Subgroups — C₄×#14582680 #25011638

Order 11008.

522 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621872	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582680	C₂	1	2	yes	maximal
3	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132678	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₄	1	1	yes	characteristic
7	C₂²×#10621872	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₄₃	C₄	1	1	yes	characteristic
9	C₂×#13063265	C₂²	1	2	yes	—
10	C₄×#10621872	—	2	1	no	—
11	C₄×#10621872	—	2	1	no	—
12	C₄×#10621872	—	2	1	no	—
13	C₄×#10621872	—	2	1	no	—
14	C₂×#13063235	C₄	1	1	yes	characteristic
15	C₂×#13063235	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
17	#14582680	C₄	1	8	yes	direct product factor
18	#14582675	—	2	1	no	—
19	C₄×C₄₃⋊C₁₆	—	2	1	no	—
20	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#346	—	2	1	no	—
23	C₄×C₁₆×C₄₃	—	2	1	no	—
24	C₂²×C₄×D₄₃	C₈	1	2	yes	—
25	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#10621872	—	2	1	no	—
27	C₂×#10621872	—	2	1	no	—
28	C₂×#10621872	—	2	2	no	—
29	C₂×#10621872	—	2	1	no	—
30	C₂×#10621872	—	2	1	no	—
31	C₂×#10621872	—	2	2	no	—
32	C₂×#10621872	C₈	1	4	yes	—
33	C₂×#10621872	C₂×C₄	1	4	yes	—
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₃	C₈	1	2	yes	—
37	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×D₄₃	—	2	1	no	—
39	C₄²×D₄₃	—	2	1	no	—
40	C₄²×D₄₃	—	2	2	no	—
41	#13063265	—	2	2	no	—
42	#13063265	C₂×C₄	1	4	yes	complemented
43	C₂×C₄₃⋊C₁₆	—	2	2	no	—
44	#13063235	—	2	1	no	—
45	#13063235	—	2	1	no	—
46	#13063235	—	2	1	no	—
47	#13063235	—	2	1	no	—
48	C₄×C₄₃⋊C₈	—	2	1	no	—
49	C₄×C₄₃⋊C₈	—	2	1	no	—
50	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₄₃	—	2	2	no	—
54	M₃₂×C₄₃	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₄₃	—	2	2	no	—
56	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
60	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	2	no	—
70	C₂×C₄×D₄₃	—	2	2	no	—
71	C₂×C₄×D₄₃	—	4	4	no	—
72	C₂×C₄×D₄₃	C₄²	1	2	yes	—
73	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
78	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	2	yes	—
82	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₄₃	M₁₆	1	2	yes	—
85	C₄×Dic₄₃	—	2	2	no	—
86	C₄×Dic₄₃	—	2	2	no	—
87	C₄×Dic₄₃	—	2	1	no	—
88	C₄×Dic₄₃	—	2	1	no	—
89	C₂×C₄₃⋊C₈	—	2	1	no	—
90	C₂×C₄₃⋊C₈	—	2	1	no	—
91	C₂×C₄₃⋊C₈	—	2	1	no	—
92	C₂×C₄₃⋊C₈	—	2	1	no	—
93	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
94	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
95	#10621872	—	2	4	no	—
96	#10621872	—	2	4	no	—
97	#10621872	—	2	4	no	—
98	#10621872	—	2	4	no	—
99	C₁₆×C₄₃	—	2	4	no	—
100	C₄₃⋊C₁₆	—	2	4	no	—
101	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
102	C₂²×D₄₃	—	2	1	no	—
103	C₂²×D₄₃	—	2	1	no	—
104	C₂²×D₄₃	—	4	2	no	—
105	C₂²×D₄₃	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	2	no	—
118	C₂×Dic₄₃	—	4	2	no	—
119	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
120	C₄×D₄₃	—	2	2	no	—
121	C₄×D₄₃	—	2	2	no	—
122	C₄×D₄₃	—	2	2	no	—
123	C₄×D₄₃	—	2	2	no	—
124	C₄×D₄₃	—	4	4	no	—
125	C₄×D₄₃	—	4	4	no	—
126	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
