Subgroups — C₂×#19136427 #25249761

Order 11008.

742 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063265	C₂	1	2	yes	maximal, complemented
2	#19136427	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063265	—	2	8	no	—
8	C₂×#13063265	C₂²	1	8	yes	complemented
9	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
10	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	complemented
13	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	8	yes	complemented
14	C₄×C₄₃⋊C₁₆	C₂²	1	4	yes	complemented
15	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₄₃	C₄	1	1	yes	characteristic
17	C₈×D₄×C₄₃	C₄	1	4	yes	—
18	C₈×D₄×C₄₃	C₂²	1	4	yes	—
19	C₂×C₄⋊C₈×C₄₃	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	#13063265	—	2	16	no	—
23	C₂×C₄₃⋊C₁₆	—	2	4	no	—
24	C₂×C₄₃⋊C₁₆	—	2	2	no	—
25	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	complemented
26	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
27	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
28	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
29	C₂²×D₄×C₄₃	C₈	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₄₃	—	2	4	no	—
32	C₂²×C₈×C₄₃	—	2	4	no	—
33	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₄₃	C₈	1	4	yes	—
39	C₄×D₄×C₄₃	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₄₃	C₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₄₃	C₈	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
44	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₃	C₂³	1	2	yes	—
47	C₂⁴×C₄₃	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₄₃	—	2	4	no	—
49	C₂×D₄×C₄₃	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₄₃	—	2	4	no	—
59	C₂×C₈×C₄₃	—	2	2	no	—
60	C₂×C₈×C₄₃	—	2	8	no	—
61	C₂×C₈×C₄₃	—	2	4	no	—
62	C₂×C₈×C₄₃	—	2	4	no	—
63	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₃	#41	1	2	yes	—
68	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
70	C₂²⋊C₄×C₄₃	C₂×C₈	1	8	yes	—
71	C₄²×C₄₃	C₂×C₈	1	2	yes	—
72	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
73	C₄₃⋊C₁₆	—	2	4	no	—
74	C₄₃⋊C₁₆	—	2	2	no	—
75	C₄₃⋊C₁₆	C₂×D₄	1	4	yes	complemented
76	C₂³×C₄₃	—	2	4	no	—
77	C₂³×C₄₃	—	2	4	no	—
78	C₂³×C₄₃	—	2	4	no	—
79	C₂³×C₄₃	C₂²×C₈	1	4	yes	—
80	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
81	D₄×C₄₃	—	2	8	no	—
82	C₂×C₄×C₄₃	—	2	4	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	—	2	4	no	—
86	C₂×C₄×C₄₃	—	2	8	no	—
87	C₂×C₄×C₄₃	M₃₂	1	4	yes	complemented
88	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
91	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₄₃	#131	1	2	yes	—
94	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₄₃	—	2	4	no	—
96	C₈×C₄₃	—	2	2	no	—
97	C₈×C₄₃	C₄×D₄	1	2	yes	—
98	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
100	C₂×#2169	—	43	1	no	maximal, Sylow
101	C₂²×C₄₃	—	2	4	no	—
102	C₂²×C₄₃	—	2	4	no	—
103	C₂²×C₄₃	—	2	4	no	—
104	C₂²×C₄₃	—	4	8	no	—
105	C₂²×C₄₃	C₈×D₄	1	2	yes	—
106	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
107	C₂²×C₄₃	#504	1	2	yes	—
108	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	2	4	no	—
111	C₄×C₄₃	C₈×D₄	1	2	yes	—
112	C₄×C₄₃	C₂×M₃₂	1	4	yes	complemented
113	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
115	C₂²×M₃₂	—	43	2	no	—
116	C₂×C₈×D₄	D₄₃	1	1	yes	characteristic
117	#2169	—	43	8	no	—
118	C₂×C₄⋊C₁₆	—	43	1	no	—
119	C₂×C₂²⋊C₁₆	—	43	2	no	—
120	C₂×C₄×C₁₆	—	43	1	no	—
121	C₂×C₄₃	—	4	4	no	—
122	C₂×C₄₃	#2169	1	4	yes	complemented
123	C₂×C₄₃	C₂×C₈×D₄	1	1	yes	characteristic
124	C₂×C₄₃	C₂²×M₃₂	1	1	yes	characteristic
125	C₂×C₄₃	C₂×#504	1	1	yes	characteristic
126	C₂³×C₈	C₂×D₄₃	1	2	yes	—
127	C₂×C₄×D₄	Dic₄₃	1	1	yes	characteristic
128	C₂×M₃₂	—	43	8	no	—
129	C₂×M₃₂	—	86	8	no	—
130	C₂²×C₁₆	—	43	1	no	—
131	C₂²×C₁₆	—	43	1	no	—
132	C₂²×C₁₆	—	43	2	no	—
133	C₈×D₄	Dic₄₃	1	4	yes	—
134	C₈×D₄	C₂×D₄₃	1	4	yes	—
135	C₂×C₄⋊C₈	Dic₄₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₈	Dic₄₃	1	2	yes	—
137	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
138	C₄⋊C₁₆	—	43	4	no	—
139	C₂²⋊C₁₆	—	43	8	no	—
140	C₄×C₁₆	—	43	4	no	—
