Subgroups — C₄×#14583010 #25095082

Order 11008.

1326 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583010	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	C₄×#13063287	C₂	1	2	yes	maximal, complemented
4	#19134495	C₂	1	2	yes	maximal, complemented
5	#19134419	C₂	1	4	yes	maximal, complemented
6	C₄×#13063245	C₂	1	2	yes	maximal, complemented
7	C₄×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₂×#13063287	C₂²	1	4	yes	—
12	C₂×C₄²×D₄₃	—	2	2	no	—
13	#14583010	C₄	1	16	yes	direct product factor
14	C₄×D₃₄₄	C₄	1	8	yes	complemented
15	C₂×#13063259	C₂²	1	1	yes	characteristic
16	C₂×#13063247	C₂²	1	2	yes	—
17	C₂×#13063245	C₂²	1	4	yes	—
18	C₂×#13063235	C₂²	1	2	yes	—
19	C₂×#13063235	C₂²	1	2	yes	—
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
22	C₄×(C₂×D₄₃)⋊C₄	—	2	2	no	—
23	#14582746	—	2	4	no	—
24	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
25	#14582661	C₂²	1	4	yes	—
26	C₂×#106×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#340	C₂²	1	2	yes	—
30	C₂²×D₁₇₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₃	—	2	2	no	—
32	C₂²×C₄×D₄₃	—	2	4	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₂×D₃₄₄	C₂×C₄	1	8	yes	—
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	C₄×D₁₇₂	—	2	4	no	—
37	C₄×D₁₇₂	C₂×C₄	1	8	yes	complemented
38	#13063287	—	2	8	no	—
39	#13063287	C₂×C₄	1	16	yes	complemented
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
42	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
43	C₂×C₄×Dic₄₃	—	2	4	no	—
44	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
47	C₄²×D₄₃	—	4	8	no	—
48	#13063259	C₂×C₄	1	4	yes	complemented
49	#13063247	C₂×C₄	1	8	yes	complemented
50	#13063245	C₂×C₄	1	16	yes	complemented
51	#13063235	—	2	4	no	—
52	#13063235	C₂×C₄	1	8	yes	complemented
53	C₄×C₄₃⋊C₈	—	2	2	no	—
54	C₄₃⋊#95	—	2	4	no	—
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#106×C₄₃	C₂×C₄	1	8	yes	complemented
61	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
62	#95×C₄₃	—	2	2	no	—
63	C₂³×D₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
70	C₂×D₁₇₂	—	2	4	no	—
71	C₂×D₁₇₂	—	2	2	no	—
72	C₂×D₁₇₂	C₂²×C₄	1	8	yes	—
73	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
74	C₂×C₄×D₄₃	—	2	8	no	—
75	C₂×C₄×D₄₃	—	2	16	no	—
76	C₂×C₄×D₄₃	—	4	8	no	—
77	C₂×C₄×D₄₃	—	4	8	no	—
78	C₂×C₄×D₄₃	—	4	4	no	—
79	C₂×C₄×D₄₃	—	4	4	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₄₃	—	2	4	no	—
85	C₄⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
86	C₄²×C₄₃	—	4	4	no	—
87	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₃	#41	1	2	yes	—
89	(C₂×D₄₃)⋊C₄	—	2	8	no	—
90	(C₂×D₄₃)⋊C₄	—	2	16	no	—
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
93	C₄×Dic₄₃	—	2	4	no	—
94	C₄×Dic₄₃	—	4	2	no	—
95	C₄×Dic₄₃	—	4	8	no	—
96	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
97	C₄×Dic₄₃	#41	1	4	yes	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
101	D₃₄₄	C₄²	1	16	yes	complemented
102	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×D₄₃	—	2	8	no	—
104	C₂²×D₄₃	—	2	4	no	—
105	C₂²×D₄₃	—	4	2	no	—
106	C₂²×D₄₃	—	4	2	no	—
107	C₂²×D₄₃	—	4	4	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	8	no	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	4	4	no	—
