Subgroups — #24989718

Order 11008.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	#19133343	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133019	C₂	1	2	yes	maximal
4	#19132490	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₃	—	2	1	no	—
7	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
8	C₈×D₁₇₂	—	2	1	no	—
9	C₈×D₁₇₂	—	2	1	no	—
10	C₈×D₁₇₂	C₄	1	2	yes	—
11	C₈×D₁₇₂	C₄	1	2	yes	—
12	#14582952	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
15	#14582652	—	2	1	no	—
16	#14582650	—	2	1	no	—
17	#14582626	—	2	4	no	—
18	C₂×#108×C₄₃	—	2	1	no	—
19	C₂×#108×C₄₃	—	2	1	no	—
20	C₄₃×#423	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×D₁₇₂	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₄₃	—	2	1	no	—
24	C₂²×C₄₃⋊C₈	—	2	1	no	—
25	C₂×C₈×D₄₃	—	2	2	no	—
26	C₂×C₈×D₄₃	—	2	2	no	—
27	C₂×C₈×D₄₃	—	4	2	no	—
28	C₂×C₈×D₄₃	—	4	2	no	—
29	C₄×D₁₇₂	—	2	1	no	—
30	C₄×D₁₇₂	—	2	1	no	—
31	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
32	C₄×D₁₇₂	D₄	1	2	yes	—
33	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
34	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
38	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
39	(C₂×D₄₃)⋊C₈	—	2	2	no	—
40	(C₂×D₄₃)⋊C₈	—	2	2	no	—
41	C₄×C₄₃⋊C₈	—	2	2	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	#108×C₄₃	—	4	2	no	—
47	#108×C₄₃	—	4	2	no	—
48	C₄⋊C₈×C₄₃	—	2	2	no	—
49	C₄⋊C₈×C₄₃	—	2	2	no	—
50	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₂³×D₄₃	—	2	1	no	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
58	C₂²×Dic₄₃	—	2	1	no	—
59	C₂×D₁₇₂	—	2	1	no	—
60	C₂×D₁₇₂	—	2	4	no	—
61	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₄₃	—	2	2	no	—
63	C₂×C₄×D₄₃	—	2	2	no	—
64	C₂×C₄×D₄₃	—	4	1	no	—
65	C₂×C₄×D₄₃	—	4	1	no	—
66	C₂×C₄×D₄₃	—	4	1	no	—
67	C₂×C₄×D₄₃	—	4	1	no	—
68	M₁₆×C₄₃	—	4	2	no	—
69	M₁₆×C₄₃	—	4	2	no	—
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	—	4	2	no	—
73	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	D₈	1	2	yes	—
78	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
79	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₃	#41	1	1	yes	characteristic
83	(C₂×D₄₃)⋊C₄	—	2	2	no	—
84	(C₂×D₄₃)⋊C₄	—	2	2	no	—
85	C₄⋊Dic₄₃	—	2	1	no	—
86	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	4	1	no	—
89	C₂×C₄₃⋊C₈	—	4	1	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₈×D₄₃	—	4	4	no	—
92	C₈×D₄₃	—	4	4	no	—
93	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂²×D₄₃	—	2	4	no	—
95	C₂²×D₄₃	—	2	2	no	—
96	C₂²×D₄₃	—	4	2	no	—
97	C₂²×D₄₃	—	4	1	no	—
98	C₂²×D₄₃	—	4	1	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	4	1	no	—
101	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
111	C₂×Dic₄₃	—	2	2	no	—
112	C₂×Dic₄₃	—	4	1	no	—
113	C₂×Dic₄₃	—	4	1	no	—
114	D₁₇₂	—	2	4	no	—
115	D₁₇₂	—	2	2	no	—
116	D₁₇₂	#108	1	4	yes	complemented
117	C₄×D₄₃	—	4	2	no	—
118	C₄×D₄₃	—	4	2	no	—
119	C₄×D₄₃	—	4	2	no	—
120	C₄×D₄₃	—	4	2	no	—
