Subgroups — C₄×#14583011 #25095080

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×#14583011	C₂	1	2	yes	maximal
2	C₄×D₄×Dic₄₃	C₂	1	2	yes	maximal, complemented
3	#19134494	C₂	1	2	yes	maximal, complemented
4	C₄×#13063257	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134422	C₂	1	4	yes	maximal, complemented
6	C₄×#13063246	C₂	1	2	yes	maximal, complemented
7	C₄×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×Dic₄₃	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₄₃	—	2	2	no	—
11	#14583011	C₄	1	16	yes	direct product factor
12	C₂×#13063257	C₂²	1	2	yes	—
13	C₂×#13063246	C₂²	1	4	yes	—
14	C₂×#13063235	C₂²	1	2	yes	—
15	C₂×#13063235	C₂²	1	2	yes	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
19	C₂²⋊C₄×Dic₄₃	—	2	4	no	—
20	C₄×C₂²⋊Dic₄₃	—	2	2	no	—
21	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
22	#14582662	C₂²	1	2	yes	—
23	#14582661	C₂²	1	4	yes	—
24	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₄₃	C₂²	1	2	yes	—
26	C₄×D₈×C₄₃	C₄	1	8	yes	complemented
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂³×Dic₄₃	—	2	4	no	—
31	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
32	D₄×Dic₄₃	—	2	8	no	—
33	D₄×Dic₄₃	C₂×C₄	1	16	yes	complemented
34	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₃	—	2	4	no	—
36	C₂×C₄×Dic₄₃	—	2	2	no	—
37	C₂×C₄×Dic₄₃	—	2	4	no	—
38	C₂×C₄×Dic₄₃	—	4	8	no	—
39	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
40	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
41	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
42	#13063257	C₂×C₄	1	8	yes	complemented
43	#13063246	C₂×C₄	1	16	yes	complemented
44	#13063235	—	2	4	no	—
45	#13063235	C₂×C₄	1	8	yes	complemented
46	C₄×C₄₃⋊C₈	—	2	2	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	4	no	—
49	C₂²×D₄×C₄₃	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₃	—	2	2	no	—
51	C₂×D₈×C₄₃	C₂×C₄	1	8	yes	—
52	C₂×D₈×C₄₃	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₄×D₄×C₄₃	—	2	4	no	—
56	C₄×D₄×C₄₃	C₂×C₄	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	#107×C₄₃	C₂×C₄	1	4	yes	complemented
61	#102×C₄₃	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
63	C₂⁴×C₄₃	—	2	2	no	—
64	C₂×D₄×C₄₃	—	2	2	no	—
65	C₂×D₄×C₄₃	—	2	4	no	—
66	C₂×D₄×C₄₃	C₂²×C₄	1	8	yes	—
67	C₂×D₄×C₄₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₃	—	2	8	no	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	4	4	no	—
71	C₂²×C₄×C₄₃	—	4	4	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₃	—	2	16	no	—
76	C₂²×Dic₄₃	—	2	4	no	—
77	C₂²×Dic₄₃	—	4	8	no	—
78	C₂²×Dic₄₃	—	4	8	no	—
79	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
80	D₈×C₄₃	C₄²	1	16	yes	complemented
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₄⋊C₄×C₄₃	—	2	2	no	—
86	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₄₃	—	2	8	no	—
88	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₃	#41	1	2	yes	—
90	C₂²⋊Dic₄₃	—	2	16	no	—
91	C₄⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
93	C₄×Dic₄₃	—	2	4	no	—
94	C₄×Dic₄₃	—	4	2	no	—
95	C₄×Dic₄₃	—	4	8	no	—
96	C₄×Dic₄₃	—	4	4	no	—
97	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
98	C₄×Dic₄₃	#41	1	4	yes	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂³×C₄₃	—	2	8	no	—
103	C₂³×C₄₃	—	2	4	no	—
104	C₂³×C₄₃	—	4	2	no	—
105	C₂³×C₄₃	—	4	2	no	—
106	C₂³×C₄₃	—	4	4	no	—