112	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₃	—	2	4	no	—
120	C₂×Dic₄₃	—	2	4	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	C₄×D₄	1	8	yes	—
125	D₁₇₂	—	2	4	no	—
126	D₁₇₂	—	2	8	no	—
127	D₁₇₂	C₂×C₄²	1	8	yes	complemented
128	C₄×D₄₃	—	4	32	no	—
129	C₄×D₄₃	—	4	16	no	—
130	C₈×C₄₃	—	2	2	no	—
131	C₈×C₄₃	C₂×C₄²	1	4	yes	complemented
132	C₄₃⋊C₈	—	2	4	no	—
133	C₄×#442	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	C₂×C₄×D₄	1	2	yes	—
135	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₂²×C₄₃	#575	1	2	yes	—
138	C₂×D₄₃	—	2	8	no	—
139	C₂×D₄₃	—	4	4	no	—
140	C₂×D₄₃	—	4	4	no	—
141	C₂×D₄₃	—	4	8	no	—
142	C₂×D₄₃	—	4	8	no	—
143	C₄×C₄₃	—	2	2	no	—
144	C₄×C₄₃	—	4	8	no	—
145	C₄×C₄₃	#442	1	4	yes	complemented
146	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
148	C₄×C₄₃	C₄×#41	1	2	yes	—
149	Dic₄₃	—	2	8	no	—
150	Dic₄₃	—	4	4	no	—
151	C₂×#442	—	43	2	no	—
152	C₂×C₄×D₈	—	43	1	no	—
153	C₄²×D₄	—	43	2	no	—
154	#1776	—	43	2	no	—
155	C₄×#106	D₄₃	1	1	yes	characteristic, complemented
156	#1762	—	43	4	no	—
157	C₄×#102	—	43	2	no	—
158	C₄×#97	—	43	1	no	—
159	C₂×C₄₃	C₄²×D₄	1	1	yes	characteristic
160	C₂×C₄₃	C₂×#442	1	1	yes	characteristic
161	C₂×C₄₃	#2943	1	1	yes	characteristic
162	C₂×C₄₃	C₄×AGL(1,ℤ/8)	1	4	yes	—
163	D₄₃	—	4	8	no	—
164	C₂²×D₈	—	43	1	no	—
165	C₂×C₄×D₄	—	43	4	no	—
166	C₂×C₄×D₄	—	43	2	no	—
167	C₂²×C₄²	—	86	2	no	—
168	#442	—	43	16	no	—
169	C₄×D₈	—	43	8	no	—
170	C₂×#107	—	43	1	no	—
171	C₂×#106	C₂×D₄₃	1	2	yes	—
172	C₂×#102	—	43	2	no	—
173	C₂×#102	—	43	4	no	—
174	C₂×#97	—	43	2	no	—
175	C₂×#97	—	43	2	no	—
176	C₂×C₄×C₈	—	43	1	no	—
177	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
178	C₄×C₄⋊C₄	—	43	4	no	—
179	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
180	C₄×C₂²⋊C₄	—	86	4	no	—
181	C₄×C₂²⋊C₄	—	86	2	no	—
182	C₄³	—	43	1	no	—
183	#340	—	43	4	no	—
184	#340	C₂×D₄₃	1	2	yes	—
185	C₄₃	C₄×#442	1	1	yes	characteristic, complemented, Sylow
186	C₂²×D₄	—	43	2	no	—
187	C₂³×C₄	—	86	4	no	—
188	C₂³×C₄	—	86	2	no	—
189	C₂×D₈	—	43	4	no	—
190	C₂×D₈	—	43	8	no	—
191	C₂²×C₈	—	43	2	no	—
192	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	C₄×D₄	—	43	16	no	—
195	C₄×D₄	—	43	8	no	—
196	C₄×D₄	—	86	8	no	—
197	C₄×D₄	—	86	4	no	—
198	C₂×C₄⋊C₄	—	43	2	no	—
199	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
200	C₂×C₂²⋊C₄	—	86	4	no	—
201	C₂×C₂²⋊C₄	—	86	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	43	2	no	—
204	C₂×C₄²	—	43	2	no	—
205	C₂×C₄²	—	43	2	no	—
206	C₂×C₄²	—	86	4	no	—
207	C₂×C₄²	—	172	8	no	—
208	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
209	#107	—	43	4	no	—
210	#106	C₄×D₄₃	1	8	yes	complemented
211	#102	—	43	8	no	—
212	#102	—	43	16	no	—
213	#97	—	43	8	no	—
214	#97	—	86	4	no	—
215	C₄×C₈	—	86	2	no	—
216	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
217	#95	—	2	2	no	—
218	#95	—	86	4	no	—
219	C₂⁴	—	86	2	no	—
220	C₂×D₄	—	43	4	no	—
221	C₂×D₄	—	43	8	no	—
222	C₂×D₄	—	86	4	no	—
223	C₂×D₄	—	86	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	43	4	no	—
226	C₂²×C₄	—	86	16	no	—
227	C₂²×C₄	—	86	2	no	—
228	C₂²×C₄	—	86	8	no	—
229	C₂²×C₄	—	172	8	no	—
