Subgroups — C₂×#19147736 #26003109

Order 11008.

2126 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583704	C₂	1	2	yes	maximal, complemented
2	#19147736	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14583337	C₂	1	6	yes	maximal, complemented
4	C₂×#14583331	C₂	1	3	yes	maximal, complemented
5	C₂²×D₆₈₈	C₂	1	3	yes	maximal, complemented
6	C₂×C₄₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063371	—	2	2	no	—
8	#14583704	—	2	8	no	—
9	#14583704	C₂²	1	16	yes	complemented
10	C₂×#13063353	—	2	6	no	—
11	C₂×#13063347	C₂²	1	6	yes	complemented
12	C₂²×D₃₄₄	C₂²	1	6	yes	complemented
13	#14583337	C₂²	1	48	yes	complemented
14	#14583331	C₂²	1	24	yes	complemented
15	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×#13063270	C₂²	1	6	yes	complemented
17	C₂×D₆₈₈	C₂²	1	3	yes	complemented
18	C₂×D₆₈₈	C₂²	1	6	yes	complemented
19	C₂×D₆₈₈	C₂²	1	24	yes	complemented
20	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#504	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₄₃	C₂²	1	3	yes	complemented
23	C₂²×C₁₆×C₄₃	C₂²	1	3	yes	complemented
24	#13063371	—	2	16	no	—
25	#13063371	—	4	8	no	—
26	C₂×#10621902	—	4	2	no	—
27	C₂×#10621898	—	2	6	no	—
28	C₂²×D₁₇₂	—	2	6	no	—
29	C₂×D₄×D₄₃	—	4	6	no	—
30	#13063353	—	2	48	no	—
31	#13063347	—	2	12	no	—
32	#13063347	C₂³	1	24	yes	complemented
33	C₂×Dic₁₇₂	C₂³	1	2	yes	—
34	C₂×#10621873	—	2	6	no	—
35	C₂×D₃₄₄	—	2	6	no	—
36	C₂×D₃₄₄	—	2	12	no	—
37	C₂×D₃₄₄	C₂³	1	6	yes	—
38	C₂×D₃₄₄	C₂³	1	24	yes	complemented
39	Dic₃₄₄	C₂³	1	4	yes	complemented
40	#13063270	C₂³	1	24	yes	complemented
41	D₆₈₈	C₂³	1	12	yes	complemented
42	D₆₈₈	C₂³	1	24	yes	complemented
43	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
44	#131×C₄₃	D₄	1	4	yes	—
45	#131×C₄₃	C₂³	1	4	yes	—
46	C₂×M₁₆×C₄₃	D₄	1	3	yes	—
47	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
48	M₃₂×C₄₃	C₂³	1	12	yes	complemented
49	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₄₃	C₂³	1	3	yes	—
51	C₂×C₁₆×C₄₃	C₂³	1	12	yes	complemented
52	C₂³×D₄₃	—	4	6	no	—
53	#41×C₄₃	D₈	1	4	yes	—
54	#41×C₄₃	C₂×D₄	1	4	yes	—
55	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
56	C₂×D₄×C₄₃	D₈	1	3	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
58	C₂×C₂²⋊D₄₃	—	4	6	no	—
59	#10621902	—	4	16	no	—
60	D₄×D₄₃	—	4	48	no	—
61	#10621898	—	2	24	no	—
62	#10621898	—	4	12	no	—
63	C₂×D₁₇₂	—	2	6	no	—
64	C₂×D₁₇₂	—	2	24	no	—
65	C₂×D₁₇₂	—	4	6	no	—
66	C₂×D₁₇₂	—	4	12	no	—
67	C₂×C₄×D₄₃	—	4	6	no	—
68	C₂×Dic₈₆	—	2	2	no	—
69	M₁₆×C₄₃	C₂×D₄	1	12	yes	—
70	C₂×C₈×C₄₃	C₂×D₄	1	3	yes	—
71	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
72	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
74	Dic₁₇₂	—	2	2	no	—
75	Dic₁₇₂	C₂⁴	1	4	yes	—
76	D₃₄₄	—	2	6	no	—
77	D₃₄₄	—	2	24	no	—
78	D₃₄₄	C₂⁴	1	12	yes	—
79	#10621873	—	2	24	no	—
80	C₁₆×C₄₃	C₂⁴	1	2	yes	—
81	C₁₆×C₄₃	C₂⁴	1	6	yes	—
82	C₂³×C₄₃	C₂×D₈	1	3	yes	—
83	C₂²×D₄₃	—	4	6	no	—
84	C₂²×D₄₃	—	4	24	no	—
85	C₂²×D₄₃	—	16	12	no	—
86	Q₈×C₄₃	C₂×D₈	1	4	yes	—