127	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
128	C₈×C₄₃	C₄×D₄	1	2	yes	—
129	C₈×C₄₃	C₄×D₄	1	2	yes	—
130	C₄₃⋊C₈	—	2	2	no	—
131	C₄₃⋊C₈	—	2	2	no	—
132	C₄×#349	—	43	1	no	maximal, Sylow
133	C₂²×C₄₃	—	2	2	no	—
134	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
135	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₄₃	#431	1	1	yes	characteristic
137	C₂×D₄₃	—	2	2	no	—
138	C₂×D₄₃	—	4	2	no	—
139	C₂×D₄₃	—	4	2	no	—
140	C₂×D₄₃	—	4	2	no	—
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	#349	1	4	yes	complemented
143	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
145	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
146	C₄×C₄₃	#431	1	1	yes	characteristic
147	Dic₄₃	—	2	2	no	—
148	Dic₄₃	—	4	2	no	—
149	C₂×C₄×M₁₆	—	43	1	no	—
150	C₂×#349	—	43	2	no	—
151	C₄×M₃₂	—	43	1	no	—
152	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
153	#1377	—	43	2	no	—
154	C₂×C₄₃	—	2	1	no	—
155	C₂×C₄₃	—	2	1	no	—
156	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₄₃	C₂×#349	1	2	yes	—
158	D₄₃	—	4	2	no	—
159	C₂²×M₁₆	—	43	1	no	—
160	C₂²×M₁₆	—	43	1	no	—
161	C₂²×C₄²	—	43	1	no	—
162	C₂×M₃₂	—	43	2	no	—
163	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
164	C₄×M₁₆	—	86	1	no	—
165	C₄×M₁₆	—	86	1	no	—
166	C₄×M₁₆	—	86	1	no	—
167	C₄×M₁₆	—	86	1	no	—
168	C₂×#97	—	43	1	no	—
169	C₂×#97	—	43	1	no	—
170	C₂×C₄×C₈	—	43	1	no	—
171	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
172	#349	—	43	8	no	—
173	#346	—	2	1	no	—
174	#346	—	86	1	no	—
175	C₄×C₁₆	—	2	1	no	—
176	C₄×C₁₆	—	86	1	no	—
177	C₄₃	C₄×#349	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	43	1	no	—
179	C₂³×C₄	—	43	2	no	—
180	C₂×M₁₆	—	43	4	no	—
181	C₂×M₁₆	—	43	4	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂×M₁₆	—	86	1	no	—
184	C₂×M₁₆	—	86	1	no	—
185	C₂×M₁₆	—	86	1	no	—
186	C₂×M₁₆	—	86	2	no	—
187	C₂×M₁₆	—	86	1	no	—
188	C₂²×C₈	—	43	1	no	—
189	C₂²×C₈	—	43	1	no	—
190	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
192	C₂×C₄²	—	43	2	no	—
193	C₂×C₄²	—	43	1	no	—
194	C₂×C₄²	—	86	2	no	—
195	C₂×C₄²	—	86	1	no	—
196	C₂×C₄²	—	86	1	no	—
197	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
198	M₃₂	—	2	2	no	—
199	M₃₂	—	43	4	no	—
200	M₃₂	—	86	2	no	—
201	M₃₂	C₄×D₄₃	1	4	yes	complemented
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	86	2	no	—
204	#97	—	86	1	no	—
205	#97	—	86	1	no	—
206	#97	—	86	1	no	—
207	#97	—	86	1	no	—
208	C₄×C₈	—	86	1	no	—
209	C₄×C₈	—	86	1	no	—
210	C₄×C₈	D₁₇₂	1	1	yes	characteristic
211	C₄×C₈	D₁₇₂	1	1	yes	characteristic
212	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
213	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
214	C₂⁴	—	43	1	no	—
215	C₂²×C₄	—	43	1	no	—
216	C₂²×C₄	—	43	2	no	—
217	C₂²×C₄	—	43	2	no	—
218	C₂²×C₄	—	43	2	no	—
219	C₂²×C₄	—	86	1	no	—
220	C₂²×C₄	—	86	1	no	—
221	C₂²×C₄	—	86	1	no	—
222	C₂²×C₄	—	86	1	no	—
223	C₂²×C₄	—	86	2	no	—
224	C₂²×C₄	—	86	2	no	—
225	C₂²×C₄	—	172	4	no	—
226	C₂²×C₄	C₈×D₄₃	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
228	M₁₆	—	86	4	no	—
229	M₁₆	—	86	4	no	—
230	M₁₆	—	86	4	no	—
231	M₁₆	—	86	4	no	—
232	C₂×C₈	—	43	2	no	—
233	C₂×C₈	—	43	2	no	—
234	C₂×C₈	—	86	1	no	—
235	C₂×C₈	—	86	1	no	—
