Subgroups — C₄×#13063357 #19143415

Order 5504.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063357	C₂	1	1	yes	maximal, characteristic
2	C₄×#10621902	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583006	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583003	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583002	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10621885	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10621873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#10621884	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582940	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582906	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582899	C₂	1	1	yes	maximal, characteristic
14	C₄×#10621872	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
18	#13063357	C₄	1	8	yes	direct product factor
19	C₂×#10621885	C₂²	1	1	yes	characteristic
20	C₂×#10621873	C₂²	1	1	yes	characteristic
21	C₂×#10621884	C₂²	1	1	yes	characteristic
22	C₂×#10621872	C₂²	1	1	yes	characteristic
23	C₂×#10621872	C₂²	1	1	yes	characteristic
24	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
26	C₄×Dic₈₆	—	2	1	no	—
27	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
28	#13063290	—	2	1	no	—
29	#13063290	C₂²	1	1	yes	characteristic
30	C₄×D₁₇₂	—	2	1	no	—
31	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
32	#13063287	—	2	1	no	—
33	#13063287	C₂²	1	1	yes	characteristic
34	#13063283	—	2	1	no	—
35	#13063283	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₄₃	—	2	1	no	—
37	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
38	C₄²×D₄₃	—	2	1	no	—
39	C₄²×D₄₃	—	2	1	no	—
40	C₄²×D₄₃	C₂²	1	1	yes	characteristic
41	#13063258	C₂²	1	1	yes	characteristic
42	#13063257	C₂²	1	1	yes	characteristic
43	#13063256	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063249	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063247	C₂²	1	1	yes	characteristic
49	#13063245	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	#13063235	C₂²	1	1	yes	characteristic
52	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
56	#107×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#103×C₄₃	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
62	#10621902	—	2	2	no	—
63	#10621902	C₂×C₄	1	4	yes	—
64	Q₈×D₄₃	—	2	2	no	—
65	Q₈×D₄₃	C₂×C₄	1	4	yes	complemented
66	C₂×D₁₇₂	—	2	1	no	—
67	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
76	C₂×Dic₈₆	—	2	1	no	—
77	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
78	Q₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	2	1	no	—
82	C₄⋊C₄×C₄₃	—	2	1	no	—
83	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
85	C₄²×C₄₃	—	2	1	no	—
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	C₂³	1	1	yes	characteristic
88	#10621885	C₂×C₄	1	4	yes	complemented
89	#10621884	C₂×C₄	1	4	yes	complemented
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₄₃	—	2	1	no	—
94	C₄⋊Dic₄₃	—	2	1	no	—
95	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	D₄	1	1	yes	characteristic
106	C₄×Dic₄₃	D₄	1	1	yes	characteristic
107	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
110	#10621873	C₂×C₄	1	4	yes	complemented
111	#10621872	—	2	2	no	—
