Subgroups — #25196561

Order 11008.

608 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×#14583022	C₂	1	1	yes	maximal, characteristic
2	#19135764	C₂	1	4	yes	maximal, complemented
3	#19135491	C₂	1	2	yes	maximal
4	#19134718	C₂	1	2	yes	maximal, complemented
5	#19134646	C₂	1	2	yes	maximal, complemented
6	#19134638	C₂	1	2	yes	maximal, complemented
7	#19134479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	C₂²	1	1	yes	characteristic
10	C₂×#13063331	C₂²	1	2	yes	—
11	#14583022	—	2	4	no	—
12	C₂×#13063261	C₂²	1	1	yes	characteristic
13	C₂×#13063261	C₂²	1	1	yes	characteristic
14	#14582937	C₂²	1	1	yes	characteristic
15	#14582917	C₂²	1	4	yes	—
16	C₂×#13063253	C₂²	1	2	yes	—
17	#14582872	C₂²	1	4	yes	—
18	#14582854	C₂²	1	2	yes	—
19	#14582669	C₂²	1	4	yes	—
20	#14582659	C₂²	1	2	yes	—
21	#14582655	C₂²	1	4	yes	—
22	C₄₃×#533	C₂²	1	2	yes	—
23	C₄₃×#501	—	2	4	no	—
24	C₄₃×#427	C₂²	1	2	yes	—
25	C₄₃×#427	C₂²	1	2	yes	—
26	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂³	1	2	yes	—
28	#13063347	—	2	2	no	—
29	#13063347	—	2	2	no	—
30	C₂×Dic₁₇₂	Q₈	1	1	yes	characteristic, complemented
31	C₂×#10621873	Q₈	1	2	yes	complemented
32	C₂×D₃₄₄	Q₈	1	1	yes	characteristic, complemented
33	#13063331	—	2	2	no	—
34	#13063331	—	2	2	no	—
35	#13063331	—	2	2	no	—
36	#13063331	—	2	2	no	—
37	C₂×#10621877	C₂³	1	2	yes	—
38	C₂×#10621877	C₂³	1	2	yes	—
39	C₂×#10621872	—	2	2	no	—
40	C₂×C₈×D₄₃	—	2	2	no	—
41	#13063284	C₂³	1	2	yes	—
42	#13063261	—	2	2	no	—
43	#13063261	—	2	2	no	—
44	#13063259	—	2	1	no	—
45	#13063257	—	2	1	no	—
46	#13063253	—	2	4	no	—
47	#13063253	—	4	4	no	—
48	#13063239	—	4	4	no	—
49	#13063235	—	2	2	no	—
50	C₈×Dic₄₃	—	2	2	no	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₄₃	—	2	2	no	—
54	#125×C₄₃	—	2	2	no	—
55	C₂²⋊Q₈×C₄₃	—	4	2	no	—
56	C₄×Q₈×C₄₃	—	4	2	no	—
57	#117×C₄₃	C₂³	1	4	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	#107×C₄₃	—	2	2	no	—
60	#107×C₄₃	—	2	2	no	—
61	#106×C₄₃	—	2	2	no	—
62	#106×C₄₃	—	2	2	no	—
63	#97×C₄₃	D₄	1	4	yes	—
64	C₂×Q₈×C₄₃	—	4	2	no	—
65	C₂²×C₄×C₄₃	—	2	1	no	—
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₂²⋊D₄₃	—	2	2	no	—
68	#10621898	—	2	2	no	—
69	#10621898	—	2	2	no	—
70	#10621898	—	2	2	no	—
71	#10621898	—	2	2	no	—
72	C₂×D₁₇₂	C₂×Q₈	1	2	yes	—
73	C₂×C₄×D₄₃	—	2	2	no	—
74	C₂×Dic₈₆	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₄₃	#41	1	2	yes	—
80	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
82	C₂×C₈×C₄₃	#41	1	2	yes	—
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄⋊C₄×C₄₃	—	4	4	no	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
86	C₂²⋊C₄×C₄₃	—	2	4	no	—
87	C₄²×C₄₃	—	2	1	no	—
88	C₄²×C₄₃	—	2	4	no	—
89	C₄²×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²⋊Dic₄₃	—	2	2	no	—
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄×Dic₄₃	—	2	2	no	—
93	#10621877	—	2	2	no	—
94	#10621877	—	2	2	no	—
95	#10621877	—	2	2	no	—
96	#10621877	—	2	2	no	—
97	C₂×C₄₃⋊C₈	—	2	2	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	Dic₁₇₂	—	2	2	no	—
