Subgroups — #25001853

Order 11008.

1422 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582863	C₂	1	2	yes	maximal
2	C₂×#14582664	C₂	1	4	yes	maximal, complemented
3	C₂⁴×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10621877	C₂²	1	4	yes	—
5	#14582863	—	2	4	no	—
6	#14582863	C₄	1	8	yes	—
7	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
8	#14582664	—	2	8	no	—
9	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₄₃	—	2	4	no	—
11	C₂³×C₈×C₄₃	—	2	8	no	—
12	C₂³×C₈×C₄₃	C₄	1	4	yes	—
13	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
14	C₂×#10621877	—	2	4	no	—
15	C₂×#10621877	—	2	8	no	—
16	C₂×#10621877	—	2	8	no	—
17	C₂×#10621877	C₂×C₄	1	8	yes	—
18	C₂²×C₄₃⋊C₈	—	2	4	no	—
19	C₂²⋊(C₄₃⋊C₈)	—	2	16	no	—
20	C₂⁵×C₄₃	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₄₃	—	2	4	no	—
22	C₂³×C₄×C₄₃	—	2	4	no	—
23	C₂³×C₄×C₄₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₃	D₄	1	2	yes	—
26	C₂³×C₄×C₄₃	D₄	1	4	yes	—
27	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₄₃	—	2	32	no	—
30	C₂²×C₈×C₄₃	—	2	8	no	—
31	C₂²×C₈×C₄₃	—	2	4	no	—
32	C₂²×C₈×C₄₃	—	2	4	no	—
33	C₂²×C₈×C₄₃	—	2	16	no	—
34	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₄₃	D₄	1	8	yes	—
36	C₂⁴×C₄₃	—	2	8	no	—
37	C₂⁴×C₄₃	—	2	4	no	—
38	C₂⁴×C₄₃	C₄⋊C₄	1	4	yes	—
39	C₂⁴×C₄₃	C₄⋊C₄	1	2	yes	—
40	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	—	2	4	no	—
42	C₂²×C₄×C₄₃	—	2	4	no	—
43	C₂²×C₄×C₄₃	—	2	16	no	—
44	C₂²×C₄×C₄₃	—	2	8	no	—
45	C₂²×C₄×C₄₃	—	2	8	no	—
46	C₂²×C₄×C₄₃	—	2	8	no	—
47	C₂²×C₄×C₄₃	—	2	4	no	—
48	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
49	C₂²×C₄×C₄₃	C₂²⋊C₄	1	8	yes	—
50	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₄₃	#41	1	4	yes	—
59	C₂×C₈×C₄₃	—	2	32	no	—
60	C₂×C₈×C₄₃	—	2	4	no	—
61	C₂×C₈×C₄₃	—	2	4	no	—
62	C₂×C₈×C₄₃	—	2	16	no	—
63	C₂×C₈×C₄₃	C₂²⋊C₄	1	8	yes	—
64	#10621877	—	2	16	no	—
65	#10621877	—	4	4	no	—
66	#10621877	—	4	4	no	—
67	C₂×C₄₃⋊C₈	—	2	8	no	—
68	C₂×C₄₃⋊C₈	—	4	8	no	—
69	C₂³×C₄₃	—	2	32	no	—
70	C₂³×C₄₃	—	2	16	no	—
71	C₂³×C₄₃	—	2	4	no	—
72	C₂³×C₄₃	—	2	8	no	—
73	C₂³×C₄₃	—	2	4	no	—
74	C₂³×C₄₃	—	2	4	no	—
75	C₂³×C₄₃	#95	1	4	yes	—
76	C₂³×C₄₃	#108	1	8	yes	—
77	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂³×C₄₃	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₄₃	—	2	4	no	—
81	C₂×C₄×C₄₃	—	2	8	no	—
82	C₂×C₄×C₄₃	—	2	8	no	—
83	C₂×C₄×C₄₃	—	2	4	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	—	2	16	no	—
86	C₂×C₄×C₄₃	#95	1	4	yes	—
87	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	#117	1	2	yes	—
92	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
93	C₂×C₄×C₄₃	#120	1	4	yes	—
94	C₂×C₄×C₄₃	#121	1	4	yes	—
95	C₂×C₄×C₄₃	#123	1	4	yes	—
96	C₈×C₄₃	—	2	8	no	—
97	C₄₃⋊C₈	—	4	8	no	—
98	#10723	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	—	2	32	no	—
100	C₂²×C₄₃	—	2	16	no	—
101	C₂²×C₄₃	—	2	8	no	—
102	C₂²×C₄₃	—	2	4	no	—
103	C₂²×C₄₃	—	2	4	no	—
104	C₂²×C₄₃	—	2	4	no	—
105	C₂²×C₄₃	#341	1	8	yes	complemented
106	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
107	C₂²×C₄₃	#380	1	2	yes	—
108	C₂²×C₄₃	#385	1	4	yes	—
109	C₂²×C₄₃	C₂×#108	1	4	yes	—
110	C₄×C₄₃	—	2	4	no	—
111	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
112	C₄×C₄₃	#379	1	1	yes	characteristic
113	C₄×C₄₃	#381	1	1	yes	characteristic
114	C₄×C₄₃	#386	1	4	yes	—
115	C₂⁴×C₈	D₄₃	1	1	yes	characteristic
116	C₂×#407	—	43	2	no	—
117	C₂×#341	—	43	4	no	—
118	C₂×C₄₃	—	2	8	no	—
119	C₂×C₄₃	—	2	4	no	—
120	C₂×C₄₃	C₂²⋊#95	1	1	yes	characteristic
121	C₂×C₄₃	C₂×#341	1	2	yes	—
