Subgroups — #24989913

Order 11008.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143570	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133121	C₂	1	2	yes	maximal
4	#19132450	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583433	C₂²	1	1	yes	characteristic
7	#14583029	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₈₆	C₄	1	1	yes	characteristic, complemented
9	#14582978	C₄	1	1	yes	characteristic, complemented
10	#14582970	—	2	1	no	—
11	C₈×D₁₇₂	C₄	1	1	yes	characteristic, complemented
12	C₈×C₂²⋊D₄₃	—	2	1	no	—
13	#14582958	—	2	1	no	—
14	#14582956	C₂²	1	1	yes	characteristic
15	C₂×#13063259	—	2	1	no	—
16	#14582853	—	2	1	no	—
17	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
18	#14582662	—	2	1	no	—
19	#14582637	—	2	2	no	—
20	#14582634	—	2	2	no	—
21	C₄₃×#432	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#335	—	2	1	no	—
24	C₄₃×#334	—	2	1	no	—
25	C₂×#10621872	—	2	1	no	—
26	C₂×C₈×D₄₃	—	2	1	no	—
27	#13063319	—	2	1	no	—
28	#13063317	Q₈	1	1	yes	characteristic
29	#13063313	Q₈	1	1	yes	characteristic
30	#13063308	—	2	1	no	—
31	#13063304	D₄	1	1	yes	characteristic
32	#13063299	D₄	1	1	yes	characteristic
33	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
34	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	#13063282	—	2	1	no	—
37	C₂×C₄⋊Dic₄₃	—	2	2	no	—
38	C₂×C₄×Dic₄₃	—	2	2	no	—
39	#13063259	—	2	4	no	—
40	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	Dic₄₃⋊C₈	—	2	1	no	—
43	Dic₄₃⋊C₈	—	2	1	no	—
44	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
45	(C₂×D₄₃)⋊C₈	—	2	1	no	—
46	(C₂×D₄₃)⋊C₈	—	2	1	no	—
47	#13063236	—	2	1	no	—
48	#13063235	—	2	1	no	—
49	C₈×Dic₄₃	—	2	1	no	—
50	C₄₃⋊#95	—	2	2	no	—
51	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	—	2	2	no	—
55	C₂²⋊C₈×C₄₃	—	2	2	no	—
56	#97×C₄₃	—	2	2	no	—
57	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	4	2	no	—
66	C₂×D₁₇₂	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×Dic₈₆	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	4	1	no	—
71	C₂×C₈×C₄₃	—	4	2	no	—
72	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₄₃	#41	1	1	yes	characteristic
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	(C₂×D₄₃)⋊C₄	—	2	1	no	—
82	(C₂×D₄₃)⋊C₄	—	2	1	no	—
83	(C₂×D₄₃)⋊C₄	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₄₃	—	2	4	no	—
86	C₄⋊Dic₄₃	—	4	2	no	—
87	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊Dic₄₃	C₄⋊C₄	1	1	yes	characteristic
89	Dic₄₃⋊C₄	—	2	1	no	—
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
92	Dic₄₃⋊C₄	—	2	1	no	—
93	C₄×Dic₄₃	—	2	1	no	—
94	C₄×Dic₄₃	—	8	2	no	—
95	C₂×C₄₃⋊C₈	—	2	1	no	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	#10621872	—	4	2	no	—
98	C₈×D₄₃	—	4	2	no	—
99	C₂³×C₄₃	#121	1	1	yes	characteristic
100	C₂²×D₄₃	—	2	1	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	—	4	1	no	—
