Subgroups — #24995135

Order 11008.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135314	C₂	1	2	yes	maximal
2	C₂×#14582650	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19132436	C₂	1	2	yes	maximal
5	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	#14582654	—	2	2	no	—
11	#14582650	—	2	2	no	—
12	C₈×C₄₃⋊C₈	—	2	2	no	—
13	C₂×#106×C₄₃	C₄	1	1	yes	characteristic
14	C₂×#106×C₄₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#340	C₄	1	2	yes	—
18	C₂²×C₄₃⋊C₈	—	2	1	no	—
19	C₂²×C₄₃⋊C₈	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	4	4	no	—
24	C₄×C₄₃⋊C₈	—	2	1	no	—
25	C₄×C₄₃⋊C₈	—	2	1	no	—
26	C₄×C₄₃⋊C₈	—	2	1	no	—
27	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₃	—	2	2	no	—
33	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	#106×C₄₃	—	2	2	no	—
35	#106×C₄₃	C₈	1	4	yes	complemented
36	C₄×C₈×C₄₃	D₄	1	2	yes	—
37	C₄×C₈×C₄₃	Q₈	1	2	yes	—
38	#95×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	2	2	no	—
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₄₃	—	2	1	no	—
45	C₂×C₈×C₄₃	—	2	1	no	—
46	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
54	C₄²×C₄₃	—	4	2	no	—
55	C₄²×C₄₃	—	4	2	no	—
56	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₄₃	#41	1	1	yes	characteristic
60	C₂×C₄₃⋊C₈	—	2	4	no	—
61	C₂×C₄₃⋊C₈	—	2	2	no	—
62	C₂×C₄₃⋊C₈	—	2	2	no	—
63	C₂×C₄₃⋊C₈	—	2	2	no	—
64	C₂×C₄₃⋊C₈	—	4	2	no	—
65	C₂×C₄₃⋊C₈	—	4	2	no	—
66	C₂×C₄₃⋊C₈	—	4	2	no	—
67	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	—
68	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	—	4	2	no	—
76	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#125	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₄₃	#135	1	2	yes	—
88	C₈×C₄₃	—	2	2	no	—
89	C₈×C₄₃	C₄⋊C₈	1	4	yes	complemented
90	C₄₃⋊C₈	—	2	4	no	—
91	C₄₃⋊C₈	—	2	4	no	—
92	C₄₃⋊C₈	—	8	2	no	—
93	#9593	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	#389	1	1	yes	characteristic
95	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
96	C₂²×C₄₃	#436	1	1	yes	characteristic
97	C₂²×C₄₃	C₄×SD₁₆	1	2	yes	—
98	C₂²×C₄₃	#443	1	2	yes	—
99	C₄×C₄₃	—	2	1	no	—
100	C₄×C₄₃	—	2	1	no	—
101	C₄×C₄₃	—	4	4	no	—
102	C₄×C₄₃	—	8	2	no	—
103	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₄₃	#424	1	1	yes	characteristic
105	C₄×C₄₃	C₈×Q₈	1	1	yes	characteristic
106	C₄×C₄₃	#446	1	1	yes	characteristic
107	C₄×C₄₃	#465	1	2	yes	—
108	C₄×C₄₃	#498	1	1	yes	characteristic
109	C₄×C₄₃	#499	1	1	yes	characteristic
110	#1939	—	43	2	no	—
111	C₄×#106	D₄₃	1	1	yes	characteristic
112	C₂×#334	—	43	1	no	—
113	C₂×C₈²	—	43	1	no	—
114	#1279	—	43	2	no	—
115	C₂×C₄₃	C₈×SD₁₆	1	2	yes	—
116	C₂×C₄₃	#1590	1	2	yes	—
117	C₂×C₄₃	#1939	1	1	yes	characteristic
118	C₂×C₄₃	#1941	1	1	yes	characteristic
119	C₂×C₄₃	#1945	1	1	yes	characteristic
120	C₂×#106	Dic₄₃	1	1	yes	characteristic
121	C₂×#106	Dic₄₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	43	2	no	—
123	C₂×C₄⋊C₈	—	86	2	no	—
124	C₂×C₄×C₈	—	43	1	no	—
125	C₂×C₄×C₈	—	43	1	no	—
126	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