122	C₂×C₄₃	#1810	1	4	yes	—
123	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
124	C₂²×M₁₆	—	43	4	no	—
125	C₂³×C₈	—	2	4	no	—
126	C₂³×C₈	—	2	8	no	—
127	C₂³×C₈	Dic₄₃	1	4	yes	—
128	C₂³×C₈	C₂×D₄₃	1	2	yes	—
129	#407	—	43	8	no	—
130	#407	—	86	4	no	—
131	C₂×C₂²⋊C₈	—	86	4	no	—
132	#341	—	86	8	no	—
133	C₄₃	#10723	1	1	yes	characteristic, complemented, Sylow
134	C₂⁵	Dic₈₆	1	1	yes	characteristic
135	C₂³×C₄	—	2	4	no	—
136	C₂³×C₄	—	2	4	no	—
137	C₂³×C₄	Dic₈₆	1	1	yes	characteristic
138	C₂³×C₄	C₄×D₄₃	1	4	yes	—
139	C₂³×C₄	D₁₇₂	1	2	yes	—
140	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
141	C₂³×C₄	C₂²⋊D₄₃	1	4	yes	—
142	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
143	C₂×M₁₆	—	43	8	no	—
144	C₂×M₁₆	—	86	4	no	—
145	C₂×M₁₆	—	86	8	no	—
146	C₂×M₁₆	—	86	8	no	—
147	C₂²×C₈	—	2	32	no	—
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	2	4	no	—
151	C₂²×C₈	—	2	16	no	—
152	C₂²×C₈	—	86	4	no	—
153	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
154	C₂²×C₈	C₂²⋊D₄₃	1	8	yes	—
155	C₂²⋊C₈	—	86	16	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	—	2	4	no	—
158	C₂⁴	Dic₄₃⋊C₄	1	4	yes	—
159	C₂⁴	C₄⋊Dic₄₃	1	2	yes	—
160	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
169	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
170	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
171	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	8	yes	—
172	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
174	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
175	C₂²×C₄	#10621898	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
179	M₁₆	—	86	16	no	—
180	M₁₆	—	172	4	no	—
181	M₁₆	—	172	4	no	—
182	C₂×C₈	—	2	32	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	16	no	—
186	C₂×C₈	—	86	8	no	—
187	C₂×C₈	—	172	8	no	—
188	C₂×C₈	C₂²⋊Dic₄₃	1	8	yes	—
189	C₂³	—	2	32	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	8	no	—
195	C₂³	C₄₃⋊#95	1	4	yes	—
196	C₂³	#13063261	1	8	yes	—
197	C₂³	C₂×Dic₄₃⋊C₄	1	2	yes	—
198	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
199	C₂³	#13063304	1	4	yes	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	C₄₃⋊#95	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
208	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
211	C₂×C₄	#13063284	1	2	yes	—
212	C₂×C₄	C₄×C₂²⋊D₄₃	1	8	yes	—
213	C₂×C₄	#13063294	1	4	yes	—
214	C₂×C₄	#13063299	1	4	yes	—
215	C₂×C₄	#13063308	1	4	yes	—
216	C₈	—	2	8	no	—
217	C₈	—	172	8	no	—
218	C₂²	—	2	32	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	16	no	—
223	C₂²	—	2	8	no	—
224	C₂²	#14582664	1	8	yes	complemented
225	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
226	C₂²	#14582759	1	2	yes	—
227	C₂²	#14582772	1	4	yes	—
228	C₂²	C₂×#13063261	1	4	yes	—
229	C₄	—	2	4	no	—
230	C₄	C₄×C₂²⋊Dic₄₃	1	2	yes	—
231	C₄	#14582755	1	1	yes	characteristic
232	C₄	#14582762	1	1	yes	characteristic
233	C₄	#14582778	1	4	yes	—
234	C₂	—	2	8	no	—
235	C₂	—	2	4	no	—
236	C₂	#19132818	1	1	yes	characteristic
237	C₂	C₂×#14582664	1	2	yes	—
238	C₂	#19134633	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂⁴×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₄₃	Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂⁴×C₈	D₄₃
hypocenter	C₄₃	#10723
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#14582664	C₂	semidirect
105. C₂²×C₄₃	#341	semidirect
133. C₄₃	#10723	semidirect
224. C₂²	#14582664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582863	2
C₂×#14582664	2
C₂⁴×C₈×C₄₃	2
#10723	43