106	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂²⋊D₄₃	—	4	1	no	—
113	C₂²⋊D₄₃	—	4	1	no	—
114	C₂×Dic₄₃	—	2	1	no	—
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	8	2	no	—
119	C₂×Dic₄₃	—	8	2	no	—
120	D₁₇₂	—	4	1	no	—
121	C₄×D₄₃	—	4	1	no	—
122	C₄×D₄₃	—	4	1	no	—
123	Dic₈₆	—	4	1	no	—
124	C₈×C₄₃	—	2	1	no	—
125	C₈×C₄₃	—	2	1	no	—
126	C₈×C₄₃	—	2	1	no	—
127	C₈×C₄₃	—	8	2	no	—
128	C₈×C₄₃	#104	1	2	yes	—
129	C₄₃⋊C₈	—	4	1	no	—
130	C₄₃⋊C₈	—	4	1	no	—
131	#8697	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	—	2	1	no	—
134	C₂²×C₄₃	#380	1	1	yes	characteristic
135	C₂²×C₄₃	#472	1	1	yes	characteristic
136	C₂²×C₄₃	#473	1	1	yes	characteristic
137	C₂×D₄₃	—	4	1	no	—
138	C₂×D₄₃	—	4	1	no	—
139	C₂×D₄₃	—	4	1	no	—
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	2	1	no	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
146	C₄×C₄₃	#421	1	1	yes	characteristic
147	Dic₄₃	—	4	1	no	—
148	Dic₄₃	—	4	1	no	—
149	Dic₄₃	—	4	1	no	—
150	Dic₄₃	—	8	4	no	—
151	#2966	—	43	1	no	—
152	C₄×#107	—	43	1	no	—
153	#1568	D₄₃	1	1	yes	characteristic, complemented
154	#1509	—	43	2	no	—
155	#1285	—	43	2	no	—
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	#1806	1	1	yes	characteristic
159	C₂×C₄₃	#1812	1	1	yes	characteristic
160	C₂×C₄₃	#1816	1	1	yes	characteristic
161	D₄₃	—	8	1	no	—
162	#529	—	43	1	no	—
163	#446	—	43	1	no	—
164	C₈×Q₈	—	43	1	no	—
165	#436	—	43	1	no	—
166	#435	—	86	1	no	—
167	C₈×D₄	—	43	1	no	—
168	C₈×D₄	—	86	1	no	—
169	#433	—	86	1	no	—
170	#432	—	43	1	no	—
171	#432	C₂×D₄₃	1	2	yes	—
172	C₂×#107	—	86	1	no	—
173	#405	—	86	1	no	—
174	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
175	C₄×C₄⋊C₄	—	43	2	no	—
176	#340	—	86	1	no	—
177	#335	—	2	1	no	—
178	#334	—	2	1	no	—
179	#328	—	86	2	no	—
180	#327	—	86	2	no	—
181	C₄₃	#8697	1	1	yes	characteristic, complemented, Sylow
182	C₂×M₁₆	—	86	1	no	—
183	C₂²×C₈	—	86	1	no	—
184	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
185	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
186	#126	—	86	1	no	—
187	#125	—	43	1	no	—
188	#124	—	43	1	no	—
189	#123	—	86	1	no	—
190	C₂²⋊Q₈	—	43	1	no	—
191	#121	—	43	1	no	—
192	C₄×Q₈	—	43	1	no	—
193	C₄×D₄	—	43	1	no	—
194	C₄×D₄	—	86	1	no	—
195	#117	—	86	1	no	—
196	C₂×C₄⋊C₄	—	86	2	no	—
197	C₂×C₄²	—	86	2	no	—
198	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
199	#107	—	86	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	43	1	no	—
202	C₄⋊C₈	—	43	1	no	—
203	C₄⋊C₈	—	86	1	no	—
204	C₄⋊C₈	—	86	1	no	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₂²⋊C₈	—	86	1	no	—
207	C₂²⋊C₈	—	86	1	no	—
208	C₂²⋊C₈	—	86	1	no	—
209	#97	—	2	2	no	—
210	#97	—	86	1	no	—
211	#97	—	86	1	no	—
212	C₄×C₈	—	86	1	no	—
213	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
214	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
217	#95	—	86	2	no	—
218	C₂×Q₈	—	43	1	no	—
219	C₂×D₄	—	43	1	no	—
220	C₂×D₄	—	86	1	no	—