87	D₄×C₄₃	C₂×D₈	1	12	yes	—
88	C₂×C₄×C₄₃	C₂×D₈	1	3	yes	—
89	C₂×C₄×C₄₃	C₂×D₈	1	6	yes	—
90	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
92	C₂²⋊D₄₃	—	4	24	no	—
93	C₂×Dic₄₃	—	4	2	no	—
94	D₁₇₂	—	2	12	no	—
95	D₁₇₂	—	4	6	no	—
96	D₁₇₂	—	4	24	no	—
97	C₄×D₄₃	—	4	24	no	—
98	Dic₈₆	—	2	4	no	—
99	Dic₈₆	—	4	2	no	—
100	C₈×C₄₃	C₂²×D₄	1	6	yes	—
101	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₄₃	C₂⁵	1	1	yes	characteristic
103	C₂×#3415	—	43	1	no	maximal, Sylow
104	C₂²×C₄₃	—	2	6	no	—
105	C₂²×C₄₃	C₂²×D₈	1	6	yes	—
106	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
107	C₂×D₄₃	—	4	12	no	—
108	C₂×D₄₃	—	8	12	no	—
109	C₂×D₄₃	—	16	24	no	—
110	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₄₃	C₂²×D₈	1	6	yes	—
112	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
113	Dic₄₃	—	4	4	no	—
114	C₂×#576	—	43	2	no	—
115	#3415	—	43	16	no	—
116	C₂×#509	—	43	6	no	—
117	C₂×#508	—	43	3	no	—
118	C₂²×D₁₆	—	43	3	no	—
119	C₂×#504	D₄₃	1	1	yes	characteristic, complemented
120	C₂×C₄₃	—	2	6	no	—
121	C₂×C₄₃	#3415	1	2	yes	complemented
122	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
123	D₄₃	—	8	12	no	—
124	C₂×H(5,𝔽₂)	—	86	2	no	—
125	#576	—	43	16	no	—
126	#576	—	86	8	no	—
127	C₂×AGL(1,ℤ/8)	—	86	6	no	—
128	C₂×#135	—	43	6	no	—
129	C₂²×D₈	—	43	6	no	—
130	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
131	#509	—	43	48	no	—
132	#508	—	43	24	no	—
133	C₂×Q₃₂	—	43	1	no	—
134	C₂×SD₃₂	—	43	6	no	—
135	C₂×D₁₆	—	43	3	no	—
136	C₂×D₁₆	—	43	6	no	—
137	C₂×D₁₆	—	43	24	no	—
138	#504	C₂×D₄₃	1	8	yes	complemented
139	C₂×M₃₂	C₂×D₄₃	1	3	yes	complemented
140	C₂²×C₁₆	C₂×D₄₃	1	3	yes	complemented
141	C₄₃	C₂×#3415	1	1	yes	characteristic, complemented, Sylow
142	H(5,𝔽₂)	—	86	16	no	—
143	H(5,𝔽₂)	—	172	8	no	—
144	C₂×#41	—	86	6	no	—
145	C₂×#41	—	172	2	no	—
146	C₂×#41	D₁₇₂	1	1	yes	characteristic
147	C₂²×D₄	—	86	6	no	—
148	C₂²×D₄	—	172	6	no	—
149	AGL(1,ℤ/8)	—	86	48	no	—
150	#135	—	43	24	no	—
151	#135	—	86	12	no	—
152	C₂×Q₁₆	—	43	2	no	—
153	C₂×SD₁₆	—	86	6	no	—
154	C₂×D₈	—	43	6	no	—
155	C₂×D₈	—	43	24	no	—
156	C₂×D₈	—	86	6	no	—
157	C₂×D₈	—	86	12	no	—
158	#131	D₁₇₂	1	4	yes	—
159	#131	C₂²×D₄₃	1	4	yes	—
160	C₂×M₁₆	D₁₇₂	1	3	yes	—
161	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
162	Q₃₂	—	43	4	no	—
163	SD₃₂	—	43	24	no	—
164	D₁₆	—	43	12	no	—
165	D₁₆	—	43	24	no	—
166	M₃₂	C₂²×D₄₃	1	12	yes	complemented
167	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₄₃	1	3	yes	—
169	C₂×C₁₆	C₂²×D₄₃	1	12	yes	complemented
170	C₂⁴	—	172	6	no	—
171	#41	—	86	24	no	—
172	#41	—	172	16	no	—
173	#41	—	172	12	no	—
174	#41	D₃₄₄	1	4	yes	—
175	#41	C₂×D₁₇₂	1	4	yes	—
176	C₂×Q₈	—	86	2	no	—
177	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
178	C₂×D₄	—	86	6	no	—
179	C₂×D₄	—	86	24	no	—
180	C₂×D₄	—	172	6	no	—
181	C₂×D₄	—	172	48	no	—
182	C₂×D₄	—	172	6	no	—
183	C₂×D₄	—	172	12	no	—
184	C₂×D₄	D₃₄₄	1	3	yes	—
185	C₂²×C₄	—	172	6	no	—
186	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
187	Q₁₆	—	43	4	no	—
