Subgroups — C₄×#14582642 #24987538

Order 11008.

708 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	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582642	C₂	1	2	yes	maximal
4	#19132479	C₂	1	2	yes	maximal, complemented
5	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063322	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	—	2	1	no	—
8	C₂×C₄×D₁₇₂	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₄	1	2	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
12	#14582801	—	2	1	no	—
13	#14582801	—	2	1	no	—
14	#14582801	C₄	1	2	yes	—
15	#14582769	C₂²	1	1	yes	characteristic
16	#14582654	—	2	1	no	—
17	#14582642	C₄	1	8	yes	direct product factor
18	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₄₃	—	2	1	no	—
20	C₄³×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#336	—	2	1	no	—
22	C₂²×D₁₇₂	—	2	1	no	—
23	C₂²×D₁₇₂	—	2	1	no	—
24	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₄₃	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	#13063322	—	2	2	no	—
28	#13063322	C₂×C₄	1	4	yes	—
29	C₄×D₁₇₂	—	4	4	no	—
30	C₄×D₁₇₂	—	4	2	no	—
31	C₄×D₁₇₂	—	4	2	no	—
32	C₄×D₁₇₂	—	4	2	no	—
33	C₄×D₁₇₂	—	4	2	no	—
34	C₂×C₄⋊Dic₄₃	—	2	1	no	—
35	C₂×C₄⋊Dic₄₃	—	2	1	no	—
36	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
40	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	complemented
42	C₄×C₄₃⋊C₈	—	4	2	no	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₄₃	—	2	2	no	—
52	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	complemented
53	C₄×C₈×C₄₃	—	4	2	no	—
54	C₂³×D₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	2	2	no	—
63	C₂×D₁₇₂	—	2	2	no	—
64	C₂×D₁₇₂	—	2	4	no	—
65	C₂×D₁₇₂	—	2	4	no	—
66	C₂×D₁₇₂	—	4	1	no	—
67	C₂×D₁₇₂	—	4	1	no	—
68	C₂×D₁₇₂	—	4	1	no	—
69	C₂×D₁₇₂	—	4	1	no	—
70	C₂×D₁₇₂	—	4	2	no	—
71	C₂×D₁₇₂	—	4	2	no	—
72	C₂×D₁₇₂	—	4	2	no	—
73	C₂×D₁₇₂	—	4	2	no	—
74	C₂×D₁₇₂	C₄²	1	4	yes	—
75	C₂×C₄×D₄₃	—	4	4	no	—
76	C₂×C₄×D₄₃	—	4	4	no	—
77	C₂×C₄×D₄₃	—	4	4	no	—
78	C₂×C₈×C₄₃	—	2	4	no	—
79	C₂×C₈×C₄₃	—	4	2	no	—
80	C₂×C₈×C₄₃	—	4	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	4	1	no	—
85	C₄²×C₄₃	D₈	1	2	yes	—
86	C₄²×C₄₃	D₈	1	2	yes	—
87	C₄²×C₄₃	SD₁₆	1	2	yes	—
88	C₄²×C₄₃	SD₁₆	1	2	yes	—
89	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₃	#41	1	2	yes	—
91	(C₂×D₄₃)⋊C₄	—	4	4	no	—
92	(C₂×D₄₃)⋊C₄	—	4	4	no	—
93	(C₂×D₄₃)⋊C₄	—	4	4	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	C₄⋊Dic₄₃	—	4	1	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄⋊Dic₄₃	—	4	2	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	—	4	2	no	—
101	C₂×C₄₃⋊C₈	—	4	2	no	—
102	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×D₄₃	—	2	4	no	—
104	C₂²×D₄₃	—	4	2	no	—
105	C₂²×D₄₃	—	4	2	no	—
106	C₂²×D₄₃	—	4	2	no	—
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	—	4	2	no	—
109	C₂²×D₄₃	—	4	2	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	#100	1	2	yes	—
120	C₂×C₄×C₄₃	#102	1	2	yes	—
121	C₂×C₄×C₄₃	#102	1	2	yes	—
122	C₂×C₄×C₄₃	#102	1	2	yes	—
123	C₂×C₄×C₄₃	#102	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#135	1	2	yes	—
133	C₂×C₄×C₄₃	#135	1	2	yes	—
134	C₂×Dic₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	4	2	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	D₁₇₂	—	4	4	no	—
138	D₁₇₂	—	4	4	no	—
