Subgroups — #25001515

Order 11008.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135359	C₂	1	2	yes	maximal
2	#19135313	C₂	1	1	yes	maximal, characteristic
3	#19134755	C₂	1	1	yes	maximal, characteristic
4	#19132479	C₂	1	2	yes	maximal
5	C₄₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	—	2	1	no	—
7	C₂×#13063259	—	2	1	no	—
8	C₂×#13063257	—	2	2	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₃⋊C₈	—	2	1	no	—
11	#14582769	C₂²	1	2	yes	—
12	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	#14582662	—	2	2	no	—
14	#14582653	—	2	1	no	—
15	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	—	2	1	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	2	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₄₃	—	2	1	no	—
24	#13063259	—	4	2	no	—
25	#13063259	—	4	2	no	—
26	#13063257	—	4	4	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	Dic₄₃⋊C₈	—	4	2	no	—
29	C₄₃⋊#95	—	2	1	no	—
30	C₂²×C₈×C₄₃	—	2	2	no	—
31	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₄₃	—	2	2	no	—
35	C₄⋊C₈×C₄₃	Q₈	1	4	yes	—
36	C₄×C₈×C₄₃	—	2	2	no	—
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
40	C₂²×Dic₄₃	—	2	1	no	—
41	C₂²×Dic₄₃	—	2	1	no	—
42	C₂²×Dic₄₃	—	4	2	no	—
43	C₂×C₈×C₄₃	—	2	1	no	—
44	C₂×C₈×C₄₃	—	2	1	no	—
45	C₂×C₈×C₄₃	—	2	1	no	—
46	C₂×C₈×C₄₃	—	2	1	no	—
47	C₂×C₈×C₄₃	—	2	4	no	—
48	C₂×C₈×C₄₃	—	4	2	no	—
49	C₂×C₈×C₄₃	—	4	2	no	—
50	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
51	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
52	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
53	C₄²×C₄₃	#41	1	2	yes	—
54	C₄⋊Dic₄₃	—	2	1	no	—
55	C₄⋊Dic₄₃	—	2	2	no	—
56	C₄⋊Dic₄₃	—	4	2	no	—
57	C₄⋊Dic₄₃	—	4	2	no	—
58	C₄⋊Dic₄₃	—	4	2	no	—
59	C₄⋊Dic₄₃	—	4	2	no	—
60	C₄⋊Dic₄₃	—	8	2	no	—
61	C₄⋊Dic₄₃	D₈	1	2	yes	—
62	C₄×Dic₄₃	—	4	1	no	—
63	C₄×Dic₄₃	—	4	1	no	—
64	C₂×C₄₃⋊C₈	—	4	1	no	—
65	C₂×C₄₃⋊C₈	—	4	1	no	—
66	C₂×C₄₃⋊C₈	—	4	1	no	—
67	C₂³×C₄₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
72	C₂×C₄×C₄₃	#125	1	2	yes	—
73	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
78	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#137	1	2	yes	—
80	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
81	C₂×Dic₄₃	—	2	2	no	—
82	C₂×Dic₄₃	—	4	1	no	—
83	C₂×Dic₄₃	—	4	1	no	—
84	C₂×Dic₄₃	—	4	1	no	—
85	C₂×Dic₄₃	—	4	1	no	—
86	C₂×Dic₄₃	—	4	1	no	—
87	C₂×Dic₄₃	—	8	2	no	—
88	C₂×Dic₄₃	—	8	2	no	—
89	C₂×Dic₄₃	—	8	2	no	—
90	C₈×C₄₃	—	2	2	no	—
91	C₈×C₄₃	—	2	2	no	—
92	C₈×C₄₃	—	4	4	no	—
93	C₄₃⋊C₈	—	8	1	no	—
94	#10651	—	43	1	no	maximal, Sylow
95	C₂²×C₄₃	#398	1	1	yes	characteristic
96	C₂²×C₄₃	#460	1	1	yes	characteristic
97	C₂²×C₄₃	#461	1	1	yes	characteristic
98	C₂²×C₄₃	#466	1	1	yes	characteristic
99	C₂²×C₄₃	#470	1	1	yes	characteristic
100	C₂²×C₄₃	#472	1	1	yes	characteristic
101	C₂²×C₄₃	#473	1	1	yes	characteristic
102	C₄×C₄₃	—	2	2	no	—
103	C₄×C₄₃	#474	1	2	yes	—
104	C₄×C₄₃	#479	1	2	yes	—
105	C₄×C₄₃	#481	1	1	yes	characteristic
106	C₄×C₄₃	#483	1	1	yes	characteristic
107	C₄×C₄₃	#498	1	1	yes	characteristic
108	C₄×C₄₃	#501	1	1	yes	characteristic
109	Dic₄₃	—	4	2	no	—
110	Dic₄₃	—	8	1	no	—
111	Dic₄₃	—	16	2	no	—
112	#1948	—	43	2	no	—
113	#1939	—	43	1	no	—
114	#1845	—	43	1	no	—
115	#1296	—	43	2	no	—
116	#1277	D₄₃	1	1	yes	characteristic
117	C₂×C₄₃	#1668	1	1	yes	characteristic
118	C₂×C₄₃	#1673	1	1	yes	characteristic
119	C₂×C₄₃	#1690	1	1	yes	characteristic
120	C₂×C₄₃	#1695	1	1	yes	characteristic
121	C₂×C₄₃	#2084	1	1	yes	characteristic
122	C₂×C₄₃	#2086	1	1	yes	characteristic
123	C₂×C₄₃	#2089	1	1	yes	characteristic
124	C₂×#107	—	86	1	no	—
125	C₂×#107	—	86	1	no	—
126	C₂×#106	—	86	2	no	—
127	C₂×C₄⋊C₈	—	43	1	no	—
