Subgroups — #24986039

Order 11008.

922 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143228	C₂	1	2	yes	maximal
2	#19132428	C₂	1	4	yes	maximal, complemented
3	C₂×C₄³×C₄₃	C₂	1	1	yes	maximal, characteristic
4	#14582956	—	2	2	no	—
5	#14582956	C₄	1	4	yes	complemented
6	#14582952	—	2	2	no	—
7	#14582952	C₂²	1	4	yes	—
8	#14582923	C₄	1	4	yes	complemented
9	#14582863	—	2	2	no	—
10	C₄×#10621877	—	2	2	no	—
11	C₂²×C₄²×C₄₃	—	2	2	no	—
12	C₂²×C₄²×C₄₃	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₄³×C₄₃	—	2	2	no	—
15	C₄³×C₄₃	C₄	1	2	yes	—
16	C₄³×C₄₃	C₂²	1	2	yes	—
17	C₂×#10621877	—	2	4	no	—
18	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	complemented
20	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
22	#13063236	—	2	4	no	—
23	C₄×C₄₃⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₄₃	—	2	2	no	—
25	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₃	D₄	1	2	yes	—
27	C₂×C₄²×C₄₃	—	2	16	no	—
28	C₂×C₄²×C₄₃	—	2	4	no	—
29	C₂×C₄²×C₄₃	—	2	2	no	—
30	C₂×C₄²×C₄₃	—	2	4	no	—
31	C₂×C₄²×C₄₃	—	2	8	no	—
32	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₄₃	D₄	1	2	yes	—
37	C₂×C₄²×C₄₃	D₄	1	2	yes	—
38	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	—	2	16	no	—
42	C₂²×C₄×C₄₃	—	2	8	no	—
43	C₂²×C₄×C₄₃	—	2	8	no	—
44	C₂²×C₄×C₄₃	—	2	4	no	—
45	C₂²×C₄×C₄₃	—	2	4	no	—
46	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
56	C₄²×C₄₃	—	2	16	no	—
57	C₄²×C₄₃	—	2	16	no	—
58	C₄²×C₄₃	—	2	8	no	—
59	C₄²×C₄₃	—	2	4	no	—
60	C₄²×C₄₃	C₄²	1	1	yes	characteristic
61	C₄²×C₄₃	C₄²	1	1	yes	characteristic
62	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
65	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
66	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₄₃	M₁₆	1	8	yes	complemented
68	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
71	#10621877	—	4	8	no	—
72	C₂×C₄₃⋊C₈	—	2	8	no	—
73	C₂³×C₄₃	—	2	4	no	—
74	C₂³×C₄₃	—	2	2	no	—
75	C₂³×C₄₃	#95	1	2	yes	—
76	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	—	2	16	no	—
78	C₂×C₄×C₄₃	—	2	32	no	—
79	C₂×C₄×C₄₃	—	2	4	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	8	no	—
82	C₂×C₄×C₄₃	—	2	4	no	—
83	C₂×C₄×C₄₃	—	2	4	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	—	2	4	no	—
86	C₂×C₄×C₄₃	—	2	4	no	—
87	C₂×C₄×C₄₃	—	2	4	no	—
88	C₂×C₄×C₄₃	#95	1	2	yes	—
89	C₂×C₄×C₄₃	#95	1	2	yes	—
90	C₂×C₄×C₄₃	#95	1	2	yes	—
91	C₂×C₄×C₄₃	#104	1	8	yes	—
92	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
97	C₄₃⋊C₈	—	4	8	no	—
98	#7867	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	—	2	8	no	—
100	C₂²×C₄₃	—	2	2	no	—
101	C₂²×C₄₃	—	2	2	no	—
102	C₂²×C₄₃	—	2	2	no	—
103	C₂²×C₄₃	#339	1	4	yes	—
104	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
105	C₂²×C₄₃	#407	1	2	yes	—
106	C₄×C₄₃	—	2	16	no	—
107	C₄×C₄₃	—	2	2	no	—
108	C₄×C₄₃	—	2	2	no	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	#339	1	4	yes	—
111	C₄×C₄₃	C₄×M₁₆	1	2	yes	—
112	C₄×C₄₃	C₂×#104	1	4	yes	—
113	C₄×C₄₃	#423	1	2	yes	—
114	#2920	—	43	2	no	—
115	C₂×C₄³	D₄₃	1	1	yes	characteristic
116	#1276	—	43	4	no	—
117	C₂×C₄₃	—	2	2	no	—
118	C₂×C₄₃	—	2	4	no	—
119	C₂×C₄₃	#1728	1	1	yes	characteristic
120	C₂×C₄₃	C₂×#339	1	2	yes	—
121	C₂²×C₄²	—	2	2	no	—
122	C₂²×C₄²	Dic₄₃	1	2	yes	—
123	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
124	#432	—	43	4	no	—
125	#432	—	86	2	no	—
126	#431	—	43	4	no	—
127	#431	—	86	2	no	—
128	#423	—	43	4	no	—
129	#407	—	86	2	no	—
130	C₄×M₁₆	—	86	2	no	—
131	C₄³	—	2	2	no	—
132	C₄³	Dic₄₃	1	2	yes	—
133	C₄³	C₂×D₄₃	1	2	yes	—
