Subgroups — #25005372

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582765	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582667	C₂	1	4	yes	maximal
3	#19134289	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
7	#14582765	—	2	2	no	—
8	#14582765	—	2	2	no	—
9	C₂×C₄₃⋊#95	C₄	1	4	yes	—
10	#14582667	—	2	8	no	—
11	#14582654	—	2	2	no	—
12	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#336	—	2	2	no	—
15	C₂³×Dic₄₃	C₂×C₄	1	4	yes	—
16	C₂×#10621877	—	2	2	no	—
17	C₂×#10621877	—	2	4	no	—
18	C₂×#10621877	—	2	4	no	—
19	C₂×#10621877	Q₈	1	4	yes	—
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	—	2	4	no	—
22	C₂×C₄⋊Dic₄₃	—	2	2	no	—
23	C₂×C₄⋊Dic₄₃	—	2	2	no	—
24	C₂×C₄⋊Dic₄₃	—	2	4	no	—
25	C₄₃⋊#95	—	2	8	no	—
26	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₄₃	—	2	2	no	—
29	C₂×M₁₆×C₄₃	—	2	4	no	—
30	C₂×M₁₆×C₄₃	—	2	4	no	—
31	C₂×M₁₆×C₄₃	D₄	1	4	yes	—
32	C₂²×C₈×C₄₃	—	2	2	no	—
33	C₂×C₄²×C₄₃	—	2	4	no	—
34	C₂×C₄²×C₄₃	D₄	1	2	yes	—
35	C₂×C₄²×C₄₃	D₄	1	2	yes	—
36	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	—	2	4	no	—
38	C₂²×C₄×C₄₃	—	4	4	no	—
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₄₃	#41	1	4	yes	—
45	C₂²×C₄×C₄₃	#41	1	2	yes	—
46	C₂²×C₄×C₄₃	#41	1	2	yes	—
47	C₂²×Dic₄₃	—	2	4	no	—
48	C₂²×Dic₄₃	—	2	8	no	—
49	C₂²×Dic₄₃	—	4	8	no	—
50	M₁₆×C₄₃	—	2	8	no	—
51	M₁₆×C₄₃	—	4	4	no	—
52	C₂×C₈×C₄₃	—	2	4	no	—
53	C₂×C₈×C₄₃	—	4	2	no	—
54	C₂×C₈×C₄₃	—	4	2	no	—
55	C₄²×C₄₃	—	2	4	no	—
56	C₄²×C₄₃	—	2	4	no	—
57	C₄⋊Dic₄₃	—	4	4	no	—
58	C₄⋊Dic₄₃	—	4	2	no	—
59	C₄⋊Dic₄₃	—	4	2	no	—
60	#10621877	—	2	8	no	—
61	#10621877	—	4	4	no	—
62	C₂×C₄₃⋊C₈	—	2	4	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	C₂×C₄₃⋊C₈	—	4	2	no	—
65	C₂³×C₄₃	—	2	4	no	—
66	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂³×C₄₃	#117	1	2	yes	—
68	C₂³×C₄₃	C₄×D₄	1	2	yes	—
69	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	2	8	no	—
72	C₂×C₄×C₄₃	—	4	8	no	—
73	C₂×C₄×C₄₃	#121	1	2	yes	—
74	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
75	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₄×C₄₃	#123	1	2	yes	—
77	C₂×C₄×C₄₃	#123	1	4	yes	—
78	C₂×C₄×C₄₃	#123	1	2	yes	—
79	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#125	1	4	yes	—
82	C₂×C₄×C₄₃	#126	1	4	yes	—
83	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×Dic₄₃	—	4	4	no	—
86	C₂×Dic₄₃	—	4	8	no	—
87	C₂×Dic₄₃	—	8	8	no	—
88	C₈×C₄₃	—	4	4	no	—
89	C₄₃⋊C₈	—	4	4	no	—
90	#11355	—	43	1	no	maximal, Sylow
91	C₂²×C₄₃	—	2	4	no	—
92	C₂²×C₄₃	—	2	4	no	—
93	C₂²×C₄₃	—	2	4	no	—
94	C₂²×C₄₃	#381	1	1	yes	characteristic
95	C₂²×C₄₃	#387	1	4	yes	—
96	C₂²×C₄₃	#390	1	1	yes	characteristic
97	C₂²×C₄₃	#391	1	1	yes	characteristic
98	C₂²×C₄₃	#413	1	4	yes	—
99	C₄×C₄₃	—	4	4	no	—
100	C₄×C₄₃	#397	1	2	yes	—
101	C₄×C₄₃	#398	1	2	yes	—
102	C₄×C₄₃	#399	1	2	yes	—
103	C₄×C₄₃	#400	1	2	yes	—
104	Dic₄₃	—	8	4	no	—
105	C₂×#382	—	43	1	no	—
106	C₂×#343	—	43	4	no	—
107	#1728	—	43	1	no	—
108	#1728	D₄₃	1	1	yes	characteristic
109	C₂×C₄₃	—	2	4	no	—
110	C₂×C₄₃	#1445	1	1	yes	characteristic
111	C₂×C₄₃	#1822	1	2	yes	—
112	C₂×C₄₃	#1967	1	2	yes	—
113	C₂×C₄₃	#1998	1	2	yes	—
114	C₂²×M₁₆	—	43	2	no	—
115	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
116	C₂²×C₄⋊C₄	—	43	2	no	—
