Subgroups — C₂×#19132569 #25006381

Order 11008.

414 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×#14583017	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582720	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132569	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063347	C₄	1	1	yes	characteristic
6	C₂×#13063331	—	2	1	no	—
7	C₂×#13063265	—	2	1	no	—
8	#14583017	—	2	1	no	—
9	#14583017	—	2	1	no	—
10	#14583017	C₄	1	2	yes	—
11	#14583017	C₂²	1	2	yes	—
12	C₂×#13063261	C₄	1	1	yes	characteristic
13	C₂×#13063253	—	2	1	no	—
14	#14582720	—	2	1	no	—
15	#14582720	C₂²	1	2	yes	complemented
16	C₂×C₄×C₈×C₄₃	—	2	1	no	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₈²×C₄₃	—	2	1	no	—
19	C₈²×C₄₃	C₂²	1	2	yes	complemented
20	C₂×#10621898	—	2	1	no	—
21	#13063347	—	2	1	no	—
22	#13063347	—	2	1	no	—
23	#13063347	C₈	1	2	yes	complemented
24	#13063347	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₇₂	C₈	1	1	yes	characteristic, complemented
26	C₂×#10621873	—	2	1	no	—
27	C₂×D₃₄₄	C₈	1	1	yes	characteristic, complemented
28	#13063331	—	2	2	no	—
29	#13063331	—	2	2	no	—
30	#13063331	—	4	1	no	—
31	#13063331	—	4	1	no	—
32	C₂×#10621877	—	2	1	no	—
33	C₂×#10621872	—	4	1	no	—
34	C₂×C₈×D₄₃	—	4	1	no	—
35	#13063265	—	2	2	no	—
36	#13063265	—	4	1	no	—
37	C₂×C₄₃⋊C₁₆	—	4	1	no	—
38	#13063261	—	2	1	no	—
39	#13063261	C₂×C₄	1	2	yes	—
40	#13063253	—	2	2	no	—
41	#13063253	—	2	2	no	—
42	C₂²×C₈×C₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₄₃	—	2	2	no	—
47	C₄×C₈×C₄₃	—	2	2	no	—
48	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	1	no	—
53	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₃	—	4	1	no	—
55	#10621898	—	2	2	no	—
56	#10621898	—	2	2	no	—
57	#10621898	—	4	1	no	—
58	#10621898	—	4	1	no	—
59	C₂×D₁₇₂	—	2	1	no	—
60	C₂×C₄×D₄₃	—	4	1	no	—
61	C₂×Dic₈₆	—	2	1	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	—	2	8	no	—
64	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₄₃	M₁₆	1	1	yes	characteristic
69	C₂×C₈×C₄₃	M₁₆	1	1	yes	characteristic
70	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
71	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
72	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
73	C₂×C₈×C₄₃	SD₁₆	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₄₃	—	2	1	no	—
77	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
79	#10621877	—	2	2	no	—
80	#10621877	—	4	1	no	—
81	C₂×C₄₃⋊C₈	—	4	1	no	—
82	Dic₁₇₂	—	2	1	no	—
83	Dic₁₇₂	C₂×C₈	1	2	yes	complemented
84	D₃₄₄	—	2	1	no	—
85	D₃₄₄	C₂×C₈	1	2	yes	complemented
86	#10621873	—	2	2	no	—
87	#10621873	—	2	2	no	—
88	#10621872	—	4	2	no	—
89	#10621872	—	4	2	no	—
90	C₈×D₄₃	—	4	2	no	—
91	C₈×D₄₃	—	4	2	no	—
92	C₄₃⋊C₁₆	—	4	2	no	—
93	C₂³×C₄₃	#104	1	1	yes	characteristic
94	C₂²×D₄₃	—	4	1	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	1	no	—
97	C₂×C₄×C₄₃	—	2	1	no	—
98	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#104	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²⋊D₄₃	—	4	2	no	—
105	C₂²⋊D₄₃	—	4	2	no	—
106	C₂×Dic₄₃	—	4	1	no	—
107	D₁₇₂	—	2	2	no	—
108	D₁₇₂	—	4	1	no	—
109	C₄×D₄₃	—	4	2	no	—
110	C₄×D₄₃	—	4	2	no	—
111	Dic₈₆	—	2	2	no	—
112	Dic₈₆	—	4	1	no	—
113	C₈×C₄₃	—	2	8	no	—
114	C₈×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₄₃	#102	1	1	yes	characteristic
117	C₈×C₄₃	#102	1	1	yes	characteristic
118	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
119	C₈×C₄₃	C₂×M₁₆	1	1	yes	characteristic
120	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
121	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₄₃⋊C₈	—	4	2	no	—
123	C₂×#1335	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	D₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₄₃	D₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
128	C₂×D₄₃	—	4	2	no	—
129	C₂×D₄₃	—	8	2	no	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	D₄⋊C₈	1	1	yes	characteristic
133	C₄×C₄₃	D₄⋊C₈	1	1	yes	characteristic
134	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
136	Dic₄₃	—	4	2	no	—
137	C₂×#443	—	43	1	no	—
138	C₂×#364	—	43	1	no	—
139	C₂×C₈²	D₄₃	1	1	yes	characteristic, complemented
140	#1335	—	43	4	no	—
141	C₂×C₄₃	—	2	1	no	—
142	C₂×C₄₃	—	2	1	no	—
143	C₂×C₄₃	#1335	1	2	yes	complemented
144	C₂×C₄₃	C₂×D₄⋊C₈	1	1	yes	characteristic
145	D₄₃	—	8	2	no	—
146	C₂×#135	—	43	1	no	—
147	C₂×#131	—	86	1	no	—
148	C₂×M₃₂	—	86	1	no	—
149	#443	—	43	2	no	—
150	#443	—	43	2	no	—
151	#443	—	86	1	no	—
152	#443	—	86	1	no	—
153	C₂×#108	—	43	1	no	—
154	C₂×#104	—	86	1	no	—
155	C₂×C₄×C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
157	#364	—	43	2	no	—
158	#364	—	86	1	no	—