134	C₄₃	#7867	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	—	2	2	no	—
136	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
137	C₂³×C₄	C₂²⋊D₄₃	1	2	yes	—
138	C₂×M₁₆	—	86	4	no	—
139	C₂×C₄²	—	2	16	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	—	2	8	no	—
144	C₂×C₄²	Dic₈₆	1	2	yes	—
145	C₂×C₄²	C₄×D₄₃	1	2	yes	—
146	C₂×C₄²	C₄×D₄₃	1	2	yes	—
147	C₂×C₄²	D₁₇₂	1	2	yes	—
148	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
149	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
150	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
151	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
152	C₄⋊C₈	—	43	8	no	—
153	C₄⋊C₈	—	86	4	no	—
154	C₂²⋊C₈	—	86	4	no	—
155	C₂²⋊C₈	—	86	4	no	—
156	#97	—	86	4	no	—
157	C₄×C₈	—	86	4	no	—
158	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	16	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
165	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
166	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
167	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
168	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
169	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
170	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
171	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
174	M₁₆	—	172	8	no	—
175	C₂×C₈	—	86	8	no	—
176	C₄²	—	2	16	no	—
177	C₄²	—	2	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	4	no	—
180	C₄²	#10621877	1	8	yes	complemented
181	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
182	C₄²	C₄×Dic₄₃	1	1	yes	characteristic
183	C₄²	Dic₄₃⋊C₄	1	2	yes	—
184	C₄²	Dic₄₃⋊C₄	1	2	yes	—
185	C₄²	C₄⋊Dic₄₃	1	2	yes	—
186	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
187	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
188	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₄₃	1	2	yes	—
190	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₄₃⋊#95	1	2	yes	—
194	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
195	C₂×C₄	—	2	16	no	—
196	C₂×C₄	—	2	32	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	C₄₃⋊#95	1	2	yes	—
207	C₂×C₄	C₄₃⋊#95	1	2	yes	—
208	C₂×C₄	C₄₃⋊#95	1	2	yes	—
209	C₂×C₄	#13063253	1	8	yes	—
210	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
211	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	2	yes	—
212	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
214	C₂×C₄	C₂×#10621877	1	4	yes	—
215	C₈	—	172	8	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	#14582659	1	4	yes	—
221	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
222	C₂²	#14582863	1	2	yes	—
223	C₄	—	2	16	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	#14582659	1	4	yes	—
228	C₄	C₄×#10621877	1	2	yes	—
229	C₄	C₂×#13063253	1	4	yes	—
230	C₄	#14582923	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	#19134289	1	1	yes	characteristic
234	C₂	C₂×#14582659	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄³×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#95
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₄³	D₄₃
hypocenter	C₄₃	#7867
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #19132428	C₂	semidirect
5. #14582956	C₄	semidirect
8. #14582923	C₄	semidirect
19. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
67. C₄²×C₄₃	M₁₆	semidirect
134. C₄₃	#7867	semidirect
180. C₄²	#10621877	semidirect

Maximal subgroups

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

subgroup	index
#19143228	2
#19132428	2
C₂×C₄³×C₄₃	2
#7867	43