107	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
108	D₄×C₄₃	—	2	8	no	—
109	D₄×C₄₃	—	2	4	no	—
110	D₄×C₄₃	C₂×C₄²	1	8	yes	complemented
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	4	16	no	—
116	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
119	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₃	—	2	4	no	—
124	C₂×Dic₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	2	8	no	—
126	C₂×Dic₄₃	—	4	32	no	—
127	C₂×Dic₄₃	—	4	4	no	—
128	C₂×Dic₄₃	—	4	4	no	—
129	C₂×Dic₄₃	C₄×D₄	1	8	yes	—
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	—	2	8	no	—
135	C₂²×C₄₃	—	4	4	no	—
136	C₂²×C₄₃	—	4	4	no	—
137	C₂²×C₄₃	—	4	8	no	—
138	C₂²×C₄₃	—	4	8	no	—
139	C₂²×C₄₃	C₂×C₄×D₄	1	2	yes	—
140	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₄₃	#575	1	2	yes	—
143	C₄×C₄₃	—	2	2	no	—
144	C₄×C₄₃	—	4	4	no	—
145	C₄×C₄₃	#442	1	4	yes	complemented
146	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₄₃	C₄×#41	1	2	yes	—
148	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
149	Dic₄₃	—	2	8	no	—
150	Dic₄₃	—	4	8	no	—
151	C₂×#442	—	43	2	no	—
152	C₂×C₄×D₈	D₄₃	1	1	yes	characteristic
153	C₄²×D₄	—	43	2	no	—
154	#1776	—	43	2	no	—
155	C₄×#106	—	43	1	no	—
156	#1762	—	43	4	no	—
157	C₄×#102	—	43	2	no	—
158	C₄×#97	—	43	1	no	—
159	C₂×C₄₃	—	4	8	no	—
160	C₂×C₄₃	C₄²×D₄	1	1	yes	characteristic
161	C₂×C₄₃	C₂×#442	1	1	yes	characteristic
162	C₂×C₄₃	#2943	1	1	yes	characteristic
163	C₂×C₄₃	C₄×AGL(1,ℤ/8)	1	4	yes	—
164	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
165	C₂×C₄×D₄	—	43	4	no	—
166	C₂×C₄×D₄	C₂×D₄₃	1	2	yes	—
167	C₂²×C₄²	—	86	2	no	—
168	#442	—	43	16	no	—
169	C₄×D₈	Dic₄₃	1	8	yes	complemented
170	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#106	—	43	2	no	—
172	C₂×#102	—	43	4	no	—
173	C₂×#102	C₂×D₄₃	1	2	yes	—
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	—	43	2	no	—
180	C₄×C₂²⋊C₄	—	86	4	no	—
181	C₄×C₂²⋊C₄	—	86	2	no	—
182	C₄³	—	43	1	no	—
183	#340	—	43	2	no	—
184	#340	—	43	4	no	—
185	C₄₃	C₄×#442	1	1	yes	characteristic, complemented, Sylow
186	C₂²×D₄	C₂²×D₄₃	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂³×C₄	—	86	4	no	—
189	C₂×D₈	C₄×D₄₃	1	8	yes	—
190	C₂×D₈	C₂×Dic₄₃	1	4	yes	—
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	—	2	4	no	—
195	C₄×D₄	—	43	16	no	—
196	C₄×D₄	—	86	8	no	—
197	C₄×D₄	C₂×Dic₄₃	1	8	yes	complemented
198	C₂×C₄⋊C₄	—	43	4	no	—
199	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	86	4	no	—
202	C₂×C₄²	—	43	2	no	—
203	C₂×C₄²	—	43	2	no	—
204	C₂×C₄²	—	43	2	no	—
205	C₂×C₄²	—	86	4	no	—
206	C₂×C₄²	—	86	2	no	—
207	C₂×C₄²	—	172	8	no	—
208	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
209	#107	C₂×Dic₄₃	1	4	yes	complemented
210	#106	—	43	8	no	—
211	#102	—	43	16	no	—
212	#102	C₂×Dic₄₃	1	8	yes	complemented
213	#97	—	43	8	no	—
214	#97	—	86	4	no	—
215	C₄×C₈	—	86	2	no	—
216	C₄×C₈	C₂×Dic₄₃	1	4	yes	complemented
217	#95	—	86	4	no	—
218	#95	—	86	2	no	—
219	C₂⁴	—	2	2	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₂×C₄×D₄₃	1	8	yes	—
223	C₂×D₄	C₂²×Dic₄₃	1	4	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	43	4	no	—
229	C₂²×C₄	—	86	16	no	—
230	C₂²×C₄	—	86	4	no	—
231	C₂²×C₄	—	172	8	no	—
232	C₂²×C₄	—	172	8	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	C₄×Dic₄₃	1	16	yes	complemented
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	4	yes	—
