Subgroups — C₂×#19133479 #25066324

Order 11008.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582976	C₂	1	2	yes	maximal, complemented
3	#19133479	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14582626	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₄₄	C₄	1	1	yes	characteristic
8	C₂²×#10621872	—	2	2	no	—
9	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
10	C₄×D₃₄₄	C₄	1	4	yes	—
11	C₄×D₃₄₄	C₂²	1	4	yes	—
12	#14582976	—	2	4	no	—
13	#14582976	C₂²	1	8	yes	complemented
14	C₂×#13063259	C₄	1	1	yes	characteristic
15	C₂×#13063247	C₄	1	2	yes	—
16	C₂×(C₂×D₄₃)⋊C₈	—	2	2	no	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	#14582626	C₂²	1	8	yes	complemented
20	C₈×C₄₃⋊C₈	C₂²	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#335	C₂²	1	4	yes	complemented
24	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₄₃	—	2	2	no	—
26	C₂×D₃₄₄	—	2	4	no	—
27	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
28	C₂×#10621872	—	2	8	no	—
29	C₂×#10621872	—	4	8	no	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
33	C₄×D₁₇₂	—	2	2	no	—
34	C₄×D₁₇₂	—	2	2	no	—
35	C₄×D₁₇₂	C₂×C₄	1	4	yes	—
36	C₄×D₁₇₂	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	#13063259	C₂×C₄	1	4	yes	—
40	#13063247	C₂×C₄	1	8	yes	—
41	(C₂×D₄₃)⋊C₈	—	2	8	no	—
42	C₄×C₄₃⋊C₈	—	2	1	no	—
43	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
44	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
45	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₄₃	—	2	2	no	—
47	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₃	—	2	2	no	—
51	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
52	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₄₃	C₂³	1	2	yes	—
54	C₂³×D₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
58	C₂²×Dic₄₃	—	2	2	no	—
59	C₂×D₁₇₂	—	2	2	no	—
60	C₂×D₁₇₂	—	2	4	no	—
61	C₂×D₁₇₂	—	2	4	no	—
62	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
63	C₂×C₄×D₄₃	—	2	4	no	—
64	C₂×C₄×D₄₃	—	2	4	no	—
65	C₂×C₄×D₄₃	—	4	4	no	—
66	C₂×C₄×D₄₃	—	4	4	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	4	4	no	—
70	C₂×C₈×C₄₃	M₁₆	1	4	yes	complemented
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₄₃	#41	1	1	yes	characteristic
76	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
77	(C₂×D₄₃)⋊C₄	—	2	4	no	—
78	(C₂×D₄₃)⋊C₄	—	2	4	no	—
79	C₄⋊Dic₄₃	—	2	2	no	—
80	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄₃⋊C₈	—	2	4	no	—
82	C₂×C₄₃⋊C₈	—	2	2	no	—
83	C₂×C₄₃⋊C₈	—	2	2	no	—
84	C₂×C₄₃⋊C₈	D₈	1	4	yes	complemented
85	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
86	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
87	D₃₄₄	—	2	8	no	—
88	#10621872	—	4	16	no	—
89	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
90	C₂²×D₄₃	—	2	4	no	—
91	C₂²×D₄₃	—	4	4	no	—
92	C₂²×D₄₃	—	4	4	no	—
93	C₂²×D₄₃	—	4	4	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₄₃	#131	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₄₃	#135	1	2	yes	—
102	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₄₃	—	2	4	no	—
106	C₂×Dic₄₃	—	4	4	no	—