141	C₄₃	C₂×#2169	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	C₄₃⋊C₈	1	1	yes	characteristic
143	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
147	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
148	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
150	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
151	C₄×D₄	C₄₃⋊C₈	1	4	yes	—
152	C₄×D₄	C₂×Dic₄₃	1	4	yes	—
153	C₂×C₄⋊C₄	C₄₃⋊C₈	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₄₃⋊C₈	1	2	yes	—
155	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
156	M₃₂	—	86	16	no	—
157	C₂×C₁₆	—	43	4	no	—
158	C₂×C₁₆	—	43	2	no	—
159	C₂×C₁₆	—	43	2	no	—
160	C₂×C₁₆	—	43	4	no	—
161	C₂×C₁₆	—	86	4	no	—
162	C₂×C₁₆	—	86	2	no	—
163	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
164	C₂²⋊C₈	C₂×Dic₄₃	1	8	yes	—
165	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
166	C₄×C₈	C₂²×D₄₃	1	2	yes	—
167	C₂⁴	C₂×C₄₃⋊C₈	1	2	yes	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	C₂×C₄₃⋊C₈	1	4	yes	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
174	C₂²×C₄	C₂×C₄₃⋊C₈	1	4	yes	—
175	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	D₄×D₄₃	1	2	yes	—
184	C₂×C₈	#10621900	1	2	yes	—
185	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
186	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
187	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
188	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
189	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
190	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	8	yes	—
191	C₄²	C₂×C₄₃⋊C₈	1	2	yes	—
192	C₄²	C₂²×Dic₄₃	1	2	yes	—
193	C₁₆	—	43	4	no	—
194	C₁₆	—	86	4	no	—
195	C₁₆	—	86	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	C₂²×C₄₃⋊C₈	1	4	yes	—
200	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
201	D₄	—	2	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	#13063265	1	4	yes	complemented
208	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
209	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
210	C₂×C₄	C₂²×C₄₃⋊C₈	1	4	yes	—
211	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
212	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
213	C₂×C₄	#13063332	1	2	yes	—
214	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	D₄×Dic₄₃	1	2	yes	—
218	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
219	C₈	C₂×#10621900	1	1	yes	characteristic
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	4	8	no	—
224	C₂²	D₄×C₄₃⋊C₈	1	2	yes	—
225	C₂²	C₂×#13063265	1	2	yes	—
226	C₂²	#14583316	1	2	yes	—
227	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×C₄₃⋊C₈	1	2	yes	—
231	C₄	C₂×#13063265	1	4	yes	complemented
232	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
233	C₄	C₂×#13063332	1	1	yes	characteristic
234	C₂	—	4	4	no	—
235	C₂	#19136427	1	4	yes	direct product factor
236	C₂	C₂×D₄×C₄₃⋊C₈	1	1	yes	characteristic
237	C₂	C₂²×#13063265	1	1	yes	characteristic
238	C₂	C₂×#14583316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₈×D₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×C₈
prime radical	C₂²×D₄×C₄₃	C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₄₃
hypocenter	C₄₃	C₂×#2169
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063265	C₂	semidirect
2. #19136427	C₂	direct (N × H)
3. C₂×C₄⋊(C₄₃⋊C₁₆)	C₂	semidirect
4. C₂×C₂²⋊(C₄₃⋊C₁₆)	C₂	semidirect
5. C₂×C₄×C₄₃⋊C₁₆	C₂	semidirect
8. C₂×#13063265	C₂²	semidirect
12. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
13. C₂²⋊(C₄₃⋊C₁₆)	C₂²	semidirect
14. C₄×C₄₃⋊C₁₆	C₂²	semidirect
25. C₂×C₄₃⋊C₁₆	D₄	semidirect
75. C₄₃⋊C₁₆	C₂×D₄	semidirect
87. C₂×C₄×C₄₃	M₃₂	semidirect
112. C₄×C₄₃	C₂×M₃₂	semidirect
122. C₂×C₄₃	#2169	semidirect
141. C₄₃	C₂×#2169	semidirect
207. C₂×C₄	#13063265	semidirect
231. C₄	C₂×#13063265	semidirect
235. C₂	#19136427	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13063265	2
#19136427	2
C₂×C₄⋊(C₄₃⋊C₁₆)	2
C₂×C₂²⋊(C₄₃⋊C₁₆)	2
C₂×C₄×C₄₃⋊C₁₆	2
C₂×C₈×D₄×C₄₃	2
C₂×#2169	43

Subgroups — C2×#19136427 #25249761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19136427 #25249761