139	D₁₇₂	—	4	4	no	—
140	D₁₇₂	—	4	4	no	—
141	D₁₇₂	—	4	4	no	—
142	D₁₇₂	—	4	4	no	—
143	C₄×D₄₃	—	8	8	no	—
144	C₈×C₄₃	—	4	4	no	—
145	C₄₃⋊C₈	—	4	4	no	—
146	C₄×#331	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
149	C₂²×C₄₃	#413	1	1	yes	characteristic
150	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
152	C₂²×C₄₃	#416	1	1	yes	characteristic
153	C₂²×C₄₃	#416	1	1	yes	characteristic
154	C₂×D₄₃	—	4	4	no	—
155	C₂×D₄₃	—	4	4	no	—
156	C₂×D₄₃	—	4	4	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₂×D₄₃	—	8	2	no	—
160	C₂×D₄₃	—	8	2	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	—	2	2	no	—
163	C₄×C₄₃	—	2	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	4	1	no	—
166	C₄×C₄₃	#331	1	4	yes	complemented
167	C₄×C₄₃	C₄×D₈	1	2	yes	—
168	C₄×C₄₃	C₄×D₈	1	2	yes	—
169	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
170	C₄×C₄₃	C₄×SD₁₆	1	2	yes	—
171	Dic₄₃	—	8	2	no	—
172	C₄×#127	—	43	1	no	—
173	C₄×C₄⋊C₈	—	43	1	no	—
174	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
175	C₂×#331	—	43	2	no	—
176	#1296	—	43	2	no	—
177	C₂×C₄₃	C₂×#331	1	1	yes	characteristic
178	C₂×C₄₃	#1540	1	1	yes	characteristic
179	C₂×C₄₃	#1540	1	1	yes	characteristic
180	C₂×C₄₃	#1543	1	1	yes	characteristic
181	C₂×C₄₃	C₄×#100	1	1	yes	characteristic
182	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
183	C₂×C₄₃	C₄×#102	1	1	yes	characteristic
184	D₄₃	—	8	4	no	—
185	C₂×#127	—	43	1	no	—
186	C₂×C₄×D₄	—	43	2	no	—
187	C₂×C₄×D₄	—	86	1	no	—
188	C₂×C₄×D₄	—	86	1	no	—
189	C₂×C₄⋊C₈	—	43	2	no	—
190	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
191	C₂×C₄×C₈	—	2	1	no	—
192	C₂×C₄×C₈	—	86	1	no	—
193	#390	—	43	2	no	—
194	#390	—	86	1	no	—
195	#390	—	86	1	no	—
196	#384	—	43	1	no	—
197	C₄³	C₂×D₄₃	1	1	yes	characteristic
198	#336	—	2	1	no	—
199	#336	—	86	1	no	—
200	#331	—	43	8	no	—
201	C₄₃	C₄×#331	1	1	yes	characteristic, complemented, Sylow
202	C₂²×D₄	—	43	2	no	—
203	C₂²×D₄	—	86	1	no	—
204	C₂²×D₄	—	86	1	no	—
205	C₂³×C₄	—	86	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	86	2	no	—
208	#127	—	43	4	no	—
209	#127	—	86	2	no	—
210	C₄×D₄	—	172	4	no	—
211	C₄×D₄	—	172	2	no	—
212	C₄×D₄	—	172	2	no	—
213	C₄×D₄	—	172	2	no	—
214	C₄×D₄	—	172	2	no	—
215	C₂×C₄⋊C₄	—	43	2	no	—
216	C₂×C₄⋊C₄	—	86	1	no	—
217	C₂×C₄⋊C₄	—	86	1	no	—
218	C₂×C₂²⋊C₄	—	86	2	no	—
219	C₂×C₂²⋊C₄	—	86	2	no	—
220	C₂×C₂²⋊C₄	—	86	2	no	—
221	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
222	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
223	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
224	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	43	4	no	—
230	C₄⋊C₈	—	86	2	no	—
231	C₄⋊C₈	C₄×D₄₃	1	4	yes	complemented
232	C₄×C₈	—	4	2	no	—
233	C₄×C₈	—	172	2	no	—
234	C₂⁴	—	86	2	no	—
235	C₂×D₄	—	43	4	no	—
236	C₂×D₄	—	86	2	no	—
237	C₂×D₄	—	86	4	no	—
238	C₂×D₄	—	86	4	no	—
239	C₂×D₄	—	172	2	no	—
240	C₂×D₄	—	172	2	no	—
241	C₂×D₄	—	172	1	no	—
242	C₂×D₄	—	172	1	no	—
243	C₂×D₄	—	172	1	no	—
244	C₂×D₄	—	172	1	no	—
245	C₂×D₄	—	172	2	no	—
246	C₂×D₄	—	172	2	no	—
247	C₂²×C₄	—	86	2	no	—
248	C₂²×C₄	—	172	4	no	—
249	C₂²×C₄	—	172	4	no	—
250	C₂²×C₄	—	172	4	no	—
251	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	86	4	no	—
262	C₂×C₈	—	172	2	no	—
263	C₂×C₈	—	172	2	no	—
