Subgroups — #25012769

Order 11008.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582727	C₂	1	2	yes	maximal, complemented
2	#19134624	C₂	1	2	yes	maximal
3	#19132687	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063259	C₂²	1	4	yes	—
6	#14582759	—	2	2	no	—
7	#14582727	—	2	4	no	—
8	#14582662	—	2	2	no	—
9	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₄₃	—	2	4	no	—
11	C₂²×C₁₆×C₄₃	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	—	2	4	no	—
15	C₂×C₂²⋊Dic₄₃	—	4	2	no	—
16	#13063259	—	2	4	no	—
17	#13063259	—	2	4	no	—
18	C₄₃⋊#95	—	4	2	no	—
19	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₃	—	2	2	no	—
21	C₂²×C₈×C₄₃	—	2	2	no	—
22	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₄₃	—	2	2	no	—
27	C₂×C₁₆×C₄₃	—	2	8	no	—
28	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
31	C₂⁴×C₄₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₃	—	4	4	no	—
39	C₂×C₈×C₄₃	—	2	2	no	—
40	C₂×C₈×C₄₃	—	2	2	no	—
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	4	no	—
43	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₄₃	#41	1	2	yes	—
48	C₂²⋊Dic₄₃	—	8	4	no	—
49	C₄⋊Dic₄₃	—	4	4	no	—
50	C₄⋊Dic₄₃	—	4	4	no	—
51	C₁₆×C₄₃	—	2	2	no	—
52	C₁₆×C₄₃	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₄₃	—	2	2	no	—
54	C₂³×C₄₃	—	2	2	no	—
55	C₂³×C₄₃	#107	1	2	yes	—
56	C₂³×C₄₃	SD₃₂	1	4	yes	—
57	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	—	2	2	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	4	no	—
62	C₂×C₄×C₄₃	#107	1	2	yes	—
63	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#121	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₄₃	#135	1	2	yes	—
67	C₂×Dic₄₃	—	8	4	no	—
68	C₂×Dic₄₃	—	8	4	no	—
69	C₂×Dic₄₃	—	8	4	no	—
70	C₈×C₄₃	—	2	2	no	—
71	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₄₃	#117	1	1	yes	characteristic
73	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
74	#12835	—	43	1	no	maximal, Sylow
75	C₂²×C₄₃	—	2	2	no	—
76	C₂²×C₄₃	—	2	2	no	—
77	C₂²×C₄₃	—	2	8	no	—
78	C₂²×C₄₃	#367	1	4	yes	complemented
79	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₄₃	#466	1	2	yes	—
81	C₂²×C₄₃	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₄₃	#508	1	2	yes	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	#380	1	1	yes	characteristic
85	C₄×C₄₃	#427	1	1	yes	characteristic
86	C₄×C₄₃	#467	1	2	yes	—
87	Dic₄₃	—	16	4	no	—
88	C₂³×C₁₆	D₄₃	1	1	yes	characteristic
89	C₂×#367	—	43	2	no	—
90	#1808	—	43	2	no	—
91	#1380	—	43	2	no	—
92	C₂×C₄₃	—	2	4	no	—
93	C₂×C₄₃	#1808	1	1	yes	characteristic
94	C₂×C₄₃	C₂×#367	1	1	yes	characteristic
95	C₂×C₄₃	#2158	1	1	yes	characteristic
96	C₂×C₄₃	#2217	1	4	yes	—
97	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
98	C₂²×C₁₆	—	2	4	no	—
99	C₂²×C₁₆	Dic₄₃	1	4	yes	complemented
100	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
102	C₂×#107	—	43	4	no	—
103	#380	—	86	2	no	—
104	#367	—	86	4	no	—
105	#340	—	86	2	no	—
106	C₄₃	#12835	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	Dic₈₆	1	2	yes	—
111	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
113	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	86	4	no	—
115	C₂×C₂²⋊C₄	—	172	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₄₃	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₄₃	1	4	yes	—
121	#107	—	86	4	no	—
122	#107	—	86	4	no	—
123	#95	—	172	2	no	—
124	C₂⁴	D₃₄₄	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	172	4	no	—
128	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
129	C₂²×C₄	Dic₁₇₂	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
131	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
137	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
138	C₂×C₈	#10621898	1	2	yes	—
139	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
141	C₄⋊C₄	—	172	4	no	—
142	C₄⋊C₄	—	172	4	no	—
143	C₂²⋊C₄	—	344	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊Dic₄₃	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#13063259	1	2	yes	—
149	C₂³	#13063270	1	4	yes	—
150	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	344	4	no	—
156	C₂×C₄	—	344	4	no	—
157	C₂×C₄	—	344	4	no	—
158	C₂×C₄	#13063259	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
160	C₂×C₄	#13063299	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
162	C₂×C₄	#13063347	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
165	C₈	#13063284	1	1	yes	characteristic
166	C₈	#13063304	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	#14582727	1	4	yes	complemented
171	C₂²	C₂×#13063259	1	1	yes	characteristic
172	C₂²	#14583135	1	2	yes	—
173	C₂²	C₂×#13063270	1	2	yes	—
174	C₂²	#14583331	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	688	4	no	—
177	C₄	#14582759	1	1	yes	characteristic
178	C₄	#14582937	1	1	yes	characteristic
179	C₄	#14583140	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	#19134624	1	1	yes	characteristic
182	C₂	C₂×#14582727	1	1	yes	characteristic
183	C₂	#19136379	1	1	yes	characteristic
184	C₂	#19136691	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂³×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂⁴×C₄₃	D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂³×C₁₆	D₄₃
hypocenter	C₄₃	#12835
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#14582727	C₂	semidirect
3. #19132687	C₂	semidirect
11. C₂²×C₁₆×C₄₃	C₄	semidirect
52. C₁₆×C₄₃	C₂²⋊C₄	semidirect
78. C₂²×C₄₃	#367	semidirect
99. C₂²×C₁₆	Dic₄₃	semidirect
106. C₄₃	#12835	semidirect
145. C₁₆	C₂²⋊Dic₄₃	semidirect
170. C₂²	#14582727	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582727	2
#19134624	2
#19132687	2
C₂³×C₁₆×C₄₃	2
#12835	43