112	#10621872	C₂×C₄	1	4	yes	complemented
113	C₂²×D₄₃	—	2	1	no	—
114	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₄₃	—	2	1	no	—
116	Q₈×C₄₃	—	2	1	no	—
117	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
118	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	D₁₇₂	—	2	4	no	—
133	D₁₇₂	—	2	1	no	—
134	D₁₇₂	C₂²×C₄	1	2	yes	—
135	C₄×D₄₃	—	2	4	no	—
136	C₄×D₄₃	—	2	2	no	—
137	C₄×D₄₃	—	2	4	no	—
138	C₄×D₄₃	—	4	1	no	—
139	C₄×D₄₃	—	4	2	no	—
140	C₄×D₄₃	—	4	2	no	—
141	C₄×D₄₃	—	4	2	no	—
142	C₄×D₄₃	C₂²×C₄	1	2	yes	—
143	C₄×D₄₃	#41	1	2	yes	—
144	Dic₈₆	—	2	1	no	—
145	Dic₈₆	—	2	4	no	—
146	Dic₈₆	C₂²×C₄	1	2	yes	—
147	C₈×C₄₃	—	2	1	no	—
148	C₈×C₄₃	C₂²×C₄	1	2	yes	—
149	C₄₃⋊C₈	—	2	1	no	—
150	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
151	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	2	2	no	—
155	C₂×D₄₃	—	4	1	no	—
156	C₂×D₄₃	—	4	1	no	—
157	C₂×D₄₃	C₄×D₄	1	2	yes	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	—	2	2	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	4	1	no	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	#137	1	2	yes	complemented
164	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
165	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	2	1	no	—
168	Dic₄₃	—	2	1	no	—
169	Dic₄₃	—	4	1	no	—
170	Dic₄₃	—	4	1	no	—
171	Dic₄₃	C₄×D₄	1	2	yes	—
172	C₄×#137	—	43	1	no	maximal, Sylow
173	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
175	C₂×C₄₃	#575	1	1	yes	characteristic
176	D₄₃	—	2	2	no	—
177	D₄₃	—	4	2	no	—
178	C₂×#137	—	43	1	no	—
179	C₄×#41	—	43	1	no	—
180	C₂×C₄×Q₈	—	43	1	no	—
181	#441	—	43	1	no	—
182	#441	—	43	1	no	—
183	#440	—	43	1	no	—
184	#440	—	43	1	no	—
185	C₄×Q₁₆	—	43	1	no	—
186	C₄×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
187	C₄×SD₁₆	—	43	1	no	—
188	C₄×SD₁₆	—	43	1	no	—
189	#428	—	43	1	no	—
190	#419	—	43	1	no	—
191	#417	—	43	1	no	—
192	C₄×M₁₆	—	43	1	no	—
193	C₄₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	43	1	no	—
195	C₂²×Q₈	—	43	1	no	—
196	#137	—	43	8	no	—
197	C₂×Q₁₆	—	43	1	no	—
198	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
199	C₂×SD₁₆	—	43	1	no	—
200	C₂×SD₁₆	—	43	1	no	—
201	C₂×M₁₆	—	43	1	no	—
202	C₂×M₁₆	—	43	1	no	—
203	C₄×Q₈	—	43	1	no	—
204	C₄×Q₈	—	43	1	no	—
205	C₄×Q₈	—	43	1	no	—
206	C₄×Q₈	—	43	1	no	—
207	C₄×Q₈	—	86	1	no	—
208	C₄×Q₈	—	86	1	no	—
209	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
211	C₄×D₄	—	43	1	no	—
212	C₄×D₄	—	43	1	no	—
213	C₄×D₄	—	86	1	no	—
214	C₄×D₄	—	86	1	no	—
215	#117	—	43	1	no	—
216	#117	—	86	1	no	—
217	C₂×C₄⋊C₄	—	43	1	no	—
218	C₂×C₄⋊C₄	—	86	1	no	—
219	C₂×C₄²	—	43	1	no	—
220	C₂×C₄²	—	86	1	no	—
221	C₂×C₄²	—	86	1	no	—
222	#107	—	43	1	no	—
223	#107	C₂×D₄₃	1	1	yes	characteristic
224	#106	—	43	1	no	—
225	#106	—	43	1	no	—
226	#103	—	43	1	no	—
227	#103	—	43	1	no	—
228	#103	—	43	1	no	—
229	#103	—	43	1	no	—
230	#103	C₂×D₄₃	1	1	yes	characteristic
231	#103	C₂×D₄₃	1	1	yes	characteristic
232	#102	—	43	1	no	—
233	#102	—	43	1	no	—
234	#97	—	43	1	no	—
235	#97	—	43	1	no	—
236	C₄×C₈	—	43	1	no	—
237	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
238	#41	—	43	4	no	—
239	#41	—	86	2	no	—
240	C₂×Q₈	—	43	4	no	—
241	C₂×Q₈	—	43	1	no	—
242	C₂×Q₈	—	86	1	no	—
243	C₂×Q₈	—	86	2	no	—