121	C₈×C₄₃	—	2	1	no	—
122	C₈×C₄₃	—	2	1	no	—
123	C₈×C₄₃	—	8	2	no	—
124	C₈×C₄₃	#102	1	2	yes	—
125	C₈×C₄₃	#102	1	2	yes	—
126	C₄₃⋊C₈	—	4	2	no	—
127	C₄₃⋊C₈	—	8	2	no	—
128	#8658	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	—	2	1	no	—
130	C₂²×C₄₃	—	2	1	no	—
131	C₂²×C₄₃	#380	1	1	yes	characteristic
132	C₂²×C₄₃	#474	1	1	yes	characteristic
133	C₂²×C₄₃	#476	1	1	yes	characteristic
134	C₂×D₄₃	—	2	4	no	—
135	C₂×D₄₃	—	4	4	no	—
136	C₂×D₄₃	—	4	4	no	—
137	C₂×D₄₃	—	4	2	no	—
138	C₂×D₄₃	—	4	2	no	—
139	C₄×C₄₃	—	2	1	no	—
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	2	1	no	—
142	C₄×C₄₃	C₂×#102	1	1	yes	characteristic
143	C₄×C₄₃	#417	1	1	yes	characteristic
144	C₄×C₄₃	C₂×#108	1	2	yes	—
145	C₄×C₄₃	#459	1	1	yes	characteristic
146	C₄×C₄₃	#461	1	1	yes	characteristic
147	Dic₄₃	—	4	2	no	—
148	C₂×C₈×D₄	—	43	1	no	—
149	#1848	D₄₃	1	1	yes	characteristic, complemented
150	#1564	—	43	1	no	—
151	#1486	—	43	2	no	—
152	#1301	—	43	2	no	—
153	C₂×C₄₃	—	2	1	no	—
154	C₂×C₄₃	—	2	1	no	—
155	C₂×C₄₃	#1802	1	1	yes	characteristic
156	C₂×C₄₃	#1810	1	1	yes	characteristic
157	C₂×C₄₃	#1814	1	1	yes	characteristic
158	D₄₃	—	4	4	no	—
159	C₂³×C₈	—	86	1	no	—
160	C₂×C₄×D₄	—	43	1	no	—
161	C₈×D₄	—	43	2	no	—
162	C₈×D₄	—	43	2	no	—
163	C₈×D₄	—	86	1	no	—
164	C₈×D₄	—	86	1	no	—
165	#431	—	43	2	no	—
166	C₂×#108	—	2	1	no	—
167	C₂×#108	—	2	1	no	—
168	#423	C₂×D₄₃	1	2	yes	—
169	C₂×C₄⋊C₈	—	43	1	no	—
170	C₂×C₂²⋊C₈	—	86	1	no	—
171	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
172	#335	—	86	1	no	—
173	#334	—	86	1	no	—
174	D₄⋊C₈	—	86	4	no	—
175	C₄₃	#8658	1	1	yes	characteristic, complemented, Sylow
176	C₂²×D₄	—	43	1	no	—
177	C₂³×C₄	—	86	1	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂²×C₈	—	86	1	no	—
180	C₂²×C₈	—	86	2	no	—
181	C₂²×C₈	—	86	2	no	—
182	C₂²×C₈	—	172	2	no	—
183	C₂²×C₈	—	172	2	no	—
184	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
186	C₄×D₄	—	43	2	no	—
187	C₄×D₄	—	43	2	no	—
188	C₄×D₄	—	86	1	no	—
189	C₄×D₄	—	86	1	no	—
190	C₂×C₄⋊C₄	—	43	1	no	—
191	C₂×C₂²⋊C₄	—	86	1	no	—
192	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
193	#108	—	4	2	no	—
194	#108	—	4	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	43	2	no	—
198	C₄⋊C₈	—	86	1	no	—
199	C₄⋊C₈	—	86	2	no	—
200	C₂²⋊C₈	—	86	2	no	—
201	C₂²⋊C₈	—	86	2	no	—
202	C₂²⋊C₈	—	86	2	no	—
203	C₄×C₈	—	86	2	no	—
204	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
205	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
206	C₄×C₈	D₁₇₂	1	1	yes	characteristic
207	C₄×C₈	D₁₇₂	1	1	yes	characteristic
208	C₂⁴	—	86	1	no	—
209	C₂×D₄	—	43	2	no	—
210	C₂×D₄	—	86	4	no	—
211	C₂×D₄	—	86	1	no	—
212	C₂²×C₄	—	86	1	no	—
213	C₂²×C₄	—	86	2	no	—
214	C₂²×C₄	—	86	2	no	—
215	C₂²×C₄	—	172	1	no	—
216	C₂²×C₄	—	172	1	no	—
217	C₂²×C₄	—	172	1	no	—
218	C₂²×C₄	—	172	1	no	—
219	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
220	C₂²×C₄	#10621902	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	86	2	no	—
