Subgroups — #19143713

Order 5504.

388 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10621898	C₂	1	1	yes	maximal, characteristic
2	#14583026	C₂	1	2	yes	maximal, complemented
3	#14582976	C₂	1	2	yes	maximal, complemented
4	#14582970	C₂	1	4	yes	maximal, complemented
5	#14582955	C₂	1	2	yes	maximal, complemented
6	#14582952	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582921	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10621898	C₄	1	1	yes	characteristic
10	C₂×#10621872	C₂²	1	4	yes	—
11	C₄×Dic₈₆	C₄	1	1	yes	characteristic
12	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
13	C₄×D₁₇₂	C₄	1	1	yes	characteristic
14	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₄₃	C₄	1	2	yes	—
16	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
17	#13063284	C₄	1	1	yes	characteristic
18	#13063282	C₄	1	2	yes	—
19	C₄²×D₄₃	C₂²	1	2	yes	—
20	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
22	Dic₄₃⋊C₈	C₂²	1	4	yes	—
23	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
24	(C₂×D₄₃)⋊C₈	C₂²	1	4	yes	—
25	#13063235	C₂²	1	4	yes	—
26	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
27	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
30	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
35	#10621898	—	2	4	no	—
36	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₄₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₄₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₄₃	C₂³	1	2	yes	—
40	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	—	4	2	no	—
42	C₂×C₈×C₄₃	C₂³	1	4	yes	—
43	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
48	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
49	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
52	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₄₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₄₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₄₃	C₂³	1	2	yes	—
57	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
58	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
59	#10621872	—	4	4	no	—
60	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	4	1	no	—
65	C₂×C₄×C₄₃	M₁₆	1	4	yes	complemented
66	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₄₃	—	2	4	no	—
73	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
76	D₁₇₂	—	2	2	no	—
77	C₄×D₄₃	—	2	4	no	—
78	C₄×D₄₃	—	2	2	no	—
79	C₄×D₄₃	—	2	2	no	—
80	C₄×D₄₃	—	4	2	no	—
81	Dic₈₆	—	2	2	no	—
82	C₈×C₄₃	—	2	4	no	—
83	C₄₃⋊C₈	—	2	2	no	—
84	C₄₃⋊C₈	—	2	2	no	—
85	C₂²×C₄₃	—	2	1	no	—
86	C₂²×C₄₃	—	2	1	no	—
87	C₂²×C₄₃	C₂×M₁₆	1	2	yes	—
88	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₄₃	—	2	2	no	—
90	C₂×D₄₃	—	2	2	no	—
91	C₂×D₄₃	—	2	2	no	—
92	C₄×C₄₃	—	2	1	no	—
93	C₄×C₄₃	—	2	1	no	—
94	C₄×C₄₃	—	2	1	no	—
95	C₄×C₄₃	C₂×M₁₆	1	2	yes	—
96	C₄×C₄₃	C₂×M₁₆	1	2	yes	—
97	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₄₃	—	2	2	no	—
100	Dic₄₃	—	2	2	no	—
101	Dic₄₃	—	2	2	no	—
102	#2989	—	43	1	no	maximal, Sylow
103	C₂×C₄₃	—	2	2	no	—
104	C₂×C₄₃	#520	1	1	yes	characteristic
105	C₂×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₄₃	#568	1	1	yes	characteristic
