Subgroups — #25005731

Order 11008.

450 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136206	C₂	1	2	yes	maximal
2	#19132535	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063265	—	2	2	no	—
5	C₂²⋊(C₄₃⋊C₁₆)	—	2	2	no	—
6	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	—
7	C₂³×C₈×C₄₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₃	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
11	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
12	C₂×C₄⋊C₈×C₄₃	C₄	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
14	#13063265	—	4	4	no	—
15	C₂×C₄₃⋊C₁₆	—	2	4	no	—
16	C₂³×C₄×C₄₃	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₄₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₄₃	—	2	1	no	—
20	C₂²×C₈×C₄₃	—	2	1	no	—
21	C₂²×C₈×C₄₃	—	4	2	no	—
22	C₂²×C₈×C₄₃	—	4	2	no	—
23	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₃	D₄	1	2	yes	—
26	C₂²×C₈×C₄₃	D₄	1	2	yes	—
27	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	—	2	1	no	—
29	C₂×C₄²×C₄₃	—	2	1	no	—
30	C₂×C₄²×C₄₃	—	2	1	no	—
31	C₂×C₄²×C₄₃	—	2	1	no	—
32	C₂×C₄²×C₄₃	C₈	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₈×C₄₃	—	2	4	no	—
35	C₄⋊C₈×C₄₃	—	2	4	no	—
36	C₂⁴×C₄₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	4	no	—
40	C₂²×C₄×C₄₃	—	2	1	no	—
41	C₂²×C₄×C₄₃	—	2	1	no	—
42	C₂²×C₄×C₄₃	—	2	1	no	—
43	C₂²×C₄×C₄₃	—	2	1	no	—
44	C₂²×C₄×C₄₃	—	4	2	no	—
45	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₃	—	2	4	no	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	—	4	1	no	—
56	C₂×C₈×C₄₃	—	4	2	no	—
57	C₂×C₈×C₄₃	—	4	1	no	—
58	C₂×C₈×C₄₃	—	4	2	no	—
59	C₂×C₈×C₄₃	—	4	2	no	—
60	C₂×C₈×C₄₃	—	4	2	no	—
61	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	—	2	2	no	—
66	C₄²×C₄₃	—	2	2	no	—
67	C₄₃⋊C₁₆	—	4	4	no	—
68	C₂³×C₄₃	—	2	2	no	—
69	C₂³×C₄₃	—	2	2	no	—
70	C₂³×C₄₃	—	2	2	no	—
71	C₂³×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
72	C₂³×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₄₃	—	2	4	no	—
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	2	no	—
79	C₂×C₄×C₄₃	—	2	2	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	—	2	1	no	—
83	C₂×C₄×C₄₃	—	2	1	no	—
84	C₂×C₄×C₄₃	—	2	1	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	4	2	no	—
87	C₂×C₄×C₄₃	—	4	2	no	—
88	C₂×C₄×C₄₃	—	4	2	no	—
89	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₄₃	#99	1	2	yes	—
92	C₂×C₄×C₄₃	#101	1	2	yes	—
93	C₂×C₄×C₄₃	M₃₂	1	4	yes	complemented
94	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
96	C₈×C₄₃	—	2	2	no	—
97	C₈×C₄₃	—	2	2	no	—
98	C₈×C₄₃	—	4	2	no	—
99	#11440	—	43	1	no	maximal, Sylow
100	C₂²×C₄₃	—	2	8	no	—
101	C₂²×C₄₃	—	2	1	no	—
102	C₂²×C₄₃	—	2	1	no	—
103	C₂²×C₄₃	—	2	1	no	—
104	C₂²×C₄₃	—	2	1	no	—
105	C₂²×C₄₃	—	2	1	no	—
106	C₂²×C₄₃	—	2	1	no	—
107	C₂²×C₄₃	#324	1	2	yes	—
108	C₂²×C₄₃	#349	1	2	yes	—
109	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₂²×C₄₃	C₂×M₃₂	1	2	yes	—
111	C₄×C₄₃	—	2	4	no	—
112	C₄×C₄₃	—	2	1	no	—
113	C₄×C₄₃	—	2	1	no	—
114	C₄×C₄₃	—	4	2	no	—
115	C₄×C₄₃	#324	1	2	yes	—
116	C₄×C₄₃	C₂×#99	1	1	yes	characteristic
117	C₄×C₄₃	C₂×#101	1	1	yes	characteristic
118	C₂²×C₄⋊C₈	D₄₃	1	1	yes	characteristic
119	#2112	—	43	2	no	—
120	#1322	—	43	4	no	—
121	C₂×C₄₃	—	2	2	no	—
122	C₂×C₄₃	—	2	2	no	—
123	C₂×C₄₃	—	2	2	no	—
124	C₂×C₄₃	C₂×#324	1	1	yes	characteristic
125	C₂×C₄₃	#2112	1	1	yes	characteristic
126	C₂×C₄₃	C₂×#349	1	1	yes	characteristic
127	C₂³×C₈	Dic₄₃	1	1	yes	characteristic
128	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
129	C₂²×C₄²	Dic₄₃	1	1	yes	characteristic
130	C₂×M₃₂	—	86	2	no	—
131	C₂×C₄⋊C₈	—	2	2	no	—
132	C₂×C₄⋊C₈	—	2	2	no	—
133	C₂×C₄⋊C₈	Dic₄₃	1	2	yes	—
134	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
135	C₂²⋊C₁₆	—	43	4	no	—
136	C₂²⋊C₁₆	—	86	2	no	—
137	C₄₃	#11440	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₄	C₄₃⋊C₈	1	1	yes	characteristic
139	C₂³×C₄	C₄₃⋊C₈	1	1	yes	characteristic
140	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	—	4	2	no	—
144	C₂²×C₈	—	4	2	no	—
145	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
146	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
148	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
149	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₄₃⋊C₈	1	2	yes	—
155	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
156	M₃₂	—	172	4	no	—
157	C₂×C₁₆	—	86	4	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₂⁴	C₂×C₄₃⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
171	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
172	C₂²×C₄	#10621877	1	1	yes	characteristic
173	C₂²×C₄	#10621877	1	1	yes	characteristic
174	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₂²⋊Dic₄₃	1	2	yes	—
186	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₁₆	—	172	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
196	C₂³	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
197	C₂³	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
214	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
215	C₂×C₄	#13063240	1	2	yes	—
216	C₂×C₄	#13063244	1	2	yes	—
217	C₂×C₄	#13063265	1	4	yes	complemented
218	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
219	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₂²	—	2	8	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#14582625	1	2	yes	—
231	C₂²	#14582681	1	2	yes	—
232	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
233	C₂²	C₂×#13063265	1	2	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	4	2	no	—
238	C₄	#14582625	1	2	yes	—
239	C₄	C₂×#13063240	1	1	yes	characteristic
240	C₄	C₂×#13063244	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	2	no	—
243	C₂	—	2	2	no	—
244	C₂	C₂×#14582625	1	1	yes	characteristic
245	C₂	#19136206	1	1	yes	characteristic
246	C₂	C₂×#14582681	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₃
commutator	C₂³×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂²×C₄⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₄₃	C₂×C₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#11440
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #19132535	C₂	semidirect
93. C₂×C₄×C₄₃	M₃₂	semidirect
137. C₄₃	#11440	semidirect
217. C₂×C₄	#13063265	semidirect

Maximal subgroups

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

subgroup	index
#19136206	2
#19132535	2
C₂²×C₄⋊C₈×C₄₃	2
#11440	43