100	D₃₄₄	—	2	2	no	—
101	#10621873	—	2	4	no	—
102	#10621872	—	4	2	no	—
103	#10621872	—	4	2	no	—
104	C₈×D₄₃	—	4	2	no	—
105	C₈×D₄₃	—	4	2	no	—
106	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	2	no	—
108	Q₈×C₄₃	—	8	4	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	—	8	1	no	—
112	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²⋊D₄₃	—	4	2	no	—
119	C₂²⋊D₄₃	—	4	2	no	—
120	C₂×Dic₄₃	—	2	2	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	D₁₇₂	—	2	2	no	—
123	D₁₇₂	—	2	2	no	—
124	C₄×D₄₃	—	4	2	no	—
125	C₄×D₄₃	—	4	2	no	—
126	Dic₈₆	—	2	2	no	—
127	Dic₈₆	—	2	2	no	—
128	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
129	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₄₃	#124	1	2	yes	—
131	C₈×C₄₃	#124	1	2	yes	—
132	C₄₃⋊C₈	—	4	2	no	—
133	C₄₃⋊C₈	—	4	2	no	—
134	#29155	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	#535	1	1	yes	characteristic
137	C₂²×C₄₃	#554	1	1	yes	characteristic
138	C₂²×C₄₃	#555	1	1	yes	characteristic
139	C₂×D₄₃	—	4	2	no	—
140	C₂×D₄₃	—	8	2	no	—
141	C₄×C₄₃	—	4	2	no	—
142	C₄×C₄₃	—	8	4	no	—
143	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₃	#554	1	1	yes	characteristic
146	C₄×C₄₃	#555	1	1	yes	characteristic
147	Dic₄₃	—	4	2	no	—
148	Dic₄₃	—	8	2	no	—
149	#3162	D₄₃	1	1	yes	characteristic, complemented
150	C₂×#444	—	43	1	no	—
151	#2031	—	43	4	no	—
152	#1976	—	43	2	no	—
153	#1833	—	43	2	no	—
154	#1814	—	43	2	no	—
155	#1812	—	43	2	no	—
156	#1773	—	43	1	no	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	—	2	1	no	—
159	C₂×C₄₃	—	2	1	no	—
160	C₂×C₄₃	#2563	1	1	yes	characteristic
161	D₄₃	—	8	2	no	—
162	C₂×#135	—	43	1	no	—
163	C₂×#131	—	43	2	no	—
164	#533	C₂×D₄₃	1	2	yes	—
165	#501	—	2	4	no	—
166	#444	—	86	4	no	—
167	C₂×#108	—	43	1	no	—
168	C₂×#108	—	43	1	no	—
169	#427	—	43	1	no	—
170	#427	C₂×D₄₃	1	2	yes	—
171	#427	C₂×D₄₃	1	2	yes	—
172	#421	—	43	4	no	—
173	C₂×#104	—	43	2	no	—
174	#410	—	43	4	no	—
175	#405	—	43	2	no	—
176	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
177	#344	—	43	4	no	—
178	#339	—	43	2	no	—
179	H(3,ℤ/4)	—	43	4	no	—
180	C₄₃	#29155	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	—	43	2	no	—
182	#135	—	86	2	no	—
183	#135	—	86	2	no	—
184	C₂×Q₁₆	—	43	1	no	—
185	C₂×SD₁₆	—	43	2	no	—
186	C₂×D₈	—	43	1	no	—
187	#131	—	86	2	no	—
188	#131	—	86	2	no	—
189	#131	—	86	2	no	—
190	#131	—	86	2	no	—
191	C₂×M₁₆	—	43	2	no	—
192	C₂×M₁₆	—	43	2	no	—
193	C₂×M₁₆	—	86	2	no	—
194	C₂²×C₈	—	86	2	no	—
195	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	#125	—	2	2	no	—
199	C₂²⋊Q₈	—	4	2	no	—
200	C₄×Q₈	—	4	2	no	—
201	#117	—	43	2	no	—
202	#117	C₂²×D₄₃	1	4	yes	—
203	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	#108	—	86	2	no	—
205	#108	—	86	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	2	no	—
208	#107	—	86	1	no	—
209	#106	—	2	2	no	—
210	#106	—	2	2	no	—
211	#106	—	86	1	no	—
212	#104	—	86	4	no	—
213	#104	—	172	4	no	—
214	#99	—	172	4	no	—
215	#97	—	86	2	no	—
