Subgroups — #24985045

Order 11008.

572 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134448	C₂	1	1	yes	maximal, characteristic, complemented
2	#19134341	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132416	C₂	1	4	yes	maximal, complemented
4	C₄₃×#2858	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
6	C₂×#13063236	—	2	1	no	—
7	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	C₂²	1	2	yes	—
10	#14582654	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₃	C₄	1	1	yes	characteristic
12	C₂×#127×C₄₃	C₄	1	1	yes	characteristic
13	C₂×#124×C₄₃	—	2	1	no	—
14	C₂×#124×C₄₃	—	2	1	no	—
15	C₄₃×#390	—	2	1	no	—
16	C₄₃×#390	—	2	1	no	—
17	C₄₃×#390	C₄	1	2	yes	—
18	C₄₃×#390	C₂²	1	2	yes	—
19	C₄³×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
23	#13063236	—	4	2	no	—
24	C₄×C₄₃⋊C₈	—	2	4	no	—
25	C₂²×Q₈×C₄₃	—	2	1	no	—
26	C₂²×D₄×C₄₃	—	2	1	no	—
27	C₂²×D₄×C₄₃	—	2	1	no	—
28	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
29	C₄⋊Q₈×C₄₃	—	4	2	no	—
30	#127×C₄₃	—	4	2	no	—
31	#124×C₄₃	—	4	4	no	—
32	#124×C₄₃	—	4	2	no	—
33	#124×C₄₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂⁴×C₄₃	—	2	2	no	—
48	C₂×Q₈×C₄₃	—	4	2	no	—
49	C₂×Q₈×C₄₃	—	4	2	no	—
50	C₂×Q₈×C₄₃	—	4	2	no	—
51	C₂×D₄×C₄₃	—	2	4	no	—
52	C₂×D₄×C₄₃	—	4	2	no	—
53	C₂×D₄×C₄₃	—	4	2	no	—
54	C₂×D₄×C₄₃	—	4	2	no	—
55	C₂×D₄×C₄₃	—	4	2	no	—
56	C₂×D₄×C₄₃	—	4	2	no	—
57	C₂×D₄×C₄₃	—	4	2	no	—
58	C₂×D₄×C₄₃	—	4	2	no	—
59	C₂×D₄×C₄₃	—	4	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₄₃	—	4	2	no	—
69	C₄⋊C₄×C₄₃	—	4	2	no	—
70	C₂²⋊C₄×C₄₃	—	4	4	no	—
71	C₂²⋊C₄×C₄₃	—	4	4	no	—
72	C₂²⋊C₄×C₄₃	—	4	2	no	—
73	C₂²⋊C₄×C₄₃	—	4	2	no	—
74	C₂²⋊C₄×C₄₃	—	4	4	no	—
75	C₄²×C₄₃	—	2	2	no	—
76	C₄²×C₄₃	—	2	2	no	—
77	C₄²×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	4	1	no	—
81	C₄²×C₄₃	D₈	1	2	yes	—
82	C₄²×C₄₃	SD₁₆	1	2	yes	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	4	2	no	—
86	C₂×C₄₃⋊C₈	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	4	2	no	—
89	C₂³×C₄₃	—	4	2	no	—
90	C₂³×C₄₃	—	4	2	no	—
91	C₂³×C₄₃	—	4	2	no	—
92	C₂³×C₄₃	—	4	2	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	—	4	2	no	—
95	C₂³×C₄₃	—	4	2	no	—
96	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	Q₈×C₄₃	—	8	4	no	—
98	D₄×C₄₃	—	8	4	no	—
99	D₄×C₄₃	—	8	4	no	—
100	D₄×C₄₃	—	8	4	no	—
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	#100	1	2	yes	—
118	C₂×C₄×C₄₃	#102	1	2	yes	—
119	C₂×C₄×C₄₃	#102	1	2	yes	—
120	C₂×C₄×C₄₃	#104	1	2	yes	—
121	C₂×C₄×C₄₃	#104	1	2	yes	—
122	C₂×C₄×C₄₃	#131	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
126	C₄₃⋊C₈	—	4	4	no	—
127	C₄₃⋊C₈	—	4	4	no	—
128	#7679	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	—	8	2	no	—
130	C₂²×C₄₃	—	8	2	no	—
131	C₂²×C₄₃	—	8	2	no	—
132	C₂²×C₄₃	—	8	2	no	—
133	C₂²×C₄₃	—	8	2	no	—
134	C₂²×C₄₃	—	8	2	no	—
135	C₂²×C₄₃	—	8	2	no	—
136	C₂²×C₄₃	#408	1	1	yes	characteristic
137	C₂²×C₄₃	#410	1	1	yes	characteristic
138	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
139	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
140	C₂²×C₄₃	#418	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
143	C₄×C₄₃	—	2	2	no	—
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	—	2	2	no	—
146	C₄×C₄₃	—	2	1	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	4	1	no	—
150	C₄×C₄₃	—	8	2	no	—
151	C₄×C₄₃	#329	1	4	yes	—
152	#2858	D₄₃	1	1	yes	characteristic
153	#1767	—	43	1	no	—
