Subgroups — #25005271

Order 11008.

790 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143185	C₂	1	1	yes	maximal, characteristic
2	#19137079	C₂	1	2	yes	maximal
3	#19134305	C₂	1	2	yes	maximal
4	#19134288	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	1	yes	characteristic
7	C₂×#13063284	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	#14582941	—	2	4	no	—
10	#14582937	—	2	2	no	—
11	#14582798	—	2	4	no	—
12	#14582774	—	2	4	no	—
13	C₄×C₄⋊Dic₄₃	—	2	4	no	—
14	#14582659	—	2	4	no	—
15	#14582653	—	2	2	no	—
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₃	C₂²	1	2	yes	—
18	C₄₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
19	C₂³×Dic₄₃	D₄	1	2	yes	—
20	C₂×#10621877	—	2	1	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	Q₈	1	2	yes	—
24	C₂²×C₄₃⋊C₈	—	2	1	no	—
25	#13063284	—	2	2	no	—
26	#13063284	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	#13063284	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	—	2	8	no	—
30	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
31	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
32	C₂×C₄×Dic₄₃	—	2	2	no	—
33	C₂×C₄×Dic₄₃	—	2	1	no	—
34	C₂×C₄×Dic₄₃	—	2	2	no	—
35	C₂×C₄×Dic₄₃	—	2	2	no	—
36	C₂×C₄×Dic₄₃	—	2	1	no	—
37	#13063259	—	2	4	no	—
38	#13063257	—	2	4	no	—
39	C₄₃⋊#95	—	2	8	no	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₃	—	2	2	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
45	C₂²×C₈×C₄₃	D₄	1	2	yes	—
46	#117×C₄₃	—	2	2	no	—
47	#117×C₄₃	—	2	2	no	—
48	#117×C₄₃	C₂×C₄	1	4	yes	—
49	#117×C₄₃	Q₈	1	4	yes	—
50	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	D₄	1	4	yes	—
53	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	2	4	no	—
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	4	4	no	—
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	complemented
74	C₄⋊C₄×C₄₃	—	2	4	no	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₂²⋊C₄×C₄₃	—	2	4	no	—
77	C₂²⋊C₄×C₄₃	—	2	4	no	—
78	C₄²×C₄₃	—	2	4	no	—
79	C₄²×C₄₃	C₄⋊C₄	1	8	yes	complemented
80	C₂²⋊Dic₄₃	—	4	2	no	—
81	C₂²⋊Dic₄₃	—	4	2	no	—
82	C₄⋊Dic₄₃	—	2	4	no	—
83	C₄⋊Dic₄₃	—	2	4	no	—
84	C₄⋊Dic₄₃	—	4	16	no	—
85	C₄×Dic₄₃	—	4	1	no	—
86	C₄×Dic₄₃	—	4	2	no	—
87	C₄×Dic₄₃	—	4	2	no	—
88	C₄×Dic₄₃	—	4	1	no	—
89	C₄×Dic₄₃	—	4	2	no	—
90	#10621877	—	2	4	no	—
91	#10621877	—	4	2	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂×C₄₃⋊C₈	—	4	1	no	—
94	C₂×C₄₃⋊C₈	—	4	1	no	—
95	C₂³×C₄₃	—	4	2	no	—
96	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
97	C₂³×C₄₃	#120	1	2	yes	—
98	C₂³×C₄₃	#121	1	1	yes	characteristic
99	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	8	no	—
102	C₂×C₄×C₄₃	—	4	8	no	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#121	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#123	1	2	yes	—
115	C₂×C₄×C₄₃	#125	1	2	yes	—
116	C₂×C₄×C₄₃	#125	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
121	C₂×Dic₄₃	—	2	8	no	—
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	4	8	no	—
127	C₂×Dic₄₃	—	4	4	no	—
128	C₈×C₄₃	—	2	4	no	—
129	C₄₃⋊C₈	—	4	2	no	—
130	#11335	—	43	1	no	maximal, Sylow
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	4	1	no	—
134	C₂²×C₄₃	—	4	2	no	—
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	#385	1	2	yes	—
137	C₂²×C₄₃	#389	1	1	yes	characteristic
138	C₂²×C₄₃	#392	1	1	yes	characteristic
139	C₂²×C₄₃	#395	1	2	yes	—
140	C₂²×C₄₃	#398	1	1	yes	characteristic
141	C₄×C₄₃	—	4	8	no	—
142	C₄×C₄₃	#384	1	1	yes	characteristic
143	C₄×C₄₃	#385	1	1	yes	characteristic
144	C₄×C₄₃	#389	1	2	yes	—
145	C₄×C₄₃	#395	1	1	yes	characteristic
146	C₄×C₄₃	#398	1	2	yes	—
147	C₄×C₄₃	#399	1	1	yes	characteristic
148	Dic₄₃	—	4	4	no	—
149	Dic₄₃	—	8	4	no	—
150	#2911	—	43	1	no	—
151	#2296	—	43	2	no	—
152	#1733	—	43	2	no	—
153	C₂×H(3,ℤ/4)	D₄₃	1	1	yes	characteristic
154	#1728	—	43	1	no	—
155	C₂×C₄₃	—	2	2	no	—
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	—	2	2	no	—
158	C₂×C₄₃	—	2	1	no	—
159	C₂×C₄₃	#1444	1	1	yes	characteristic
160	C₂×C₄₃	#2010	1	2	yes	—
161	C₂²×M₁₆	—	43	1	no	—
162	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
163	C₂×#117	—	43	2	no	—
164	C₂×#117	C₂×D₄₃	1	2	yes	—
165	C₂²×C₄²	—	43	1	no	—
166	#428	—	86	4	no	—
167	#427	—	86	2	no	—
168	#389	—	86	4	no	—
169	#385	—	86	4	no	—
170	C₄×C₄⋊C₄	—	86	4	no	—
171	#339	—	86	4	no	—
172	H(3,ℤ/4)	Dic₄₃	1	4	yes	complemented
173	#336	—	86	2	no	—