221	C₂²×C₄	—	86	1	no	—
222	C₂²×C₄	—	172	2	no	—
223	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
225	C₂²×C₄	#10621902	1	1	yes	characteristic
226	M₁₆	—	172	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	86	1	no	—
231	C₂×C₈	—	86	1	no	—
232	C₂×C₈	—	172	2	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	C₄⋊C₄	—	43	1	no	—
238	C₄⋊C₄	—	43	1	no	—
239	C₄⋊C₄	—	86	4	no	—
240	C₄⋊C₄	—	86	1	no	—
241	C₄⋊C₄	—	86	1	no	—
242	C₄⋊C₄	—	86	1	no	—
243	C₄⋊C₄	—	86	1	no	—
244	C₄⋊C₄	—	172	2	no	—
245	C₂²⋊C₄	—	86	1	no	—
246	C₂²⋊C₄	—	86	1	no	—
247	C₂²⋊C₄	—	86	1	no	—
248	C₂²⋊C₄	—	86	1	no	—
249	C₂²⋊C₄	—	86	1	no	—
250	C₄²	—	86	1	no	—
251	C₄²	—	344	2	no	—
252	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
253	C₄²	#10621900	1	1	yes	characteristic
254	C₄²	Q₈×D₄₃	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
256	C₂³	—	86	1	no	—
257	C₂³	#13063300	1	1	yes	characteristic
258	Q₈	—	172	1	no	—
259	D₄	—	172	1	no	—
260	D₄	—	172	1	no	—
261	D₄	—	172	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	—	4	1	no	—
267	C₂×C₄	—	86	1	no	—
268	C₂×C₄	—	86	1	no	—
269	C₂×C₄	—	86	1	no	—
270	C₂×C₄	—	344	2	no	—
271	C₂×C₄	—	344	2	no	—
272	C₂×C₄	—	172	4	no	—
273	C₂×C₄	—	172	1	no	—
274	C₂×C₄	—	172	1	no	—
275	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
277	C₂×C₄	#13063283	1	1	yes	characteristic
278	C₂×C₄	#13063296	1	1	yes	characteristic
279	C₂×C₄	#13063301	1	1	yes	characteristic
280	C₂×C₄	#13063305	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	8	2	no	—
285	C₈	—	172	1	no	—
286	C₈	—	172	1	no	—
287	C₈	#13063251	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	172	1	no	—
291	C₂²	—	172	1	no	—
292	C₂²	—	172	1	no	—
293	C₂²	#14582756	1	1	yes	characteristic
294	C₂²	#14583170	1	1	yes	characteristic
295	C₂²	#14583175	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	344	4	no	—
302	C₄	—	172	1	no	—
303	C₄	—	172	1	no	—
304	C₄	—	172	1	no	—
305	C₄	C₂×#13063251	1	1	yes	characteristic
306	C₄	#14582915	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	344	1	no	—
310	C₂	#19134612	1	1	yes	characteristic
311	C₂	#19134638	1	1	yes	characteristic
312	C₂	#19134656	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582756
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1568	C₂
socle	C₂²×C₄₃	#380
prime radical	#13063299	D₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1568	D₄₃
hypocenter	C₄₃	#8697
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. C₄×#13063259	C₂	semidirect
4. #19132450	C₂	semidirect
5. C₄₃×#1568	C₂	semidirect
7. #14583029	C₄	semidirect
8. C₈×Dic₈₆	C₄	semidirect
9. #14582978	C₄	semidirect
11. C₈×D₁₇₂	C₄	semidirect
153. #1568	D₄₃	semidirect
181. C₄₃	#8697	semidirect

Maximal subgroups

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

subgroup	index
#19143570	2
C₄×#13063259	2
#19133121	2
#19132450	2
C₄₃×#1568	2
#8697	43