Subgroups — #24996753

Order 11008.

506 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134712	C₂	1	1	yes	maximal, characteristic
2	#19134448	C₂	1	1	yes	maximal, characteristic
3	#19132478	C₂	1	4	yes	maximal
4	C₄₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13063236	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	—	2	1	no	—
10	#14582654	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₄₃	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₄₃	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₄₃	C₄	1	2	yes	—
14	C₄₃×#384	—	2	1	no	—
15	C₄₃×#384	—	2	1	no	—
16	C₄₃×#384	C₄	1	2	yes	—
17	C₄₃×#384	C₂²	1	2	yes	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
23	C₄⋊(C₄₃⋊C₈)	Q₈	1	4	yes	complemented
24	#13063236	—	4	2	no	—
25	C₂²×Q₈×C₄₃	—	2	2	no	—
26	C₄⋊Q₈×C₄₃	—	4	4	no	—
27	C₄⋊Q₈×C₄₃	—	4	4	no	—
28	C₄⋊Q₈×C₄₃	—	4	2	no	—
29	C₄⋊Q₈×C₄₃	—	4	2	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₄₃	—	4	4	no	—
44	C₂×Q₈×C₄₃	—	4	4	no	—
45	C₂×Q₈×C₄₃	—	4	4	no	—
46	C₂²×C₄×C₄₃	—	2	4	no	—
47	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₄₃	—	4	4	no	—
55	C₄⋊C₄×C₄₃	—	4	4	no	—
56	C₄⋊C₄×C₄₃	—	4	4	no	—
57	C₄⋊C₄×C₄₃	—	4	4	no	—
58	C₄⋊C₄×C₄₃	—	4	2	no	—
59	C₄⋊C₄×C₄₃	—	4	2	no	—
60	C₄⋊C₄×C₄₃	—	4	4	no	—
61	C₄²×C₄₃	—	2	2	no	—
62	C₄²×C₄₃	—	2	2	no	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	—	2	2	no	—
66	C₄²×C₄₃	D₈	1	2	yes	—
67	C₄²×C₄₃	SD₁₆	1	2	yes	—
68	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
69	C₄²×C₄₃	#41	1	2	yes	—
70	C₂×C₄₃⋊C₈	—	2	4	no	—
71	C₂×C₄₃⋊C₈	—	4	2	no	—
72	C₂×C₄₃⋊C₈	—	4	2	no	—
73	C₂×C₄₃⋊C₈	—	4	2	no	—
74	C₂×C₄₃⋊C₈	—	4	2	no	—
75	C₂×C₄₃⋊C₈	—	4	2	no	—
76	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Q₈×C₄₃	—	8	8	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	1	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	1	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	4	4	no	—
89	C₂×C₄×C₄₃	—	4	4	no	—
90	C₂×C₄×C₄₃	—	4	4	no	—
91	C₂×C₄×C₄₃	#100	1	2	yes	—
92	C₂×C₄×C₄₃	#101	1	2	yes	—
93	C₂×C₄×C₄₃	#102	1	2	yes	—
94	C₂×C₄×C₄₃	#102	1	2	yes	—
95	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₄₃	#137	1	2	yes	—
103	C₂×C₄×C₄₃	#137	1	2	yes	—
104	C₄₃⋊C₈	—	4	4	no	—
105	C₄₃⋊C₈	—	8	2	no	—
106	#9905	—	43	1	no	maximal, Sylow
107	C₂²×C₄₃	#391	1	1	yes	characteristic
108	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
109	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
110	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
111	C₂²×C₄₃	#418	1	1	yes	characteristic
112	C₂²×C₄₃	#419	1	1	yes	characteristic
113	C₂²×C₄₃	#419	1	1	yes	characteristic
114	C₄×C₄₃	—	2	2	no	—
115	C₄×C₄₃	—	2	2	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	—	8	4	no	—
120	C₄×C₄₃	#474	1	2	yes	—
121	C₄×C₄₃	#475	1	2	yes	—
122	C₄×C₄₃	#489	1	2	yes	—
123	C₄×C₄₃	#501	1	2	yes	—
124	#2868	D₄₃	1	1	yes	characteristic
125	#1831	—	43	1	no	—
126	#1767	—	43	1	no	—
127	#1296	—	43	4	no	—
128	C₂×C₄₃	#1545	1	1	yes	characteristic
129	C₂×C₄₃	#1546	1	1	yes	characteristic
130	C₂×C₄₃	#1556	1	1	yes	characteristic
131	C₂×C₄₃	#1558	1	1	yes	characteristic