107	D₄₃	—	4	2	no	—
108	C₄×#41	—	43	1	no	—
109	#446	—	43	2	no	—
110	#436	—	43	2	no	—
111	#435	—	43	4	no	—
112	#432	—	43	2	no	—
113	#431	—	43	1	no	—
114	#423	—	43	2	no	—
115	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
116	C₄₃	#2989	1	1	yes	characteristic, complemented, Sylow
117	C₂×#41	—	43	1	no	—
118	C₂×M₁₆	—	43	4	no	—
119	C₂²×C₈	C₂×D₄₃	1	2	yes	—
120	C₄×Q₈	—	43	1	no	—
121	C₄×Q₈	—	43	1	no	—
122	C₄×D₄	—	43	2	no	—
123	C₄×D₄	—	43	2	no	—
124	C₄×D₄	—	43	1	no	—
125	C₄×D₄	—	43	1	no	—
126	#117	—	43	1	no	—
127	#117	—	43	2	no	—
128	C₂×C₄²	—	43	2	no	—
129	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
130	C₄⋊C₈	—	43	4	no	—
131	C₄⋊C₈	—	43	1	no	—
132	C₄⋊C₈	—	43	1	no	—
133	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
134	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
135	C₂²⋊C₈	—	43	2	no	—
136	C₂²⋊C₈	—	43	4	no	—
137	#97	—	43	4	no	—
138	C₄×C₈	—	43	2	no	—
139	#41	—	86	4	no	—
140	C₂×Q₈	—	43	1	no	—
141	C₂×D₄	—	43	2	no	—
142	C₂×D₄	—	43	1	no	—
143	C₂²×C₄	—	43	2	no	—
144	C₂²×C₄	—	43	2	no	—
145	C₂²×C₄	—	43	2	no	—
146	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
149	M₁₆	—	172	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	43	2	no	—
152	C₂×C₈	—	43	2	no	—
153	C₂×C₈	C₂²×D₄₃	1	4	yes	—
154	C₄⋊C₄	—	43	2	no	—
155	C₄⋊C₄	—	43	2	no	—
156	C₄⋊C₄	—	43	1	no	—
157	C₄⋊C₄	—	43	1	no	—
158	C₂²⋊C₄	—	43	2	no	—
159	C₂²⋊C₄	—	43	2	no	—
160	C₂²⋊C₄	—	43	2	no	—
161	C₄²	—	43	2	no	—
162	C₄²	—	43	2	no	—
163	C₄²	—	43	2	no	—
164	C₄²	C₄×D₄₃	1	1	yes	characteristic
165	C₄²	C₄×D₄₃	1	1	yes	characteristic
166	C₄²	C₂²×D₄₃	1	1	yes	characteristic
167	C₄²	C₂²×D₄₃	1	1	yes	characteristic
168	C₂³	—	43	2	no	—
169	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
170	Q₈	—	86	2	no	—
171	D₄	—	86	4	no	—
172	D₄	—	86	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	43	2	no	—
177	C₂×C₄	—	43	2	no	—
178	C₂×C₄	—	43	2	no	—
179	C₂×C₄	—	86	4	no	—
180	C₂×C₄	—	86	2	no	—
181	C₂×C₄	—	86	2	no	—
182	C₂×C₄	—	172	2	no	—
183	C₂×C₄	#10621872	1	4	yes	complemented
184	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	86	2	no	—
192	C₈	—	86	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	86	2	no	—
196	C₂²	—	86	2	no	—
197	C₂²	—	86	2	no	—
198	C₂²	C₂×#10621872	1	2	yes	—
199	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	86	2	no	—
204	C₄	—	86	2	no	—
205	C₄	—	86	2	no	—
206	C₄	C₂×#10621872	1	2	yes	—
207	C₄	C₂×#10621872	1	2	yes	—
208	C₄	#13063370	1	1	yes	characteristic
209	C₄	#13063373	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	172	2	no	—
212	C₂	#14583380	1	1	yes	characteristic
213	C₂	C₂²×#10621872	1	1	yes	characteristic
214	C₂	#14583665	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#2989
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14583026	C₂	semidirect
3. #14582976	C₂	semidirect
4. #14582970	C₂	semidirect
5. #14582955	C₂	semidirect
6. #14582952	C₂	semidirect
7. #14582921	C₂	semidirect
8. C₂×C₄⋊C₈×C₄₃	C₂	semidirect
65. C₂×C₄×C₄₃	M₁₆	semidirect
115. C₂×C₄⋊C₈	D₄₃	semidirect
116. C₄₃	#2989	semidirect
183. C₂×C₄	#10621872	semidirect