Subgroups — #25002387

Order 11008.

950 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#14582654	C₂	1	4	yes	maximal
3	C₂×C₄₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₃⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
6	C₂×C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	8	yes	complemented
8	#14582654	—	2	8	no	—
9	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×#376	—	2	4	no	—
12	C₂×#95×C₄₃	C₄	1	4	yes	—
13	C₂²×C₄₃⋊C₈	—	2	8	no	—
14	C₂²×C₄₃⋊C₈	—	2	4	no	—
15	C₂²×C₄₃⋊C₈	—	2	4	no	—
16	C₂²×C₄₃⋊C₈	—	2	8	no	—
17	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
18	C₄⋊(C₄₃⋊C₈)	—	2	16	no	—
19	C₂³×C₄×C₄₃	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	—	2	4	no	—
23	C₂×C₄²×C₄₃	—	2	4	no	—
24	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₄₃	D₄	1	4	yes	—
26	C₂×C₄²×C₄₃	D₄	1	2	yes	—
27	C₂×C₄²×C₄₃	D₄	1	4	yes	—
28	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
29	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
30	#95×C₄₃	—	2	8	no	—
31	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	—	2	4	no	—
33	C₂²×C₄×C₄₃	—	2	4	no	—
34	C₂²×C₄×C₄₃	—	2	8	no	—
35	C₂²×C₄×C₄₃	—	2	16	no	—
36	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
37	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₄₃	#41	1	4	yes	—
47	C₄²×C₄₃	—	2	8	no	—
48	C₄²×C₄₃	—	2	8	no	—
49	C₄²×C₄₃	C₂²⋊C₄	1	8	yes	—
50	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
51	C₂×C₄₃⋊C₈	—	2	8	no	—
52	C₂×C₄₃⋊C₈	—	2	16	no	—
53	C₂×C₄₃⋊C₈	—	4	8	no	—
54	C₂×C₄₃⋊C₈	—	4	4	no	—
55	C₂×C₄₃⋊C₈	—	4	4	no	—
56	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
57	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
60	C₂³×C₄₃	#117	1	2	yes	—
61	C₂³×C₄₃	#131	1	8	yes	—
62	C₂×C₄×C₄₃	—	2	8	no	—
63	C₂×C₄×C₄₃	—	2	8	no	—
64	C₂×C₄×C₄₃	—	2	8	no	—
65	C₂×C₄×C₄₃	—	2	8	no	—
66	C₂×C₄×C₄₃	—	2	16	no	—
67	C₂×C₄×C₄₃	—	4	8	no	—
68	C₂×C₄×C₄₃	—	4	8	no	—
69	C₂×C₄×C₄₃	#95	1	8	yes	—
70	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
74	C₂×C₄×C₄₃	#121	1	4	yes	—
75	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
76	C₂×C₄×C₄₃	#124	1	2	yes	—
77	C₂×C₄×C₄₃	#125	1	2	yes	—
78	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₄₃⋊C₈	—	4	8	no	—
82	#10833	—	43	1	no	maximal, Sylow
83	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
84	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
85	C₂²×C₄₃	#380	1	2	yes	—
86	C₂²×C₄₃	#382	1	1	yes	characteristic
87	C₂²×C₄₃	#384	1	1	yes	characteristic
88	C₂²×C₄₃	#405	1	4	yes	—
89	C₂²×C₄₃	#408	1	4	yes	—
90	C₂²×C₄₃	#424	1	8	yes	—
91	C₂²×C₄₃	#424	1	4	yes	—
92	C₂²×C₄₃	#433	1	8	yes	—
93	C₄×C₄₃	—	2	8	no	—
94	C₄×C₄₃	—	4	8	no	—
95	C₄×C₄₃	#389	1	4	yes	—
96	C₄×C₄₃	#390	1	2	yes	—
97	C₄×C₄₃	#391	1	2	yes	—
98	C₂²×C₄⋊C₈	—	43	2	no	—
99	C₂×#376	D₄₃	1	1	yes	characteristic
100	C₂×#336	—	43	4	no	—
101	C₂×C₄₃	C₄⋊#95	1	1	yes	characteristic
102	C₂×C₄₃	#1729	1	2	yes	—
103	C₂×C₄₃	#1768	1	4	yes	—
104	C₂×C₄₃	#1798	1	4	yes	—
105	C₂×C₄₃	#1832	1	2	yes	—
106	C₂×C₄₃	#1843	1	2	yes	—
107	C₂³×C₈	—	43	4	no	—
108	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
109	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	43	8	no	—
111	C₂×C₄⋊C₈	—	86	4	no	—
112	C₂×C₄⋊C₈	—	86	4	no	—