107	D₁₇₂	—	2	4	no	—
108	D₁₇₂	—	2	4	no	—
109	D₁₇₂	—	4	4	no	—
110	C₄×D₄₃	—	4	4	no	—
111	C₄×D₄₃	—	4	4	no	—
112	C₄×D₄₃	—	4	8	no	—
113	C₈×C₄₃	—	2	2	no	—
114	C₈×C₄₃	—	4	4	no	—
115	C₈×C₄₃	C₂×M₁₆	1	4	yes	complemented
116	C₄₃⋊C₈	—	2	4	no	—
117	C₄₃⋊C₈	—	2	2	no	—
118	C₄₃⋊C₈	C₂×D₈	1	4	yes	complemented
119	C₂×#1589	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	#436	1	2	yes	—
121	C₂²×C₄₃	C₄×D₈	1	2	yes	—
122	C₂²×C₄₃	#443	1	2	yes	—
123	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂×D₄₃	—	4	4	no	—
125	C₂×D₄₃	—	4	4	no	—
126	C₂×D₄₃	—	4	4	no	—
127	C₂×D₄₃	—	8	8	no	—
128	C₄×C₄₃	—	2	1	no	—
129	C₄×C₄₃	—	2	1	no	—
130	C₄×C₄₃	#436	1	2	yes	—
131	C₄×C₄₃	C₄×D₈	1	2	yes	—
132	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
133	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
136	Dic₄₃	—	4	4	no	—
137	C₂×C₄×D₈	—	43	1	no	—
138	C₂×#436	—	43	2	no	—
139	#1589	—	43	8	no	—
140	C₂×#335	D₄₃	1	1	yes	characteristic, complemented
141	C₂×D₄⋊C₈	—	43	2	no	—
142	C₂×C₈²	—	43	1	no	—
143	C₂×C₄₃	#1589	1	4	yes	complemented
144	C₂×C₄₃	C₂×#436	1	1	yes	characteristic
145	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
146	C₂×C₄₃	C₂×#443	1	1	yes	characteristic
147	D₄₃	—	8	4	no	—
148	C₂²×D₈	—	43	1	no	—
149	C₂²×M₁₆	—	86	2	no	—
150	C₂×C₄×D₄	—	43	2	no	—
151	C₄×D₈	—	43	4	no	—
152	C₄×D₈	—	43	4	no	—
153	#436	—	43	8	no	—
154	#436	—	86	4	no	—
155	C₂×#107	—	43	1	no	—
156	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
157	C₂×#102	—	43	2	no	—
158	C₂×C₂²⋊C₈	—	86	2	no	—
159	C₂×C₄×C₈	—	43	1	no	—
160	C₂×C₄×C₈	—	43	1	no	—
161	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
162	#335	C₂×D₄₃	1	4	yes	complemented
163	D₄⋊C₈	—	43	8	no	—
164	C₈²	—	43	4	no	—
165	C₄₃	C₂×#1589	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₄	—	43	2	no	—
167	C₂³×C₄	—	86	2	no	—
168	C₂×D₈	—	43	4	no	—
169	C₂×D₈	—	86	4	no	—
170	C₂×M₁₆	—	86	8	no	—
171	C₂×M₁₆	—	172	8	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	43	1	no	—
174	C₂²×C₈	—	43	1	no	—
175	C₂²×C₈	—	86	1	no	—
176	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
177	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
178	C₄×D₄	—	43	4	no	—
179	C₄×D₄	—	43	4	no	—
180	C₄×D₄	—	86	2	no	—
181	C₄×D₄	—	86	2	no	—
182	C₂×C₄⋊C₄	—	43	2	no	—
183	C₂×C₂²⋊C₄	—	86	2	no	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	#107	—	43	4	no	—
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
188	#102	—	43	8	no	—
189	C₂²⋊C₈	—	86	8	no	—
190	C₄×C₈	—	43	2	no	—
191	C₄×C₈	—	43	2	no	—
192	C₄×C₈	—	43	2	no	—
193	C₄×C₈	—	86	1	no	—
194	C₄×C₈	C₄×D₄₃	1	2	yes	—
195	C₄×C₈	C₂²×D₄₃	1	2	yes	—
196	C₂⁴	—	86	2	no	—
197	C₂×D₄	—	43	4	no	—
198	C₂×D₄	—	86	2	no	—
199	C₂×D₄	—	86	4	no	—
200	C₂×D₄	—	86	4	no	—
201	C₂²×C₄	—	86	2	no	—
202	C₂²×C₄	—	86	4	no	—
203	C₂²×C₄	—	86	4	no	—
204	C₂²×C₄	—	172	4	no	—
205	C₂²×C₄	—	172	4	no	—
206	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
208	C₂²×C₄	#10621902	1	1	yes	characteristic
209	D₈	—	86	8	no	—
210	M₁₆	—	172	16	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	43	4	no	—