188	Q₁₆	—	86	2	no	—
189	SD₁₆	—	86	24	no	—
190	D₈	—	43	12	no	—
191	D₈	—	86	6	no	—
192	D₈	—	86	24	no	—
193	M₁₆	C₂×D₁₇₂	1	12	yes	—
194	C₂×C₈	C₂×D₁₇₂	1	3	yes	—
195	C₂×C₈	C₂×D₁₇₂	1	6	yes	—
196	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
197	C₂×C₈	C₂³×D₄₃	1	6	yes	—
198	C₁₆	C₂³×D₄₃	1	2	yes	—
199	C₁₆	C₂³×D₄₃	1	6	yes	—
200	C₂³	—	172	6	no	—
201	C₂³	—	172	24	no	—
202	C₂³	—	688	12	no	—
203	C₂³	C₂×D₃₄₄	1	3	yes	—
204	Q₈	—	86	4	no	—
205	Q₈	—	172	2	no	—
206	Q₈	C₂×D₃₄₄	1	4	yes	—
207	D₄	—	86	12	no	—
208	D₄	—	172	24	no	—
209	D₄	—	172	24	no	—
210	D₄	—	172	6	no	—
211	D₄	C₂×D₃₄₄	1	12	yes	—
212	C₂×C₄	—	172	2	no	—
213	C₂×C₄	—	172	24	no	—
214	C₂×C₄	C₂×D₃₄₄	1	6	yes	—
215	C₂×C₄	C₂×D₃₄₄	1	3	yes	—
216	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
217	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
218	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
219	C₈	C₂²×D₁₇₂	1	6	yes	—
220	C₈	C₂⁴×D₄₃	1	1	yes	characteristic
221	C₂²	—	2	6	no	—
222	C₂²	—	344	12	no	—
223	C₂²	—	172	12	no	—
224	C₂²	—	688	24	no	—
225	C₂²	C₂²×D₃₄₄	1	6	yes	—
226	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
227	C₄	—	172	4	no	—
228	C₄	C₂²×D₃₄₄	1	6	yes	—
229	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
230	C₄	C₂³×D₁₇₂	1	1	yes	characteristic
231	C₂	—	2	6	no	—
232	C₂	—	344	12	no	—
233	C₂	#19147736	1	2	yes	direct product factor
234	C₂	C₂³×D₃₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₈×C₄₃	C₂⁵
Frattini	C₈	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#504	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#504	D₄₃
hypocenter	C₄₃	C₂×#3415
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#14583704	C₂	semidirect
2. #19147736	C₂	direct (N × H)
3. C₂×#14583337	C₂	semidirect
4. C₂×#14583331	C₂	semidirect
5. C₂²×D₆₈₈	C₂	semidirect
6. C₂×C₄₃×#504	C₂	semidirect
9. #14583704	C₂²	semidirect
11. C₂×#13063347	C₂²	semidirect
12. C₂²×D₃₄₄	C₂²	semidirect
13. #14583337	C₂²	semidirect
14. #14583331	C₂²	semidirect
15. C₂×Dic₃₄₄	C₂²	semidirect
16. C₂×#13063270	C₂²	semidirect
17. C₂×D₆₈₈	C₂²	semidirect
18. C₂×D₆₈₈	C₂²	semidirect
19. C₂×D₆₈₈	C₂²	semidirect
21. C₄₃×#504	C₂²	semidirect
22. C₂×M₃₂×C₄₃	C₂²	semidirect
23. C₂²×C₁₆×C₄₃	C₂²	semidirect
32. #13063347	C₂³	semidirect
38. C₂×D₃₄₄	C₂³	semidirect
39. Dic₃₄₄	C₂³	semidirect
40. #13063270	C₂³	semidirect
41. D₆₈₈	C₂³	semidirect
42. D₆₈₈	C₂³	semidirect
48. M₃₂×C₄₃	C₂³	semidirect
51. C₂×C₁₆×C₄₃	C₂³	semidirect
119. C₂×#504	D₄₃	semidirect
121. C₂×C₄₃	#3415	semidirect
138. #504	C₂×D₄₃	semidirect
139. C₂×M₃₂	C₂×D₄₃	semidirect
140. C₂²×C₁₆	C₂×D₄₃	semidirect
141. C₄₃	C₂×#3415	semidirect
166. M₃₂	C₂²×D₄₃	semidirect
169. C₂×C₁₆	C₂²×D₄₃	semidirect
233. C₂	#19147736	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×#14583704	2
#19147736	2
C₂×#14583337	2
C₂×#14583331	2
C₂²×D₆₈₈	2
C₂×C₄₃×#504	2
C₂×#3415	43

Subgroups — C2×#19147736 #26003109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19147736 #26003109