174	C₄₃	#11335	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	43	2	no	—
176	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	43	2	no	—
180	C₂×M₁₆	—	86	1	no	—
181	C₂×M₁₆	—	86	2	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂×M₁₆	Dic₈₆	1	2	yes	—
184	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
185	C₂²×C₈	—	86	1	no	—
186	C₂²×C₈	D₁₇₂	1	2	yes	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	86	2	no	—
190	#117	—	86	2	no	—
191	#117	—	86	2	no	—
192	#117	—	86	2	no	—
193	#117	Dic₈₆	1	4	yes	—
194	#117	C₂×Dic₄₃	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	43	4	no	—
197	C₂×C₄⋊C₄	—	86	8	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	86	2	no	—
200	C₂×C₄²	—	86	2	no	—
201	C₂×C₄²	—	86	1	no	—
202	C₂×C₄²	—	86	2	no	—
203	C₂×C₄²	—	86	2	no	—
204	C₂×C₄²	—	86	1	no	—
205	C₂×C₄²	D₁₇₂	1	4	yes	—
206	#107	—	86	4	no	—
207	#106	—	86	4	no	—
208	#95	—	86	8	no	—
209	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	86	4	no	—
212	C₂²×C₄	—	86	2	no	—
213	C₂²×C₄	—	86	4	no	—
214	C₂²×C₄	—	172	4	no	—
215	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
217	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
219	C₂²×C₄	D₄×D₄₃	1	2	yes	—
220	C₂²×C₄	#10621900	1	2	yes	—
221	C₂²×C₄	#10621900	1	1	yes	characteristic
222	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	#10621902	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	86	4	no	—
228	M₁₆	—	172	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	86	2	no	—
232	C₂×C₈	—	172	1	no	—
233	C₂×C₈	—	172	1	no	—
234	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	complemented
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	86	4	no	—
238	C₄⋊C₄	—	86	4	no	—
239	C₄⋊C₄	—	172	16	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	172	2	no	—
243	C₂²⋊C₄	—	172	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	172	1	no	—
246	C₄²	—	172	2	no	—
247	C₄²	—	172	2	no	—
248	C₄²	—	172	1	no	—
249	C₄²	—	172	2	no	—
250	C₄²	C₄⋊Dic₄₃	1	8	yes	complemented
251	C₂³	—	4	2	no	—
252	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
253	C₂³	#13063295	1	2	yes	—
254	C₂³	#13063300	1	1	yes	characteristic
255	C₂³	#13063302	1	2	yes	—
256	C₂³	#13063324	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	86	8	no	—
260	C₂×C₄	—	86	4	no	—
261	C₂×C₄	—	172	8	no	—
262	C₂×C₄	—	172	4	no	—
263	C₂×C₄	—	172	4	no	—
264	C₂×C₄	—	172	2	no	—
265	C₂×C₄	—	172	2	no	—
266	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
268	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
269	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
270	C₂×C₄	#13063295	1	1	yes	characteristic
271	C₂×C₄	#13063300	1	2	yes	—
272	C₂×C₄	#13063302	1	2	yes	—
273	C₂×C₄	#13063302	1	2	yes	—
274	C₂×C₄	#13063305	1	2	yes	—
275	C₂×C₄	#13063305	1	1	yes	characteristic
276	C₂×C₄	#13063309	1	2	yes	—
277	C₂×C₄	#13063309	1	1	yes	characteristic
278	C₂×C₄	#13063316	1	2	yes	—
279	C₂×C₄	#13063316	1	1	yes	characteristic
280	C₂×C₄	#13063322	1	1	yes	characteristic
281	C₂×C₄	#13063324	1	2	yes	—
282	C₂×C₄	#13063325	1	1	yes	characteristic
283	C₂×C₄	#13063325	1	2	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	172	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	#14582774	1	2	yes	—
292	C₂²	#14582798	1	1	yes	characteristic
293	C₂²	#14582809	1	1	yes	characteristic
294	C₂²	#14582820	1	2	yes	—
295	C₂²	#14582831	1	1	yes	characteristic
296	C₄	—	4	8	no	—
297	C₄	—	344	4	no	—
298	C₄	—	172	4	no	—
299	C₄	#14582769	1	1	yes	characteristic
300	C₄	#14582774	1	1	yes	characteristic
301	C₄	#14582798	1	2	yes	—
302	C₄	#14582820	1	1	yes	characteristic
303	C₄	#14582831	1	2	yes	—
304	C₄	#14582836	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	#19132860	1	1	yes	characteristic
310	C₂	#19135663	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063322
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×H(3,ℤ/4)	C₂
socle	C₂²×C₄₃	#392
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₃
hypocenter	C₄₃	#11335
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
18. C₄₃×H(3,ℤ/4)	C₄	semidirect
73. C₂×C₈×C₄₃	C₄⋊C₄	semidirect
79. C₄²×C₄₃	C₄⋊C₄	semidirect
172. H(3,ℤ/4)	Dic₄₃	semidirect
174. C₄₃	#11335	semidirect
234. C₂×C₈	C₄⋊Dic₄₃	semidirect
250. C₄²	C₄⋊Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19143185	2
#19137079	2
#19134305	2
#19134288	2
C₂×C₄₃×H(3,ℤ/4)	2
#11335	43