154	#1744	—	43	1	no	—
155	#1270	—	43	4	no	—
156	C₂×C₄₃	—	16	2	no	—
157	C₂×C₄₃	#1469	1	1	yes	characteristic
158	C₂×C₄₃	#1484	1	1	yes	characteristic
159	C₂×C₄₃	#1492	1	1	yes	characteristic
160	C₂×C₄₃	#1502	1	1	yes	characteristic
161	C₂×C₄₃	#1505	1	1	yes	characteristic
162	C₂×C₄₃	C₂×#329	1	1	yes	characteristic
163	C₂×C₄₃	#1545	1	1	yes	characteristic
164	C₂×C₄⋊Q₈	Dic₄₃	1	1	yes	characteristic
165	C₂×#127	Dic₄₃	1	1	yes	characteristic
166	C₂×#124	—	2	1	no	—
167	C₂×#124	—	2	1	no	—
168	C₂×C₄⋊C₈	—	86	1	no	—
169	C₂×#97	—	86	1	no	—
170	C₂×C₄×C₈	—	86	1	no	—
171	#390	—	2	1	no	—
172	#390	—	2	1	no	—
173	#390	Dic₄₃	1	2	yes	—
174	#390	C₂×D₄₃	1	2	yes	—
175	C₄³	C₂×D₄₃	1	1	yes	characteristic
176	#336	—	43	2	no	—
177	#336	—	43	2	no	—
178	#336	—	86	1	no	—
179	C₄₃	#7679	1	1	yes	characteristic, complemented, Sylow
180	C₂²×Q₈	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	—	2	1	no	—
183	C₂²×D₄	C₂×Dic₄₃	1	2	yes	—
184	C₂²×C₈	—	86	2	no	—
185	C₂²×C₈	—	86	2	no	—
186	C₄⋊Q₈	—	4	2	no	—
187	#127	—	4	2	no	—
188	#124	—	4	4	no	—
189	#124	—	4	2	no	—
190	#124	—	4	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	C₄⋊C₈	—	86	4	no	—
205	#97	—	172	2	no	—
206	C₄×C₈	—	86	4	no	—
207	C₂⁴	—	2	2	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
228	C₂×C₈	—	86	4	no	—
229	C₂×C₈	—	86	4	no	—
230	C₂×C₈	—	172	2	no	—
231	C₂×C₈	—	172	2	no	—
232	C₂×C₈	—	172	2	no	—
233	C₂×C₈	—	172	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	#10621882	1	2	yes	—
248	C₄²	#10621883	1	2	yes	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
257	Q₈	—	8	4	no	—
258	D₄	—	8	4	no	—
259	D₄	—	8	4	no	—
260	D₄	—	8	4	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#13063242	1	2	yes	—
278	C₂×C₄	#13063246	1	2	yes	—
279	C₂×C₄	#13063246	1	2	yes	—
280	C₂×C₄	#13063252	1	2	yes	—
281	C₂×C₄	#13063252	1	2	yes	—
282	C₂×C₄	#13063332	1	2	yes	—
283	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
284	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
285	C₂×C₄	#13063351	1	2	yes	—
286	C₈	—	172	4	no	—
287	C₈	—	172	4	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	#14582866	1	1	yes	characteristic
296	C₂²	#14582873	1	1	yes	characteristic
297	C₂²	C₂×#13063242	1	1	yes	characteristic
298	C₂²	C₂×#13063246	1	1	yes	characteristic
299	C₂²	#14582904	1	1	yes	characteristic
300	C₂²	C₂×#13063252	1	1	yes	characteristic
301	C₂²	C₂×#13063252	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	2	no	—
310	C₄	#14582639	1	4	yes	—
311	C₂	—	16	2	no	—
312	C₂	#19132947	1	1	yes	characteristic
313	C₂	#19133012	1	1	yes	characteristic
314	C₂	#19133046	1	1	yes	characteristic
315	C₂	#19133090	1	1	yes	characteristic
316	C₂	#19133103	1	1	yes	characteristic
317	C₂	C₂×#14582639	1	1	yes	characteristic
318	C₂	#19133263	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×#2858	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₃	C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2858	D₄₃
hypocenter	C₄₃	#7679
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2858

Complements

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

N	H ≅ G/N	type
1. #19134448	C₂	semidirect
2. #19134341	C₂	semidirect
3. #19132416	C₂	semidirect
179. C₄₃	#7679	semidirect

Maximal subgroups

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

subgroup	index
#19134448	2
#19134341	2
#19132416	2
C₄₃×#2858	2
#7679	43