117	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
118	#382	—	86	2	no	—
119	#382	—	86	2	no	—
120	C₂×#95	—	43	4	no	—
121	#343	—	86	8	no	—
122	#336	—	2	2	no	—
123	#336	—	86	2	no	—
124	C₄₃	#11355	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₄	—	43	4	no	—
126	C₂³×C₄	C₄×D₄₃	1	2	yes	—
127	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	—	43	4	no	—
132	C₂×M₁₆	—	86	2	no	—
133	C₂×M₁₆	—	86	4	no	—
134	C₂×M₁₆	—	86	4	no	—
135	C₂×M₁₆	D₁₇₂	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	86	2	no	—
138	C₂×C₄⋊C₄	—	86	4	no	—
139	C₂×C₄⋊C₄	—	86	2	no	—
140	C₂×C₄⋊C₄	—	86	4	no	—
141	C₂×C₄⋊C₄	—	86	2	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	D₁₇₂	1	2	yes	—
144	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
145	#95	—	86	8	no	—
146	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	86	8	no	—
150	C₂²×C₄	—	86	4	no	—
151	C₂²×C₄	—	172	8	no	—
152	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
153	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
154	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
155	C₂²×C₄	#10621898	1	2	yes	—
156	C₂²×C₄	#10621900	1	4	yes	—
157	C₂²×C₄	#10621900	1	2	yes	—
158	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	M₁₆	—	4	4	no	—
162	M₁₆	—	86	8	no	—
163	M₁₆	—	172	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	86	4	no	—
168	C₂×C₈	—	172	2	no	—
169	C₂×C₈	—	172	2	no	—
170	C₄⋊C₄	—	172	4	no	—
171	C₄⋊C₄	—	172	2	no	—
172	C₄⋊C₄	—	172	2	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	4	no	—
176	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
177	C₂³	#13063281	1	2	yes	—
178	C₂³	C₄×D₁₇₂	1	2	yes	—
179	C₂³	#13063290	1	2	yes	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	344	8	no	—
184	C₂×C₄	—	172	4	no	—
185	C₂×C₄	—	172	8	no	—
186	C₂×C₄	#13063299	1	2	yes	—
187	C₂×C₄	#13063305	1	4	yes	—
188	C₂×C₄	#13063305	1	2	yes	—
189	C₂×C₄	#13063306	1	2	yes	—
190	C₂×C₄	#13063309	1	4	yes	—
191	C₂×C₄	#13063309	1	2	yes	—
192	C₂×C₄	#13063312	1	1	yes	characteristic
193	C₂×C₄	#13063313	1	1	yes	characteristic
194	C₂×C₄	#13063315	1	4	yes	—
195	C₂×C₄	#13063318	1	4	yes	—
196	C₂×C₄	#13063322	1	1	yes	characteristic
197	C₂×C₄	#13063323	1	1	yes	characteristic
198	C₈	—	4	4	no	—
199	C₈	—	172	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	#14582760	1	1	yes	characteristic
204	C₂²	#14582784	1	4	yes	—
205	C₂²	#14582801	1	1	yes	characteristic
206	C₂²	#14582804	1	1	yes	characteristic
207	C₂²	#14582881	1	4	yes	—
208	C₄	—	4	4	no	—
209	C₄	—	344	4	no	—
210	C₄	#14582824	1	2	yes	—
211	C₄	#14582829	1	2	yes	—
212	C₄	#14582836	1	2	yes	—
213	C₄	#14582839	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	#19132862	1	1	yes	characteristic
216	C₂	#19134676	1	2	yes	—
217	C₂	#19135444	1	2	yes	—
218	C₂	#19135587	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₃)⋊C₄
commutator, prime radical	C₂⁴×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1728	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1728	D₄₃
hypocenter	C₄₃	#11355
LCS	G ▷ C₂⁴×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
124. C₄₃	#11355	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582765	2
C₂×#14582667	2
#19134289	2
C₄₃×#1728	2
#11355	43