132	C₂×C₄₃	#1988	1	1	yes	characteristic
133	C₂×C₄₃	#1993	1	1	yes	characteristic
134	C₂×C₄₃	#1997	1	1	yes	characteristic
135	C₂×C₄⋊Q₈	—	2	1	no	—
136	C₂×C₄⋊Q₈	—	2	1	no	—
137	C₂×C₄⋊Q₈	Dic₄₃	1	2	yes	—
138	C₂×C₄⋊C₈	—	43	2	no	—
139	C₂×C₄⋊C₈	—	43	2	no	—
140	C₂×#97	—	86	1	no	—
141	#384	—	2	1	no	—
142	#384	—	2	1	no	—
143	#384	Dic₄₃	1	2	yes	—
144	#384	C₂×D₄₃	1	2	yes	—
145	C₄³	C₂×D₄₃	1	1	yes	characteristic
146	#336	—	86	1	no	—
147	#336	—	86	1	no	—
148	#336	—	86	1	no	—
149	C₄₃	#9905	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×C₈	—	86	2	no	—
152	C₂²×C₈	—	86	2	no	—
153	C₄⋊Q₈	—	4	4	no	—
154	C₄⋊Q₈	—	4	4	no	—
155	C₄⋊Q₈	—	4	2	no	—
156	C₄⋊Q₈	—	4	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
163	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	C₄⋊C₈	—	43	4	no	—
171	C₄⋊C₈	—	86	2	no	—
172	C₄⋊C₈	—	86	4	no	—
173	#97	—	172	2	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
182	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
185	C₂×C₈	—	86	4	no	—
186	C₂×C₈	—	172	2	no	—
187	C₂×C₈	—	172	2	no	—
188	C₂×C₈	—	172	2	no	—
189	C₂×C₈	—	172	2	no	—
190	C₂×C₈	—	172	2	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	#10621882	1	2	yes	—
204	C₄²	#10621883	1	2	yes	—
205	C₄²	Q₈×D₄₃	1	2	yes	—
206	C₄²	#10621902	1	2	yes	—
207	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
208	Q₈	—	8	8	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#13063242	1	2	yes	—
223	C₂×C₄	#13063244	1	2	yes	—
224	C₂×C₄	#13063246	1	2	yes	—
225	C₂×C₄	#13063246	1	2	yes	—
226	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
227	C₂×C₄	#13063301	1	2	yes	—
228	C₂×C₄	#13063314	1	1	yes	characteristic
229	C₂×C₄	#13063323	1	1	yes	characteristic
230	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
231	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
232	C₂×C₄	#13063351	1	2	yes	—
233	C₂×C₄	#13063356	1	2	yes	—
234	C₂×C₄	#13063356	1	2	yes	—
235	C₈	—	344	2	no	—
236	C₈	—	172	4	no	—
237	C₂²	#14582806	1	1	yes	characteristic
238	C₂²	C₂×#13063242	1	1	yes	characteristic
239	C₂²	C₂×#13063244	1	1	yes	characteristic
240	C₂²	C₂×#13063246	1	1	yes	characteristic
241	C₂²	#14582904	1	1	yes	characteristic
242	C₂²	#14582908	1	1	yes	characteristic
243	C₂²	#14582908	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#14583176	1	2	yes	—
251	C₄	#14583181	1	2	yes	—
252	C₄	#14583261	1	2	yes	—
253	C₄	#14583305	1	2	yes	—
254	C₂	#19133263	1	1	yes	characteristic
255	C₂	#19133267	1	1	yes	characteristic
256	C₂	#19133312	1	1	yes	characteristic
257	C₂	#19133321	1	1	yes	characteristic
258	C₂	#19135548	1	1	yes	characteristic
259	C₂	#19135569	1	1	yes	characteristic
260	C₂	#19135585	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2868	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2868	D₄₃
hypocenter	C₄₃	#9905
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
23. C₄⋊(C₄₃⋊C₈)	Q₈	semidirect
149. C₄₃	#9905	semidirect

Maximal subgroups

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

subgroup	index
#19134712	2
#19134448	2
#19132478	2
C₄₃×#2868	2
#9905	43