264	C₄⋊C₄	—	172	2	no	—
265	C₄⋊C₄	—	172	1	no	—
266	C₄⋊C₄	—	172	1	no	—
267	C₄⋊C₄	—	172	1	no	—
268	C₄⋊C₄	—	172	1	no	—
269	C₂²⋊C₄	—	172	4	no	—
270	C₂²⋊C₄	—	172	4	no	—
271	C₂²⋊C₄	—	172	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	4	1	no	—
276	C₄²	#10621873	1	2	yes	—
277	C₄²	D₃₄₄	1	2	yes	—
278	C₄²	#10621882	1	2	yes	—
279	C₄²	#10621884	1	2	yes	—
280	C₄²	C₂×C₄×D₄₃	1	2	yes	—
281	C₄²	#10621898	1	2	yes	—
282	C₂³	—	86	4	no	—
283	C₂³	—	172	2	no	—
284	C₂³	—	172	2	no	—
285	C₂³	—	172	2	no	—
286	C₂³	—	172	2	no	—
287	C₂³	—	172	2	no	—
288	C₂³	—	172	2	no	—
289	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
290	D₄	—	172	4	no	—
291	D₄	—	172	4	no	—
292	D₄	—	172	4	no	—
293	D₄	—	172	4	no	—
294	D₄	—	172	4	no	—
295	D₄	—	172	4	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	344	8	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	—	172	2	no	—
309	C₂×C₄	#13063241	1	2	yes	—
310	C₂×C₄	#13063245	1	2	yes	—
311	C₂×C₄	#13063245	1	2	yes	—
312	C₂×C₄	#13063247	1	2	yes	—
313	C₂×C₄	#13063247	1	2	yes	—
314	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
315	C₂×C₄	#13063282	1	1	yes	characteristic
316	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
317	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
318	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
319	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
320	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
321	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
322	C₂×C₄	#13063345	1	2	yes	—
323	C₂×C₄	#13063347	1	2	yes	—
324	C₈	—	4	4	no	—
325	C₈	—	172	4	no	—
326	C₂²	—	344	2	no	—
327	C₂²	—	344	2	no	—
328	C₂²	—	344	2	no	—
329	C₂²	—	344	2	no	—
330	C₂²	—	172	4	no	—
331	C₂²	—	172	4	no	—
332	C₂²	—	172	4	no	—
333	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
334	C₂²	C₂×#13063241	1	1	yes	characteristic
335	C₂²	#14582881	1	1	yes	characteristic
336	C₂²	C₂×#13063245	1	1	yes	characteristic
337	C₂²	C₂×#13063247	1	1	yes	characteristic
338	C₂²	#14582893	1	1	yes	characteristic
339	C₂²	#14582896	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	344	2	no	—
346	C₄	#14582642	1	4	yes	direct product factor
347	C₄	C₄×#10621882	1	2	yes	—
348	C₄	C₄×D₃₄₄	1	2	yes	—
349	C₄	C₄×#10621884	1	2	yes	—
350	C₄	C₄×#10621873	1	2	yes	—
351	C₂	—	344	4	no	—
352	C₂	C₂×#14582642	1	1	yes	characteristic
353	C₂	#19133242	1	1	yes	characteristic
354	C₂	#19133245	1	1	yes	characteristic
355	C₂	#19133255	1	1	yes	characteristic
356	C₂	C₄×#13063241	1	1	yes	characteristic
357	C₂	C₄×#13063247	1	1	yes	characteristic
358	C₂	C₄×#13063245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	C₄×#331
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
4. #19132479	C₂	semidirect
5. C₄×C₄⋊C₈×C₄₃	C₂	semidirect
17. #14582642	C₄	direct (N × H)
41. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
52. C₄⋊C₈×C₄₃	C₂×C₄	semidirect
166. C₄×C₄₃	#331	semidirect
174. C₄×C₄⋊C₈	D₄₃	semidirect
201. C₄₃	C₄×#331	semidirect
231. C₄⋊C₈	C₄×D₄₃	semidirect
346. C₄	#14582642	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#13063322	2
C₄×C₄⋊(C₄₃⋊C₈)	2
C₂×#14582642	2
#19132479	2
C₄×C₄⋊C₈×C₄₃	2
C₄×#331	43

Subgroups — C4×#14582642 #24987538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582642 #24987538