113	#376	—	2	4	no	—
114	C₂×#95	Dic₄₃	1	4	yes	—
115	#336	—	86	8	no	—
116	C₄₃	#10833	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₄	C₂×Dic₄₃	1	4	yes	—
118	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
119	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
120	C₂²×C₈	—	43	8	no	—
121	C₂²×C₈	—	86	8	no	—
122	C₂²×C₈	—	86	4	no	—
123	C₂²×C₈	—	86	8	no	—
124	C₂²×C₈	—	86	4	no	—
125	C₂×C₄²	—	2	4	no	—
126	C₂×C₄²	—	2	4	no	—
127	C₂×C₄²	Dic₈₆	1	4	yes	—
128	C₂×C₄²	Dic₈₆	1	2	yes	—
129	C₂×C₄²	C₄×D₄₃	1	4	yes	—
130	C₂×C₄²	D₁₇₂	1	4	yes	—
131	C₂×C₄²	D₁₇₂	1	2	yes	—
132	C₂×C₄²	C₂²⋊D₄₃	1	4	yes	—
133	C₄⋊C₈	—	86	16	no	—
134	#95	—	2	8	no	—
135	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	16	no	—
140	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
141	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
142	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
143	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
144	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
147	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
149	C₂²×C₄	#10621898	1	4	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
151	C₂×C₈	—	86	8	no	—
152	C₂×C₈	—	86	16	no	—
153	C₂×C₈	—	172	8	no	—
154	C₂×C₈	—	172	4	no	—
155	C₂×C₈	—	172	4	no	—
156	C₄²	—	2	8	no	—
157	C₄²	—	2	8	no	—
158	C₄²	Dic₄₃⋊C₄	1	8	yes	—
159	C₄²	(C₂×D₄₃)⋊C₄	1	8	yes	—
160	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
162	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
163	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
164	C₂³	#13063284	1	2	yes	—
165	C₂³	#13063332	1	8	yes	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	C₄₃⋊#95	1	8	yes	—
174	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
175	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
176	C₂×C₄	C₄×D₁₇₂	1	4	yes	—
177	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
178	C₂×C₄	#13063299	1	4	yes	—
179	C₂×C₄	#13063304	1	4	yes	—
180	C₂×C₄	#13063313	1	2	yes	—
181	C₂×C₄	#13063317	1	2	yes	—
182	C₂×C₄	#13063322	1	1	yes	characteristic
183	C₂×C₄	#13063325	1	2	yes	—
184	C₂×C₄	#13063325	1	1	yes	characteristic
185	C₈	—	172	8	no	—
186	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
187	C₂²	C₄×C₄⋊Dic₄₃	1	2	yes	—
188	C₂²	#14582759	1	2	yes	—
189	C₂²	#14582765	1	1	yes	characteristic
190	C₂²	#14582769	1	1	yes	characteristic
191	C₂²	#14582855	1	4	yes	—
192	C₂²	#14582866	1	4	yes	—
193	C₂²	#14582926	1	4	yes	—
194	C₂²	#14582926	1	8	yes	—
195	C₂²	#14582960	1	8	yes	—
196	C₄	—	2	8	no	—
197	C₄	—	4	8	no	—
198	C₄	#14582797	1	4	yes	—
199	C₄	#14582801	1	2	yes	—
200	C₄	#14582805	1	2	yes	—
201	C₂	#19132825	1	1	yes	characteristic
202	C₂	#19134292	1	2	yes	—
203	C₂	#19134452	1	4	yes	—
204	C₂	#19134579	1	4	yes	—
205	C₂	#19134715	1	2	yes	—
206	C₂	#19134749	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₃
commutator	C₂³×C₄₃	C₂×C₄²
Fitting	C₂×C₄₃×#376	C₂
socle, prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#376	D₄₃
hypocenter	C₄₃	#10833
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
7. C₂×C₄⋊(C₄₃⋊C₈)	C₄	semidirect
116. C₄₃	#10833	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₃⋊C₈)	2
C₂×#14582654	2
C₂×C₄₃×#376	2
#10833	43