236	C₂×C₈	—	86	1	no	—
237	C₂×C₈	—	86	1	no	—
238	C₂×C₈	C₈×D₄₃	1	2	yes	—
239	C₂×C₈	#10621872	1	2	yes	—
240	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
241	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
243	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
244	C₂×C₈	#10621898	1	2	yes	—
245	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
246	C₄²	—	86	2	no	—
247	C₄²	—	86	2	no	—
248	C₄²	—	86	1	no	—
249	C₄²	—	86	1	no	—
250	C₄²	#10621872	1	2	yes	—
251	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
252	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
253	C₁₆	—	2	4	no	—
254	C₁₆	—	86	4	no	—
255	C₂³	—	43	2	no	—
256	C₂³	—	86	1	no	—
257	C₂³	—	86	1	no	—
258	C₂³	—	172	2	no	—
259	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	43	2	no	—
262	C₂×C₄	—	86	1	no	—
263	C₂×C₄	—	86	1	no	—
264	C₂×C₄	—	86	2	no	—
265	C₂×C₄	—	86	2	no	—
266	C₂×C₄	—	86	2	no	—
267	C₂×C₄	—	86	2	no	—
268	C₂×C₄	—	86	2	no	—
269	C₂×C₄	—	172	4	no	—
270	C₂×C₄	—	172	4	no	—
271	C₂×C₄	—	172	2	no	—
272	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
273	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
274	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
275	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	#13063282	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
279	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
280	C₈	—	86	2	no	—
281	C₈	—	86	2	no	—
282	C₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
283	C₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
284	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
285	C₈	C₄×D₁₇₂	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	86	2	no	—
288	C₂²	—	172	2	no	—
289	C₂²	—	172	2	no	—
290	C₂²	—	172	2	no	—
291	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
292	C₂²	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
293	C₂²	#14582950	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	86	2	no	—
296	C₄	—	172	2	no	—
297	C₄	#14582680	1	4	yes	direct product factor
298	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
299	C₄	C₄×#10621872	1	1	yes	characteristic
300	C₄	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
301	C₄	#14582950	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	172	2	no	—
305	C₂	C₄×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
306	C₂	C₂×#14582680	1	2	yes	—

	subgroup	G/H
center	C₄²	(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₄₃	C₂×C₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	C₄×#349
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×#13063265	C₂	semidirect
4. #19132678	C₂	semidirect
5. C₄×M₃₂×C₄₃	C₂	semidirect
17. #14582680	C₄	direct (N × H)
42. #13063265	C₂×C₄	semidirect
54. M₃₂×C₄₃	C₂×C₄	semidirect
142. C₄×C₄₃	#349	semidirect
152. C₄×M₃₂	D₄₃	semidirect
177. C₄₃	C₄×#349	semidirect
201. M₃₂	C₄×D₄₃	semidirect
297. C₄	#14582680	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×C₄×#10621872	2
C₂×#14582680	2
C₄×#13063265	2
#19132678	2
C₄×M₃₂×C₄₃	2
C₄×#349	43

Subgroups — C4×#14582680 #25011638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582680 #25011638