230	C₂²×C₄	—	172	8	no	—
231	C₂²×C₄	—	172	4	no	—
232	C₂²×C₄	—	172	4	no	—
233	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	D₈	—	43	16	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	43	4	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	43	4	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	C₂×C₄×D₄₃	1	8	yes	—
248	C₂²⋊C₄	—	86	16	no	—
249	C₂²⋊C₄	—	86	8	no	—
250	C₄²	—	4	4	no	—
251	C₄²	—	43	4	no	—
252	C₄²	—	43	4	no	—
253	C₄²	—	86	4	no	—
254	C₄²	—	172	2	no	—
255	C₄²	—	172	8	no	—
256	C₄²	C₂×C₄×D₄₃	1	2	yes	—
257	C₄²	#10621902	1	2	yes	—
258	C₂³	—	86	8	no	—
259	C₂³	—	86	4	no	—
260	C₂³	—	172	2	no	—
261	C₂³	—	172	2	no	—
262	C₂³	—	172	4	no	—
263	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
264	D₄	—	43	8	no	—
265	D₄	—	86	4	no	—
266	D₄	—	86	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	43	8	no	—
272	C₂×C₄	—	86	4	no	—
273	C₂×C₄	—	86	4	no	—
274	C₂×C₄	—	86	4	no	—
275	C₂×C₄	—	172	32	no	—
276	C₂×C₄	—	172	2	no	—
277	C₂×C₄	—	172	2	no	—
278	C₂×C₄	—	172	16	no	—
279	C₂×C₄	#13063287	1	2	yes	—
280	C₂×C₄	#13063287	1	2	yes	—
281	C₂×C₄	#13063352	1	4	yes	—
282	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
284	C₂×C₄	#41×D₄₃	1	2	yes	—
285	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	86	4	no	—
288	C₈	C₄²×D₄₃	1	4	yes	complemented
289	C₂²	—	86	8	no	—
290	C₂²	—	172	4	no	—
291	C₂²	—	172	4	no	—
292	C₂²	—	172	8	no	—
293	C₂²	—	172	8	no	—
294	C₂²	C₄×D₄×D₄₃	1	2	yes	—
295	C₂²	C₂×#13063287	1	1	yes	characteristic
296	C₂²	C₂×#13063352	1	2	yes	—
297	C₂²	#14583696	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	4	8	no	—
300	C₄	—	86	8	no	—
301	C₄	—	172	4	no	—
302	C₄	#14583010	1	4	yes	direct product factor
303	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
304	C₄	#14583367	1	1	yes	characteristic
305	C₄	C₄×#10621902	1	2	yes	—
306	C₂	—	172	8	no	—
307	C₂	C₄×#13063287	1	1	yes	characteristic
308	C₂	C₂×#14583010	1	1	yes	characteristic
309	C₂	#19143395	1	1	yes	characteristic
310	C₂	C₄×#13063352	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	C₄×#442
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
3. C₄×#13063287	C₂	semidirect
4. #19134495	C₂	semidirect
5. #19134419	C₂	semidirect
6. C₄×#13063245	C₂	semidirect
7. C₄×#13063235	C₂	semidirect
8. C₄×#106×C₄₃	C₂	semidirect
13. #14583010	C₄	direct (N × H)
14. C₄×D₃₄₄	C₄	semidirect
37. C₄×D₁₇₂	C₂×C₄	semidirect
39. #13063287	C₂×C₄	semidirect
48. #13063259	C₂×C₄	semidirect
49. #13063247	C₂×C₄	semidirect
50. #13063245	C₂×C₄	semidirect
52. #13063235	C₂×C₄	semidirect
60. #106×C₄₃	C₂×C₄	semidirect
61. C₄×C₈×C₄₃	C₂×C₄	semidirect
101. D₃₄₄	C₄²	semidirect
127. D₁₇₂	C₂×C₄²	semidirect
131. C₈×C₄₃	C₂×C₄²	semidirect
145. C₄×C₄₃	#442	semidirect
155. C₄×#106	D₄₃	semidirect
185. C₄₃	C₄×#442	semidirect
210. #106	C₄×D₄₃	semidirect
216. C₄×C₈	C₄×D₄₃	semidirect
288. C₈	C₄²×D₄₃	semidirect
302. C₄	#14583010	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#14583010	2
C₂×C₄×D₃₄₄	2
C₄×#13063287	2
#19134495	2
#19134419	2
C₄×#13063245	2
C₄×#13063235	2
C₄×#106×C₄₃	2
C₄×#442	43

Subgroups — C4×#14583010 #25095082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14583010 #25095082