228	C₂×C₈	—	172	2	no	—
229	C₂×C₈	—	172	1	no	—
230	C₂×C₈	—	172	1	no	—
231	C₂×C₈	—	172	4	no	—
232	C₂×C₈	—	172	4	no	—
233	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
234	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
235	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
236	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
237	C₂×C₈	#10621882	1	2	yes	—
238	C₂×C₈	#10621884	1	2	yes	—
239	C₄⋊C₄	—	43	2	no	—
240	C₄⋊C₄	—	86	1	no	—
241	C₂²⋊C₄	—	86	2	no	—
242	C₂²⋊C₄	—	86	2	no	—
243	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
244	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
245	C₄²	D₄×D₄₃	1	1	yes	characteristic
246	C₄²	#10621900	1	1	yes	characteristic
247	C₂³	—	86	4	no	—
248	C₂³	—	86	2	no	—
249	C₂³	—	172	2	no	—
250	C₂³	—	172	1	no	—
251	C₂³	—	172	1	no	—
252	C₂³	#13063301	1	1	yes	characteristic
253	D₄	—	43	4	no	—
254	D₄	—	86	4	no	—
255	D₄	—	86	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	86	2	no	—
259	C₂×C₄	—	172	1	no	—
260	C₂×C₄	—	172	1	no	—
261	C₂×C₄	—	172	2	no	—
262	C₂×C₄	—	172	2	no	—
263	C₂×C₄	—	172	2	no	—
264	C₂×C₄	—	172	2	no	—
265	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
267	C₂×C₄	#13063283	1	1	yes	characteristic
268	C₂×C₄	#13063296	1	1	yes	characteristic
269	C₂×C₄	#13063300	1	1	yes	characteristic
270	C₂×C₄	#13063305	1	1	yes	characteristic
271	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
272	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
273	C₂×C₄	#13063351	1	1	yes	characteristic
274	C₂×C₄	#13063356	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	344	2	no	—
279	C₈	—	172	2	no	—
280	C₈	#13063245	1	2	yes	—
281	C₈	#13063245	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	86	4	no	—
285	C₂²	—	172	4	no	—
286	C₂²	—	172	4	no	—
287	C₂²	—	172	2	no	—
288	C₂²	—	172	2	no	—
289	C₂²	#14582756	1	1	yes	characteristic
290	C₂²	#14583176	1	1	yes	characteristic
291	C₂²	#14583186	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	172	2	no	—
296	C₄	C₂×#13063245	1	1	yes	characteristic
297	C₄	#14582898	1	1	yes	characteristic
298	C₄	#108×D₄₃	1	2	yes	—
299	C₄	#14583091	1	1	yes	characteristic
300	C₄	#14583101	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	172	4	no	—
304	C₂	#19134592	1	1	yes	characteristic
305	C₂	#19134630	1	1	yes	characteristic
306	C₂	#19134650	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063296
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1848	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂²×D₁₇₂	Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1848	D₄₃
hypocenter	C₄₃	#8658
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #19133343	C₂	semidirect
4. #19132490	C₂	semidirect
5. C₄₃×#1848	C₂	semidirect
116. D₁₇₂	#108	semidirect
149. #1848	D₄₃	semidirect
175. C₄₃	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₇₂	2
#19133343	2
#19133019	2
#19132490	2
C₄₃×#1848	2
#8658	43