159	C₈²	—	2	1	no	—
160	C₈²	C₂×D₄₃	1	2	yes	complemented
161	C₄₃	C₂×#1335	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	86	1	no	—
163	#135	—	43	2	no	—
164	#135	—	43	2	no	—
165	#135	—	86	1	no	—
166	#135	—	86	1	no	—
167	C₂×Q₁₆	—	43	1	no	—
168	C₂×SD₁₆	—	86	1	no	—
169	C₂×D₈	—	43	1	no	—
170	#131	—	86	2	no	—
171	#131	—	86	2	no	—
172	#131	—	172	1	no	—
173	#131	—	172	1	no	—
174	C₂×M₁₆	—	86	1	no	—
175	C₂×M₁₆	—	172	1	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	172	1	no	—
178	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
179	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
181	M₃₂	—	86	2	no	—
182	M₃₂	—	172	1	no	—
183	C₂×C₁₆	—	172	1	no	—
184	#108	—	43	2	no	—
185	#108	—	86	1	no	—
186	#104	—	86	2	no	—
187	#104	—	86	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
191	C₄×C₈	D₁₇₂	1	1	yes	characteristic
192	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	#41	—	86	2	no	—
195	#41	—	86	2	no	—
196	#41	—	172	1	no	—
197	#41	—	172	1	no	—
198	C₂×Q₈	—	86	1	no	—
199	C₂×D₄	—	86	1	no	—
200	C₂×D₄	—	172	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	172	1	no	—
203	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
204	Q₁₆	—	43	2	no	—
205	Q₁₆	—	86	1	no	—
206	SD₁₆	—	86	2	no	—
207	SD₁₆	—	86	2	no	—
208	D₈	—	43	2	no	—
209	D₈	—	86	1	no	—
210	M₁₆	—	86	2	no	—
211	M₁₆	—	172	1	no	—
212	M₁₆	—	172	2	no	—
213	M₁₆	—	172	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	172	1	no	—
217	C₂×C₈	—	172	2	no	—
218	C₂×C₈	—	172	2	no	—
219	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
220	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
221	C₂×C₈	#10621872	1	1	yes	characteristic
222	C₂×C₈	#10621872	1	1	yes	characteristic
223	C₂×C₈	#10621873	1	1	yes	characteristic
224	C₂×C₈	#10621873	1	1	yes	characteristic
225	C₂×C₈	D₃₄₄	1	1	yes	characteristic
226	C₂×C₈	D₃₄₄	1	1	yes	characteristic
227	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
228	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
231	C₄²	—	2	1	no	—
232	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
234	C₁₆	—	172	2	no	—
235	C₂³	—	172	1	no	—
236	C₂³	#13063253	1	1	yes	characteristic
237	Q₈	—	86	2	no	—
238	Q₈	—	172	1	no	—
239	D₄	—	86	2	no	—
240	D₄	—	172	2	no	—
241	D₄	—	172	2	no	—
242	D₄	—	172	1	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	172	1	no	—
247	C₂×C₄	—	172	2	no	—
248	C₂×C₄	—	172	2	no	—
249	C₂×C₄	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
250	C₂×C₄	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
251	C₂×C₄	#13063247	1	1	yes	characteristic
252	C₂×C₄	#13063247	1	1	yes	characteristic
253	C₂×C₄	#13063253	1	1	yes	characteristic
254	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	172	2	no	—
257	C₈	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
258	C₈	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
259	C₈	#13063247	1	1	yes	characteristic
260	C₈	#13063247	1	1	yes	characteristic
261	C₈	C₂×C₈×D₄₃	1	1	yes	characteristic
262	C₈	C₂×#10621872	1	1	yes	characteristic
263	C₈	C₂×D₃₄₄	1	1	yes	characteristic
264	C₈	C₂×#10621873	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	344	2	no	—
267	C₂²	—	172	2	no	—
268	C₂²	#14582628	1	1	yes	characteristic
269	C₂²	#14582628	1	1	yes	characteristic
270	C₂²	C₂×#13063253	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	172	2	no	—
274	C₄	#14582628	1	1	yes	characteristic
275	C₄	#14582628	1	1	yes	characteristic
276	C₄	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
277	C₄	C₂×#13063247	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	344	2	no	—
281	C₂	#19132569	1	2	yes	direct product factor
282	C₂	C₂×#14582628	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₃₄₄
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
socle	C₂²×C₄₃	D₄⋊C₈
prime radical	C₂×#13063347	C₄
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	C₂×#1335
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. C₂×#14582720	C₂	semidirect
3. #19132569	C₂	direct (N × H)
4. C₂×C₈²×C₄₃	C₂	semidirect
15. #14582720	C₂²	semidirect
19. C₈²×C₄₃	C₂²	semidirect
23. #13063347	C₈	semidirect
25. C₂×Dic₁₇₂	C₈	semidirect
27. C₂×D₃₄₄	C₈	semidirect
83. Dic₁₇₂	C₂×C₈	semidirect
85. D₃₄₄	C₂×C₈	semidirect
139. C₂×C₈²	D₄₃	semidirect
143. C₂×C₄₃	#1335	semidirect
160. C₈²	C₂×D₄₃	semidirect
161. C₄₃	C₂×#1335	semidirect
281. C₂	#19132569	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×#14583017	2
C₂×#14582720	2
#19132569	2
C₂×C₈²×C₄₃	2
C₂×#1335	43

Subgroups — C2×#19132569 #25006381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19132569 #25006381