Subgroups — #19134327

Order 5504.

420 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063284	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063284	C₂	1	1	yes	maximal, characteristic
3	#14582665	C₂	1	4	yes	maximal
4	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
5	#13063284	—	2	1	no	—
6	#13063284	—	2	1	no	—
7	#13063284	—	2	2	no	—
8	#13063284	—	2	1	no	—
9	#13063284	—	2	1	no	—
10	#13063284	C₄	1	4	yes	complemented
11	C₂×C₄⋊Dic₄₃	—	2	1	no	—
12	C₂×C₄⋊Dic₄₃	C₄	1	2	yes	complemented
13	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₃	—	2	1	no	—
16	C₂×C₄×Dic₄₃	C₄	1	2	yes	complemented
17	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
18	#117×C₄₃	—	2	2	no	—
19	#117×C₄₃	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
21	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₄₃	—	2	1	no	—
23	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₃	—	2	2	no	—
25	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	D₄	1	2	yes	—
34	C₂²×C₄×C₄₃	Q₈	1	2	yes	—
35	C₂²×Dic₄₃	—	2	2	no	—
36	C₂²×Dic₄₃	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₄₃	—	2	4	no	—
38	C₂²⋊C₄×C₄₃	—	2	2	no	—
39	C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
40	C₄²×C₄₃	—	2	4	no	—
41	C₂²⋊Dic₄₃	—	2	2	no	—
42	C₂²⋊Dic₄₃	—	2	2	no	—
43	C₂²⋊Dic₄₃	—	2	2	no	—
44	C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₄₃	—	2	4	no	—
46	C₄⋊Dic₄₃	—	2	2	no	—
47	C₄⋊Dic₄₃	—	2	2	no	—
48	C₄×Dic₄₃	—	2	4	no	—
49	C₄×Dic₄₃	—	2	2	no	—
50	C₄×Dic₄₃	—	2	2	no	—
51	C₂³×C₄₃	—	4	2	no	—
52	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	2	4	no	—
60	C₂×C₄×C₄₃	—	4	4	no	—
61	C₂×C₄×C₄₃	C₄²	1	2	yes	—
62	C₂×C₄×C₄₃	C₄²	1	2	yes	—
63	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
70	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
77	C₂×Dic₄₃	—	2	4	no	—
78	C₂×Dic₄₃	—	2	4	no	—
79	C₂×Dic₄₃	—	4	4	no	—
80	C₂×Dic₄₃	—	4	4	no	—
81	C₂²×C₄₃	—	2	2	no	—
82	C₂²×C₄₃	—	2	2	no	—
83	C₂²×C₄₃	—	2	2	no	—
84	C₂²×C₄₃	—	4	1	no	—
85	C₂²×C₄₃	—	4	1	no	—
86	C₂²×C₄₃	—	4	1	no	—
87	C₂²×C₄₃	—	4	1	no	—
88	C₂²×C₄₃	C₂×C₄²	1	1	yes	characteristic
89	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₄₃	—	4	4	no	—
96	C₄×C₄₃	#95	1	2	yes	—
97	C₄×C₄₃	#95	1	2	yes	—
98	C₄×C₄₃	#95	1	2	yes	—
99	C₄×C₄₃	#95	1	2	yes	—
100	Dic₄₃	—	4	4	no	—
101	Dic₄₃	—	4	4	no	—
102	#1740	—	43	1	no	maximal, Sylow
103	C₂×C₄₃	—	2	1	no	—
104	C₂×C₄₃	—	2	1	no	—
105	C₂×C₄₃	—	2	1	no	—
106	C₂×C₄₃	—	2	1	no	—
107	C₂×C₄₃	—	2	1	no	—
108	C₂×C₄₃	—	2	1	no	—
109	C₂×C₄₃	C₂×#95	1	1	yes	characteristic
110	C₂×C₄₃	#410	1	2	yes	—
111	C₂×#117	—	43	1	no	—
112	C₂×#117	—	43	1	no	—
113	C₂×#117	D₄₃	1	1	yes	characteristic
114	#342	—	43	4	no	—
115	C₄₃	#1740	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
117	#117	—	2	2	no	—
118	#117	—	43	4	no	—
119	#117	—	86	1	no	—
120	#117	—	86	1	no	—
121	#117	—	86	1	no	—
122	#117	—	86	2	no	—
123	#117	—	86	1	no	—
124	#117	Dic₄₃	1	4	yes	complemented
125	C₂×C₄⋊C₄	—	2	1	no	—
126	C₂×C₄⋊C₄	—	43	2	no	—
127	C₂×C₄⋊C₄	—	86	1	no	—
128	C₂×C₂²⋊C₄	—	43	2	no	—
129	C₂×C₂²⋊C₄	—	43	2	no	—
130	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	—	43	2	no	—
133	C₂×C₄²	—	86	1	no	—
134	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	43	4	no	—
137	C₂²×C₄	—	86	2	no	—
138	C₂²×C₄	Dic₈₆	1	2	yes	—
139	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
140	C₂²×C₄	C₄×D₄₃	1	2	yes	—
141	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
142	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
143	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
147	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	86	2	no	—
150	C₄⋊C₄	—	86	2	no	—
151	C₄⋊C₄	—	86	4	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	43	4	no	—
154	C₂²⋊C₄	—	86	2	no	—
155	C₂²⋊C₄	—	86	2	no	—
156	C₂²⋊C₄	—	86	2	no	—
157	C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	86	2	no	—
160	C₄²	—	86	4	no	—
161	C₄²	—	86	2	no	—
162	C₂³	—	4	2	no	—
163	C₂³	C₂×Dic₈₆	1	1	yes	characteristic
164	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
166	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
167	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
168	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
169	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	86	4	no	—
173	C₂×C₄	—	86	4	no	—
174	C₂×C₄	—	172	4	no	—
175	C₂×C₄	—	172	4	no	—
176	C₂×C₄	C₄×Dic₄₃	1	2	yes	—
177	C₂×C₄	C₄×Dic₄₃	1	2	yes	—
178	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
179	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
180	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
181	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
182	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
183	C₂×C₄	C₄⋊Dic₄₃	1	2	yes	—
184	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
186	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
187	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
188	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
189	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
191	C₂×C₄	C₂²⋊Dic₄₃	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	4	1	no	—
199	C₂²	C₂×C₄×Dic₄₃	1	1	yes	characteristic
200	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
201	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
202	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
203	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
204	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
205	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
206	C₄	—	4	4	no	—
207	C₄	—	172	4	no	—
208	C₄	—	172	4	no	—
209	C₄	C₄₃⋊#95	1	2	yes	—
210	C₄	C₄₃⋊#95	1	2	yes	—
211	C₄	C₄₃⋊#95	1	2	yes	—
212	C₄	C₄₃⋊#95	1	2	yes	—
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	C₂×C₄₃⋊#95	1	1	yes	characteristic
220	C₂	#14582872	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄₃	C₂×C₄²
Frattini	C₂⁴	C₂²×D₄₃
Fitting	C₂×#117×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#1740
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
10. #13063284	C₄	semidirect
12. C₂×C₄⋊Dic₄₃	C₄	semidirect
16. C₂×C₄×Dic₄₃	C₄	semidirect
19. #117×C₄₃	C₄	semidirect
115. C₄₃	#1740	semidirect
124. #117	Dic₄₃	semidirect