127	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
128	#340	Dic₄₃	1	2	yes	—
129	#336	—	86	2	no	—
130	#334	—	86	2	no	—
131	C₈²	—	86	2	no	—
132	C₄₃	#9593	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	43	2	no	—
134	C₂²×C₈	—	86	1	no	—
135	C₂²×C₈	—	86	2	no	—
136	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
142	#106	—	2	2	no	—
143	#106	C₄₃⋊C₈	1	4	yes	complemented
144	C₄⋊C₈	—	86	2	no	—
145	C₄⋊C₈	—	86	2	no	—
146	C₄⋊C₈	—	172	4	no	—
147	C₄×C₈	—	43	2	no	—
148	C₄×C₈	—	86	1	no	—
149	C₄×C₈	—	86	1	no	—
150	C₄×C₈	—	86	1	no	—
151	C₄×C₈	Dic₈₆	1	2	yes	—
152	C₄×C₈	D₁₇₂	1	2	yes	—
153	#95	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
157	C₂²×C₄	#10621900	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	43	4	no	—
162	C₂×C₈	—	86	4	no	—
163	C₂×C₈	—	86	2	no	—
164	C₂×C₈	—	86	2	no	—
165	C₂×C₈	—	86	2	no	—
166	C₂×C₈	—	172	2	no	—
167	C₂×C₈	—	172	2	no	—
168	C₂×C₈	—	172	2	no	—
169	C₂×C₈	C₂×C₄₃⋊C₈	1	2	yes	—
170	C₂×C₈	#10621877	1	2	yes	—
171	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
172	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
180	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
181	C₄²	Q₈×D₄₃	1	1	yes	characteristic
182	C₄²	#10621902	1	1	yes	characteristic
183	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
192	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
193	C₂×C₄	#13063300	1	1	yes	characteristic
194	C₂×C₄	#13063305	1	1	yes	characteristic
195	C₂×C₄	#13063316	1	1	yes	characteristic
196	C₂×C₄	#13063324	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
199	C₂×C₄	#13063332	1	1	yes	characteristic
200	C₂×C₄	#13063332	1	1	yes	characteristic
201	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
202	C₂×C₄	#13063346	1	2	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	86	4	no	—
205	C₈	—	86	4	no	—
206	C₈	—	344	2	no	—
207	C₈	C₄⋊(C₄₃⋊C₈)	1	4	yes	complemented
208	C₂²	#14582798	1	1	yes	characteristic
209	C₂²	D₄×C₄₃⋊C₈	1	1	yes	characteristic
210	C₂²	#14582977	1	1	yes	characteristic
211	C₂²	SD₁₆×Dic₄₃	1	2	yes	—
212	C₂²	#14583016	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	C₂×C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
218	C₄	#14582926	1	1	yes	characteristic
219	C₄	Q₈×C₄₃⋊C₈	1	1	yes	characteristic
220	C₄	#14583030	1	1	yes	characteristic
221	C₄	#14583129	1	2	yes	—
222	C₄	#14583296	1	1	yes	characteristic
223	C₄	#14583299	1	1	yes	characteristic
224	C₂	SD₁₆×C₄₃⋊C₈	1	2	yes	—
225	C₂	#19133486	1	2	yes	—
226	C₂	#19135314	1	1	yes	characteristic
227	C₂	#19135324	1	1	yes	characteristic
228	C₂	#19135344	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₄×D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	#9593
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
35. #106×C₄₃	C₈	semidirect
89. C₈×C₄₃	C₄⋊C₈	semidirect
132. C₄₃	#9593	semidirect
143. #106	C₄₃⋊C₈	semidirect
207. C₈	C₄⋊(C₄₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#19135314	2
C₂×#14582650	2
C₂×C₈×C₄₃⋊C₈	2
#19132436	2
C₄×#106×C₄₃	2
#9593	43