128	C₂×C₄⋊C₈	—	86	1	no	—
129	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
131	#384	—	43	2	no	—
132	C₄×C₄⋊C₄	—	43	1	no	—
133	#340	—	86	2	no	—
134	#336	—	86	1	no	—
135	C₄₃	#10651	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	86	1	no	—
138	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
139	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	43	1	no	—
141	C₂×C₄⋊C₄	—	43	1	no	—
142	C₂×C₄⋊C₄	—	86	2	no	—
143	C₂×C₄⋊C₄	—	86	2	no	—
144	C₂×C₄⋊C₄	—	86	2	no	—
145	C₂×C₄²	—	86	1	no	—
146	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
147	#107	—	172	2	no	—
148	#107	—	172	2	no	—
149	#106	—	172	4	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	86	2	no	—
152	C₄⋊C₈	—	172	2	no	—
153	C₄⋊C₈	Dic₈₆	1	4	yes	—
154	C₄×C₈	—	2	2	no	—
155	#95	—	86	1	no	—
156	C₂²×C₄	—	86	1	no	—
157	C₂²×C₄	—	86	1	no	—
158	C₂²×C₄	—	172	2	no	—
159	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
160	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
161	C₂²×C₄	#10621902	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	172	1	no	—
170	C₂×C₈	—	172	1	no	—
171	C₂×C₈	—	172	1	no	—
172	C₂×C₈	#10621873	1	2	yes	—
173	C₂×C₈	#10621873	1	2	yes	—
174	C₄⋊C₄	—	43	2	no	—
175	C₄⋊C₄	—	86	1	no	—
176	C₄⋊C₄	—	86	2	no	—
177	C₄⋊C₄	—	344	2	no	—
178	C₄⋊C₄	—	172	2	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₄⋊C₄	—	172	2	no	—
181	C₄⋊C₄	—	172	2	no	—
182	C₄²	—	172	1	no	—
183	C₄²	—	172	1	no	—
184	C₄²	C₂×Dic₈₆	1	2	yes	—
185	C₄²	#10621900	1	2	yes	—
186	C₂³	#13063300	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	86	2	no	—
190	C₂×C₄	—	344	2	no	—
191	C₂×C₄	—	344	2	no	—
192	C₂×C₄	—	344	2	no	—
193	C₂×C₄	—	172	1	no	—
194	C₂×C₄	—	172	1	no	—
195	C₂×C₄	—	172	1	no	—
196	C₂×C₄	—	172	1	no	—
197	C₂×C₄	—	172	1	no	—
198	C₂×C₄	#13063302	1	2	yes	—
199	C₂×C₄	#13063309	1	1	yes	characteristic
200	C₂×C₄	#13063316	1	2	yes	—
201	C₂×C₄	#13063325	1	1	yes	characteristic
202	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
203	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
204	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
205	C₂×C₄	#13063348	1	1	yes	characteristic
206	C₂×C₄	#13063353	1	1	yes	characteristic
207	C₂×C₄	#13063354	1	2	yes	—
208	C₂×C₄	#13063358	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	4	no	—
212	C₈	—	344	1	no	—
213	C₂²	#14582831	1	1	yes	characteristic
214	C₂²	#14583100	1	1	yes	characteristic
215	C₂²	#14583105	1	1	yes	characteristic
216	C₂²	#14583136	1	1	yes	characteristic
217	C₂²	#14583158	1	1	yes	characteristic
218	C₂²	#14583170	1	1	yes	characteristic
219	C₂²	#14583175	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	344	1	no	—
222	C₄	—	172	2	no	—
223	C₄	—	688	2	no	—
224	C₄	#14583177	1	2	yes	—
225	C₄	#14583208	1	2	yes	—
226	C₄	#14583218	1	1	yes	characteristic
227	C₄	#14583230	1	1	yes	characteristic
228	C₄	#14583297	1	1	yes	characteristic
229	C₄	#14583308	1	1	yes	characteristic
230	C₂	#19133956	1	1	yes	characteristic
231	C₂	#19133985	1	1	yes	characteristic
232	C₂	#19134090	1	1	yes	characteristic
233	C₂	#19134119	1	1	yes	characteristic
234	C₂	#19136073	1	1	yes	characteristic
235	C₂	#19136085	1	1	yes	characteristic
236	C₂	#19136102	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063300
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1277	C₂
socle, prime radical	C₂³×C₄₃	#121
hypercenter	#1277	D₄₃
hypocenter	C₄₃	#10651
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
135. C₄₃	#10651	semidirect

Maximal subgroups

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

subgroup	index
#19135359	2
#19135313	2
#19134755	2
#19132479	2
C₄₃×#1277	2
#10651	43