Subgroups — #19137507

Order 5504.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063290	C₂	1	2	yes	maximal
3	#14582795	C₂	1	2	yes	maximal
4	#14582794	C₂	1	4	yes	maximal
5	#14582768	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₄₃	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#384	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₈₆	C₄	1	8	yes	complemented
11	#13063290	—	2	8	no	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
16	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
18	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
19	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
21	C₄₃⋊#95	C₂²	1	2	yes	—
22	C₄₃⋊#95	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₃	C₂³	1	2	yes	—
32	C₂²×Dic₄₃	C₂³	1	2	yes	—
33	C₂²×Dic₄₃	C₂³	1	2	yes	—
34	C₂²×Dic₄₃	C₂³	1	2	yes	—
35	C₂×Dic₈₆	C₂×C₄	1	4	yes	—
36	C₂×Dic₈₆	D₄	1	4	yes	—
37	C₂×Dic₈₆	Q₈	1	4	yes	—
38	C₄⋊C₄×C₄₃	—	2	4	no	—
39	C₄⋊C₄×C₄₃	—	2	4	no	—
40	C₄⋊C₄×C₄₃	—	4	2	no	—
41	C₄²×C₄₃	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₄₃	—	2	2	no	—
43	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₄₃	Q₈	1	4	yes	complemented
45	Dic₄₃⋊C₄	—	2	8	no	—
46	Dic₄₃⋊C₄	—	4	4	no	—
47	Dic₄₃⋊C₄	C₂×C₄	1	8	yes	—
48	C₄×Dic₄₃	—	2	4	no	—
49	C₄×Dic₄₃	—	2	8	no	—
50	C₄×Dic₄₃	C₂×C₄	1	8	yes	—
51	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	4	no	—
55	C₂×C₄×C₄₃	—	2	4	no	—
56	C₂×C₄×C₄₃	—	2	4	no	—
57	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₄₃	#41	1	2	yes	—
64	C₂×Dic₄₃	—	2	2	no	—
65	C₂×Dic₄₃	—	2	2	no	—
66	C₂×Dic₄₃	—	2	2	no	—
67	C₂×Dic₄₃	—	2	2	no	—
68	C₂×Dic₄₃	—	2	2	no	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
73	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
74	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
75	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
76	Dic₈₆	C₄⋊C₄	1	16	yes	complemented
77	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
78	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
82	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₄₃	—	2	2	no	—
85	C₄×C₄₃	—	4	4	no	—
86	C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₄×C₄₃	C₄×Q₈	1	4	yes	complemented
88	Dic₄₃	—	2	4	no	—
89	Dic₄₃	—	2	4	no	—
90	Dic₄₃	—	4	2	no	—
91	Dic₄₃	C₂×C₄⋊C₄	1	8	yes	—
92	Q₈×C₄⋊C₄	—	43	1	no	maximal, Sylow
93	C₂×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄₃	#520	1	1	yes	characteristic
96	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
97	C₂×C₄₃	#550	1	1	yes	characteristic
98	C₂×C₄₃	#557	1	1	yes	characteristic
99	C₂×C₄₃	Q₈²	1	1	yes	characteristic
100	C₂×C₄×Q₈	—	43	2	no	—
101	C₂×C₄×Q₈	—	43	1	no	—
102	#389	—	43	4	no	—
103	#389	—	43	2	no	—
104	#384	—	43	2	no	—
105	#384	D₄₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	43	2	no	—
107	C₄×C₄⋊C₄	—	43	1	no	—
108	C₄₃	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
109	C₂²×Q₈	—	43	1	no	—
110	C₄×Q₈	—	43	8	no	—
111	C₄×Q₈	—	86	8	no	—
112	C₂×C₄⋊C₄	—	43	2	no	—
113	C₂×C₄⋊C₄	—	43	1	no	—
114	C₂×C₄⋊C₄	—	43	1	no	—
115	C₂×C₄⋊C₄	—	43	4	no	—
116	C₂×C₄⋊C₄	—	43	1	no	—
117	C₂×C₄⋊C₄	—	43	4	no	—
118	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
121	C₂×C₄²	—	43	2	no	—
122	C₂×C₄²	—	43	4	no	—
123	C₂×C₄²	—	43	2	no	—
124	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
125	#95	—	43	4	no	—
126	#95	—	43	2	no	—
127	C₂×Q₈	—	43	4	no	—
128	C₂×Q₈	—	43	4	no	—
129	C₂×Q₈	—	43	4	no	—
130	C₂²×C₄	—	43	2	no	—
131	C₂²×C₄	—	43	2	no	—
132	C₂²×C₄	—	43	2	no	—
133	C₂²×C₄	—	43	2	no	—
134	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
135	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	43	8	no	—
142	C₄⋊C₄	—	43	4	no	—
143	C₄⋊C₄	—	43	4	no	—
144	C₄⋊C₄	—	86	8	no	—
145	C₄⋊C₄	—	86	2	no	—
146	C₄⋊C₄	—	172	4	no	—
147	C₄²	—	43	8	no	—
148	C₄²	—	86	8	no	—
149	C₄²	—	86	4	no	—
150	C₄²	C₄×D₄₃	1	4	yes	—
151	C₂³	C₂³×D₄₃	1	1	yes	characteristic
152	Q₈	—	43	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	43	4	no	—
159	C₂×C₄	—	43	4	no	—
160	C₂×C₄	—	43	4	no	—
161	C₂×C₄	—	43	4	no	—
162	C₂×C₄	—	43	4	no	—
163	C₂×C₄	—	86	2	no	—
164	C₂×C₄	—	86	2	no	—
165	C₂×C₄	—	86	2	no	—
166	C₂×C₄	—	86	2	no	—
167	C₂×C₄	—	86	2	no	—
168	C₂×C₄	—	86	2	no	—
169	C₂×C₄	—	86	2	no	—
170	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
173	C₂×C₄	D₄×D₄₃	1	2	yes	—
174	C₂×C₄	#10621900	1	2	yes	—
175	C₂×C₄	Q₈×D₄₃	1	2	yes	—
176	C₂×C₄	Q₈×D₄₃	1	2	yes	—
177	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
180	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
181	C₂²	C₂×#10621900	1	1	yes	characteristic
182	C₂²	#13063370	1	1	yes	characteristic
183	C₂²	#13063373	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	43	8	no	—
187	C₄	—	86	4	no	—
188	C₄	—	86	4	no	—
189	C₄	—	172	2	no	—
190	C₄	C₄⋊C₄×D₄₃	1	4	yes	—
191	C₄	#13063290	1	4	yes	complemented
192	C₂	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
193	C₂	C₂×#13063290	1	1	yes	characteristic
194	C₂	#14583380	1	1	yes	characteristic
195	C₂	#14583564	1	1	yes	characteristic
196	C₂	#14583569	1	1	yes	characteristic
197	C₂	#14583624	1	1	yes	characteristic
198	C₂	#14583629	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#384	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	#384	D₄₃
hypocenter	C₄₃	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₈₆	C₄	semidirect
44. C₄⋊Dic₄₃	Q₈	semidirect
76. Dic₈₆	C₄⋊C₄	semidirect
87. C₄×C₄₃	C₄×Q₈	semidirect
108. C₄₃	Q₈×C₄⋊C₄	semidirect
191. C₄	#13063290	semidirect