242	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
243	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	43	8	no	—
246	C₄⋊C₄	—	86	4	no	—
247	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₂²⋊C₄	—	86	16	no	—
250	C₄²	—	43	4	no	—
251	C₄²	—	43	4	no	—
252	C₄²	—	86	4	no	—
253	C₄²	—	172	2	no	—
254	C₄²	—	172	8	no	—
255	C₄²	—	172	4	no	—
256	C₄²	#10621900	1	2	yes	—
257	C₄²	C₂²×Dic₄₃	1	2	yes	—
258	C₂³	—	2	8	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
264	D₄	—	2	8	no	—
265	D₄	—	2	4	no	—
266	D₄	C₂×C₄×Dic₄₃	1	8	yes	complemented
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	—	43	8	no	—
273	C₂×C₄	—	86	4	no	—
274	C₂×C₄	—	86	4	no	—
275	C₂×C₄	—	86	8	no	—
276	C₂×C₄	—	172	32	no	—
277	C₂×C₄	—	172	4	no	—
278	C₂×C₄	—	172	4	no	—
279	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
280	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
281	C₂×C₄	#13063349	1	4	yes	—
282	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
283	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
284	C₂×C₄	#41×D₄₃	1	2	yes	—
285	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	86	4	no	—
288	C₈	C₂×C₄×Dic₄₃	1	4	yes	complemented
289	C₂²	—	2	8	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	8	no	—
293	C₂²	—	4	8	no	—
294	C₂²	C₄×D₄×D₄₃	1	2	yes	—
295	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
296	C₂²	C₂×#13063349	1	2	yes	—
297	C₂²	#14583696	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	—	86	8	no	—
301	C₄	—	172	8	no	—
302	C₄	#14583011	1	4	yes	direct product factor
303	C₄	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
304	C₄	C₄×#10621900	1	2	yes	—
305	C₄	#41×Dic₄₃	1	1	yes	characteristic
306	C₂	—	4	8	no	—
307	C₂	C₄×D₄×Dic₄₃	1	1	yes	characteristic
308	C₂	C₂×#14583011	1	1	yes	characteristic
309	C₂	#19143396	1	1	yes	characteristic
310	C₂	C₄×#13063349	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄×D₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₈×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₃
hypocenter	C₄₃	C₄×#442
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×Dic₄₃	C₂	semidirect
3. #19134494	C₂	semidirect
4. C₄×#13063257	C₂	semidirect
5. #19134422	C₂	semidirect
6. C₄×#13063246	C₂	semidirect
7. C₄×#13063235	C₂	semidirect
11. #14583011	C₄	direct (N × H)
26. C₄×D₈×C₄₃	C₄	semidirect
33. D₄×Dic₄₃	C₂×C₄	semidirect
42. #13063257	C₂×C₄	semidirect
43. #13063246	C₂×C₄	semidirect
45. #13063235	C₂×C₄	semidirect
56. C₄×D₄×C₄₃	C₂×C₄	semidirect
60. #107×C₄₃	C₂×C₄	semidirect
61. #102×C₄₃	C₂×C₄	semidirect
62. C₄×C₈×C₄₃	C₂×C₄	semidirect
80. D₈×C₄₃	C₄²	semidirect
110. D₄×C₄₃	C₂×C₄²	semidirect
131. C₈×C₄₃	C₂×C₄²	semidirect
145. C₄×C₄₃	#442	semidirect
169. C₄×D₈	Dic₄₃	semidirect
185. C₄₃	C₄×#442	semidirect
197. C₄×D₄	C₂×Dic₄₃	semidirect
209. #107	C₂×Dic₄₃	semidirect
212. #102	C₂×Dic₄₃	semidirect
216. C₄×C₈	C₂×Dic₄₃	semidirect
236. D₈	C₄×Dic₄₃	semidirect
266. D₄	C₂×C₄×Dic₄₃	semidirect
288. C₈	C₂×C₄×Dic₄₃	semidirect
302. C₄	#14583011	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×#14583011	2
C₄×D₄×Dic₄₃	2
#19134494	2
C₄×#13063257	2
#19134422	2
C₄×#13063246	2
C₄×#13063235	2
C₂×C₄×D₈×C₄₃	2
C₄×#442	43

Subgroups — C4×#14583011 #25095080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14583011 #25095080