	—	2	1	no	—
103	C₂³×C₄₃	—	2	1	no	—
104	C₂³×C₄₃	—	2	1	no	—
105	C₂³×C₄₃	C₂×D₄	1	2	yes	—
106	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
107	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
108	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
109	Q₈×C₄₃	—	4	4	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	—	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	—	4	4	no	—
126	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	#41	1	2	yes	—
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	—	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	—	4	4	no	—
149	C₂×Dic₄₃	—	4	4	no	—
150	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
151	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
152	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
153	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
154	C₂×Dic₄₃	#41	1	2	yes	—
155	Dic₈₆	—	4	4	no	—
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	2	2	no	—
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	2	2	no	—
161	C₂²×C₄₃	—	2	2	no	—
162	C₂²×C₄₃	C₄⋊Q₈	1	4	yes	—
163	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
164	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
165	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
166	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
167	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
168	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	2	2	no	—
172	C₄×C₄₃	—	2	2	no	—
173	C₄×C₄₃	—	4	1	no	—
174	C₄×C₄₃	—	4	1	no	—
175	C₄×C₄₃	—	4	1	no	—
176	Dic₄₃	—	2	2	no	—
177	Dic₄₃	—	2	2	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	#2549	—	43	1	no	maximal, Sylow
185	C₂×C₄₃	—	2	4	no	—
186	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄₃	#533	1	1	yes	characteristic
188	C₂×C₄₃	#543	1	1	yes	characteristic
189	C₂×C₄₃	#546	1	1	yes	characteristic
190	C₂×C₄₃	#546	1	1	yes	characteristic
191	C₂×C₄₃	#554	1	1	yes	characteristic
192	C₂×C₄₃	#554	1	1	yes	characteristic
193	C₂×C₂²⋊Q₈	—	43	1	no	—
194	C₂×C₂²⋊Q₈	D₄₃	1	1	yes	characteristic
195	#398	—	43	1	no	—
196	#398	—	43	1	no	—
197	#395	—	43	1	no	—
198	#395	—	43	1	no	—
199	#391	—	43	1	no	—
200	#391	—	43	1	no	—
201	#389	—	43	1	no	—
202	#389	—	43	1	no	—
203	#385	—	43	1	no	—
204	#385	—	43	1	no	—
205	#380	—	43	1	no	—
206	C₄×C₂²⋊C₄	—	43	1	no	—
207	C₂×#95	—	43	1	no	—
208	C₄₃	#2549	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	43	1	no	—
210	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
211	C₂³×C₄	—	43	1	no	—
212	C₂³×C₄	—	43	1	no	—
213	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
214	C₂²⋊Q₈	—	2	4	no	—
215	C₂²⋊Q₈	—	86	4	no	—
216	C₂×C₄⋊C₄	—	43	1	no	—
217	C₂×C₄⋊C₄	—	43	1	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	C₂×D₄₃	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	43	1	no	—
229	C₂×C₂²⋊C₄	—	43	1	no	—
230	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
232	C₂×C₄²	—	43	1	no	—
233	C₂×C₄²	—	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	—	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	—	43	1	no	—
246	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	86	2	no	—
251	C₂×Q₈	—	86	2	no	—
252	C₂×Q₈	—	172	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	43	1	no	—
257	C₂²×C₄	—	43	1	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	43	1	no	—
260	C₂²×C₄	—	43	1	no	—
261	C₂²×C₄	—	43	4	no	—
262	C₂²×C₄	—	43	1	no	—
263	C₂²×C₄	—	43	1	no	—
264	C₂²×C₄	—	43	1	no	—
265	C₂²×C₄	—	86	2	no	—
266	C₂²×C₄	—	86	2	no	—
267	C₂²×C₄	—	172	1	no	—
268	C₂²×C₄	—	172	1	no	—
269	C₂²×C₄	—	172	1	no	—
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	—	2	2	no	—
278	C₄⋊C₄	—	2	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	—	172	1	no	—
285	C₄⋊C₄	—	172	1	no	—
286	C₄⋊C₄	—	172	1	no	—
287	C₄⋊C₄	—	172	1	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	43	4	no	—
290	C₂²⋊C₄	—	86	2	no	—
291	C₂²⋊C₄	C₂²⋊D₄₃	1	4	yes	—
292	C₄²	—	172	1	no	—
293	C₄²	—	172	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	Q₈×D₄₃	1	2	yes	—
299	C₂³	Q₈×D₄₃	1	2	yes	—
300	C₂³	C₂×C₂²⋊D₄₃	1	2	yes	—
301	C₂³	C₂³×D₄₃	1	1	yes	characteristic
302	Q₈	—	4	4	no	—
303	Q₈	—	172	4	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	—	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	—	4	4	no	—
320	C₂×C₄	—	43	2	no	—
321	C₂×C₄	—	43	2	no	—
322	C₂×C₄	—	43	2	no	—
323	C₂×C₄	—	43	2	no	—
324	C₂×C₄	—	43	2	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	—	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	—	172	4	no	—
345	C₂×C₄	—	172	4	no	—
346	C₂×C₄	#10621900	1	2	yes	—
347	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
348	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	#13063323	1	4	yes	—
356	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
357	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
358	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
359	C₂²	#41×D₄₃	1	1	yes	characteristic
360	C₂²	C₂×#10621900	1	1	yes	characteristic
361	C₂²	#13063370	1	1	yes	characteristic
362	C₂²	#13063371	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	86	2	no	—
370	C₄	—	86	2	no	—
371	C₄	—	86	2	no	—
372	C₄	—	86	2	no	—
373	C₄	—	86	2	no	—
374	C₄	—	172	1	no	—
375	C₄	—	172	1	no	—
376	C₄	—	172	1	no	—
377	C₂	—	2	4	no	—
378	C₂	C₂×#13063323	1	1	yes	characteristic
379	C₂	#14583451	1	1	yes	characteristic
380	C₂	#14583524	1	1	yes	characteristic
381	C₂	#14583537	1	1	yes	characteristic
382	C₂	#14583543	1	1	yes	characteristic
383	C₂	#14583603	1	1	yes	characteristic
384	C₂	#14583607	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₂²⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₂²⋊Q₈	D₄₃
hypocenter	C₄₃	#2549
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #14582830	C₂	semidirect
3. #14582829	C₂	semidirect
4. #14582819	C₂	semidirect
5. #14582818	C₂	semidirect
6. #14582806	C₂	semidirect
7. #14582804	C₂	semidirect
8. #14582794	C₂	semidirect
9. #14582794	C₂	semidirect
71. C₂²×Dic₄₃	Q₈	semidirect
89. C₂²⋊Dic₄₃	Q₈	semidirect
208. C₄₃	#2549	semidirect