Subgroups — #19137073

Order 5504.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14582806	C₂	1	2	yes	maximal
2	#14582798	C₂	1	2	yes	maximal
3	#14582785	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊Dic₄₃	C₂	1	2	yes	maximal
6	#14582745	C₂	1	2	yes	maximal
7	C₄²×Dic₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Dic₄₃	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
13	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
16	C₄₃⋊#95	C₂²	1	2	yes	—
17	C₄₃⋊#95	C₂²	1	4	yes	—
18	C₄₃⋊#95	C₂²	1	4	yes	—
19	C₄₃⋊#95	C₂²	1	4	yes	—
20	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₄₃	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
26	C₂×Q₈×C₄₃	—	2	4	no	—
27	C₂×Q₈×C₄₃	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₃	C₂³	1	4	yes	—
34	C₂²×Dic₄₃	C₂³	1	2	yes	—
35	C₂²×Dic₄₃	C₂³	1	2	yes	—
36	C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
37	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
38	C₄²×C₄₃	C₂×C₄	1	8	yes	—
39	C₄²×C₄₃	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₄₃	—	2	4	no	—
41	C₄⋊Dic₄₃	—	2	8	no	—
42	C₄×Dic₄₃	—	2	4	no	—
43	C₄×Dic₄₃	—	2	2	no	—
44	C₄×Dic₄₃	—	2	4	no	—
45	C₄×Dic₄₃	—	2	4	no	—
46	C₄×Dic₄₃	Q₈	1	4	yes	complemented
47	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
48	Q₈×C₄₃	—	2	8	no	—
49	C₂×C₄×C₄₃	—	2	2	no	—
50	C₂×C₄×C₄₃	—	2	4	no	—
51	C₂×C₄×C₄₃	—	2	4	no	—
52	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₄₃	#41	1	4	yes	—
59	C₂×C₄×C₄₃	#41	1	2	yes	—
60	C₂×Dic₄₃	—	2	2	no	—
61	C₂×Dic₄₃	—	2	2	no	—
62	C₂×Dic₄₃	—	2	2	no	—
63	C₂×Dic₄₃	—	2	2	no	—
64	C₂×Dic₄₃	—	2	4	no	—
65	C₂×Dic₄₃	—	2	4	no	—
66	C₂×Dic₄₃	—	2	4	no	—
67	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
68	C₂×Dic₄₃	#41	1	4	yes	—
69	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
70	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂²×C₄₃	C₂×#41	1	2	yes	—
72	C₂²×C₄₃	C₂×#41	1	2	yes	—
73	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
74	C₄×C₄₃	—	2	4	no	—
75	C₄×C₄₃	—	2	2	no	—
76	C₄×C₄₃	—	2	4	no	—
77	C₄×C₄₃	#117	1	4	yes	—
78	C₄×C₄₃	C₄×Q₈	1	4	yes	—
79	Dic₄₃	—	2	4	no	—
80	Dic₄₃	—	2	4	no	—
81	Dic₄₃	—	4	4	no	—
82	#2295	—	43	1	no	maximal, Sylow
83	C₂×C₄₃	C₂×#117	1	1	yes	characteristic
84	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄₃	C₄×#41	1	1	yes	characteristic
86	C₂×C₄₃	#529	1	2	yes	—
87	C₂×C₄₃	#533	1	2	yes	—
88	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic
89	#391	—	43	2	no	—
90	#389	—	43	2	no	—
91	#387	—	43	4	no	—
92	C₄×C₄⋊C₄	—	43	1	no	—
93	C₄×C₄⋊C₄	—	43	2	no	—
94	#376	—	43	2	no	—
95	C₄³	—	43	1	no	—
96	C₄₃	#2295	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
98	C₄×Q₈	Dic₄₃	1	8	yes	complemented
99	C₂×C₄⋊C₄	—	43	4	no	—
100	C₂×C₄⋊C₄	—	43	2	no	—
101	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
102	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
103	C₂×C₄²	—	43	1	no	—
104	C₂×C₄²	—	43	2	no	—
105	C₂×C₄²	—	43	2	no	—
106	C₂×C₄²	—	43	1	no	—
107	C₂×C₄²	—	43	2	no	—
108	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
109	C₂×C₄²	C₂×D₄₃	1	2	yes	—
110	#95	—	43	4	no	—
111	#95	—	43	4	no	—
112	#95	—	43	2	no	—
113	#95	—	43	4	no	—
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×Dic₄₃	1	4	yes	—
116	C₂²×C₄	—	43	2	no	—
117	C₂²×C₄	—	43	2	no	—
118	C₂²×C₄	—	43	4	no	—
119	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
124	C₄⋊C₄	—	86	8	no	—
125	C₄⋊C₄	—	86	4	no	—
126	C₄⋊C₄	C₂×Dic₄₃	1	8	yes	—
127	C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
128	C₄²	—	43	4	no	—
129	C₄²	—	86	4	no	—
130	C₄²	—	86	4	no	—
131	C₄²	—	86	2	no	—
132	C₄²	—	86	4	no	—
133	C₄²	C₂×Dic₄₃	1	8	yes	—
134	C₄²	C₂×Dic₄₃	1	4	yes	—
135	C₂³	C₂³×D₄₃	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	43	4	no	—
141	C₂×C₄	—	43	4	no	—
142	C₂×C₄	—	86	2	no	—
143	C₂×C₄	—	86	2	no	—
144	C₂×C₄	—	86	2	no	—
145	C₂×C₄	—	86	2	no	—
146	C₂×C₄	—	86	4	no	—
147	C₂×C₄	—	86	4	no	—
148	C₂×C₄	—	86	4	no	—
149	C₂×C₄	#10621898	1	4	yes	—
150	C₂×C₄	Q₈×D₄₃	1	2	yes	—
151	C₂×C₄	#10621902	1	2	yes	—
152	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
153	C₂×C₄	C₂²×Dic₄₃	1	4	yes	—
154	C₂×C₄	C₂²×Dic₄₃	1	4	yes	—
155	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
156	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
157	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
158	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
159	C₂²	#41×D₄₃	1	2	yes	—
160	C₂²	C₂×#10621898	1	2	yes	—
161	C₂²	C₂×#10621902	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	86	4	no	—
166	C₄	—	86	4	no	—
167	C₄	—	172	4	no	—
168	C₄	#13063284	1	4	yes	—
169	C₄	Q₈×Dic₄₃	1	4	yes	—
170	C₂	C₂×#13063284	1	1	yes	characteristic
171	C₂	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
172	C₂	#41×Dic₄₃	1	1	yes	characteristic
173	C₂	#14583436	1	2	yes	—
174	C₂	#14583453	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Frattini	C₂³	C₂³×D₄₃
Fitting	C₂×C₄×Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₃
hypocenter	C₄₃	#2295
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₄×Q₈×C₄₃	C₄	semidirect
46. C₄×Dic₄₃	Q₈	semidirect
96. C₄₃	#2295	semidirect
98. C₄×Q₈	Dic₄₃	semidirect