244	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂×D₄	—	43	1	no	—
247	C₂×D₄	—	86	1	no	—
248	C₂²×C₄	—	43	1	no	—
249	C₂²×C₄	—	43	1	no	—
250	C₂²×C₄	—	43	1	no	—
251	C₂²×C₄	—	86	1	no	—
252	C₂²×C₄	—	86	1	no	—
253	C₂²×C₄	—	86	1	no	—
254	C₂²×C₄	—	86	1	no	—
255	C₂²×C₄	—	86	1	no	—
256	Q₁₆	—	43	4	no	—
257	Q₁₆	C₄×D₄₃	1	4	yes	complemented
258	SD₁₆	—	43	4	no	—
259	SD₁₆	—	43	4	no	—
260	M₁₆	—	43	4	no	—
261	M₁₆	—	86	2	no	—
262	C₂×C₈	—	43	1	no	—
263	C₂×C₈	—	43	1	no	—
264	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	43	1	no	—
269	C₄⋊C₄	—	43	1	no	—
270	C₄⋊C₄	—	86	1	no	—
271	C₄⋊C₄	—	86	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	86	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	43	1	no	—
283	C₄²	—	43	1	no	—
284	C₄²	—	43	1	no	—
285	C₄²	—	86	1	no	—
286	C₄²	—	86	1	no	—
287	C₄²	—	86	1	no	—
288	C₄²	—	86	1	no	—
289	C₄²	—	86	1	no	—
290	C₄²	C₂²×D₄₃	1	1	yes	characteristic
291	C₂³	—	43	1	no	—
292	C₂³	—	86	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	43	2	no	—
296	Q₈	—	86	4	no	—
297	Q₈	—	86	1	no	—
298	Q₈	C₂×C₄×D₄₃	1	2	yes	—
299	Q₈	C₂×C₄×D₄₃	1	2	yes	—
300	D₄	—	43	2	no	—
301	D₄	—	86	4	no	—
302	D₄	—	86	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	43	1	no	—
308	C₂×C₄	—	43	2	no	—
309	C₂×C₄	—	43	2	no	—
310	C₂×C₄	—	43	1	no	—
311	C₂×C₄	—	43	1	no	—
312	C₂×C₄	—	86	4	no	—
313	C₂×C₄	—	86	4	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	2	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	172	1	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
323	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	43	2	no	—
327	C₈	—	86	1	no	—
328	C₈	C₂×C₄×D₄₃	1	2	yes	—
329	C₂²	—	43	2	no	—
330	C₂²	—	86	1	no	—
331	C₂²	—	86	1	no	—
332	C₂²	—	86	2	no	—
333	C₂²	—	172	1	no	—
334	C₂²	—	172	1	no	—
335	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	43	2	no	—
342	C₄	—	86	2	no	—
343	C₄	—	86	1	no	—
344	C₄	—	86	1	no	—
345	C₄	—	172	1	no	—
346	C₄	—	172	1	no	—
347	C₄	#13063357	1	2	yes	direct product factor
348	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
349	C₄	#41×D₄₃	1	1	yes	characteristic
350	C₂	—	86	2	no	—
351	C₂	—	172	2	no	—
352	C₂	C₄×D₄×D₄₃	1	1	yes	characteristic
353	C₂	C₂×#13063357	1	1	yes	characteristic
354	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×Q₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×Q₁₆	D₄₃
hypocenter	C₄₃	C₄×#137
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Q₈×D₄₃	C₂	semidirect
4. #14583006	C₂	semidirect
5. #14583003	C₂	semidirect
6. #14583002	C₂	semidirect
7. #14582998	C₂	semidirect
8. C₄×#10621885	C₂	semidirect
9. C₄×#10621873	C₂	semidirect
10. C₄×#10621884	C₂	semidirect
11. #14582940	C₂	semidirect
12. #14582906	C₂	semidirect
14. C₄×#10621872	C₂	semidirect
15. C₄×Q₁₆×C₄₃	C₂	semidirect
18. #13063357	C₄	direct (N × H)
65. Q₈×D₄₃	C₂×C₄	semidirect
78. Q₁₆×C₄₃	C₂×C₄	semidirect
88. #10621885	C₂×C₄	semidirect
89. #10621884	C₂×C₄	semidirect
110. #10621873	C₂×C₄	semidirect
112. #10621872	C₂×C₄	semidirect
163. C₄×C₄₃	#137	semidirect
186. C₄×Q₁₆	D₄₃	semidirect
193. C₄₃	C₄×#137	semidirect
257. Q₁₆	C₄×D₄₃	semidirect
347. C₄	#13063357	direct (N × H)

Subgroups — C4×#13063357 #19143415

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13063357 #19143415