215	C₂×C₈	—	43	2	no	—
216	C₂×C₈	—	43	2	no	—
217	C₂×C₈	—	86	4	no	—
218	C₂×C₈	—	86	2	no	—
219	C₂×C₈	—	86	2	no	—
220	C₂×C₈	#10621872	1	4	yes	complemented
221	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
223	C₄⋊C₄	—	43	4	no	—
224	C₄⋊C₄	—	86	2	no	—
225	C₂²⋊C₄	—	86	4	no	—
226	C₂²⋊C₄	—	86	4	no	—
227	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
228	C₄²	D₄×D₄₃	1	1	yes	characteristic
229	C₄²	#10621902	1	1	yes	characteristic
230	C₄²	C₂³×D₄₃	1	1	yes	characteristic
231	C₂³	—	86	4	no	—
232	C₂³	—	172	4	no	—
233	C₂³	—	172	4	no	—
234	C₂³	—	172	4	no	—
235	C₂³	#13063287	1	1	yes	characteristic
236	D₄	—	86	4	no	—
237	D₄	—	86	4	no	—
238	D₄	—	172	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	86	4	no	—
241	C₂×C₄	—	172	4	no	—
242	C₂×C₄	—	172	4	no	—
243	C₂×C₄	—	172	4	no	—
244	C₂×C₄	—	172	8	no	—
245	C₂×C₄	#13063287	1	1	yes	characteristic
246	C₂×C₄	#13063287	1	1	yes	characteristic
247	C₂×C₄	#13063287	1	1	yes	characteristic
248	C₂×C₄	C₂×#10621872	1	2	yes	—
249	C₂×C₄	#13063333	1	2	yes	—
250	C₂×C₄	D₈×D₄₃	1	2	yes	—
251	C₂×C₄	#13063348	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	4	no	—
257	C₈	—	43	4	no	—
258	C₈	—	86	4	no	—
259	C₈	—	86	2	no	—
260	C₈	C₂×#10621872	1	4	yes	complemented
261	C₂²	—	344	8	no	—
262	C₂²	—	172	4	no	—
263	C₂²	—	172	4	no	—
264	C₂²	—	172	4	no	—
265	C₂²	#14582976	1	2	yes	—
266	C₂²	#14582981	1	2	yes	—
267	C₂²	#14583015	1	2	yes	—
268	C₂²	C₂×#13063287	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	172	4	no	—
272	C₄	#14582976	1	2	yes	—
273	C₄	#14582981	1	2	yes	—
274	C₄	C₂²×#10621872	1	1	yes	characteristic
275	C₄	C₂×#13063333	1	1	yes	characteristic
276	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
277	C₄	C₂×#13063348	1	1	yes	characteristic
278	C₂	—	344	4	no	—
279	C₂	#19133479	1	4	yes	direct product factor
280	C₂	C₂×#14582976	1	1	yes	characteristic
281	C₂	C₂×#14582981	1	1	yes	characteristic
282	C₂	C₂×#14583015	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#335	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂²×D₃₄₄	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₂×#335	D₄₃
hypocenter	C₄₃	C₂×#1589
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. C₂×#14582976	C₂	semidirect
3. #19133479	C₂	direct (N × H)
4. C₂×#14582626	C₂	semidirect
5. C₂×C₈×C₄₃⋊C₈	C₂	semidirect
6. C₂×C₄₃×#335	C₂	semidirect
13. #14582976	C₂²	semidirect
19. #14582626	C₂²	semidirect
20. C₈×C₄₃⋊C₈	C₂²	semidirect
23. C₄₃×#335	C₂²	semidirect
70. C₂×C₈×C₄₃	M₁₆	semidirect
84. C₂×C₄₃⋊C₈	D₈	semidirect
115. C₈×C₄₃	C₂×M₁₆	semidirect
118. C₄₃⋊C₈	C₂×D₈	semidirect
140. C₂×#335	D₄₃	semidirect
143. C₂×C₄₃	#1589	semidirect
162. #335	C₂×D₄₃	semidirect
165. C₄₃	C₂×#1589	semidirect
220. C₂×C₈	#10621872	semidirect
260. C₈	C₂×#10621872	semidirect
279. C₂	#19133479	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×D₃₄₄	2
C₂×#14582976	2
#19133479	2
C₂×#14582626	2
C₂×C₈×C₄₃⋊C₈	2
C₂×C₄₃×#335	2
C₂×#1589	43

Subgroups — C2×#19133479 #25066324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19133479 #25066324