216	#97	C₂²⋊D₄₃	1	4	yes	—
217	C₄×C₈	—	86	2	no	—
218	#41	—	86	2	no	—
219	#41	—	86	2	no	—
220	#41	—	86	2	no	—
221	#41	—	86	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	—	43	2	no	—
224	C₂×D₄	—	43	2	no	—
225	C₂×D₄	—	86	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	86	2	no	—
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	Q₁₆	—	86	2	no	—
230	SD₁₆	—	86	4	no	—
231	D₈	—	86	2	no	—
232	M₁₆	—	86	2	no	—
233	M₁₆	—	86	2	no	—
234	M₁₆	—	86	2	no	—
235	M₁₆	—	86	2	no	—
236	M₁₆	—	172	2	no	—
237	M₁₆	—	172	2	no	—
238	C₂×C₈	—	86	2	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	172	2	no	—
241	C₂×C₈	—	172	2	no	—
242	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
243	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
244	C₂×C₈	#10621902	1	2	yes	—
245	C₂×C₈	#10621902	1	1	yes	characteristic
246	C₂×C₈	#10621902	1	1	yes	characteristic
247	C₂×C₈	#10621902	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	86	2	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	86	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	86	2	no	—
259	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
260	C₂³	—	86	2	no	—
261	C₂³	#13063370	1	1	yes	characteristic
262	Q₈	—	8	4	no	—
263	Q₈	—	86	2	no	—
264	Q₈	—	86	2	no	—
265	D₄	—	86	2	no	—
266	D₄	—	86	2	no	—
267	D₄	—	172	2	no	—
268	D₄	—	172	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	86	2	no	—
273	C₂×C₄	—	172	2	no	—
274	C₂×C₄	—	172	2	no	—
275	C₂×C₄	—	172	2	no	—
276	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
277	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
279	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
280	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
281	C₂×C₄	#13063370	1	1	yes	characteristic
282	C₈	—	172	2	no	—
283	C₈	—	172	2	no	—
284	C₈	#13063301	1	2	yes	—
285	C₈	#13063301	1	2	yes	—
286	C₈	#13063314	1	2	yes	—
287	C₈	#13063314	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	344	2	no	—
290	C₂²	—	172	2	no	—
291	C₂²	#14583466	1	1	yes	characteristic
292	C₂²	#14583610	1	1	yes	characteristic
293	C₂²	#14583615	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	8	4	no	—
296	C₄	—	344	2	no	—
297	C₄	—	172	2	no	—
298	C₄	C₂×#13063301	1	1	yes	characteristic
299	C₄	C₂×#13063314	1	1	yes	characteristic
300	C₄	#14583610	1	1	yes	characteristic
301	C₄	#14583615	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	344	2	no	—
306	C₂	#19139769	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583615
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#3162	C₂
socle	C₂×C₄₃	#2563
prime radical	C₂×#13063347	C₂²
hypercenter	#3162	D₄₃
hypocenter	C₄₃	#29155
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3162

Complements

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

N	H ≅ G/N	type
2. #19135764	C₂	semidirect
4. #19134718	C₂	semidirect
5. #19134646	C₂	semidirect
6. #19134638	C₂	semidirect
7. #19134479	C₂	semidirect
8. C₄₃×#3162	C₂	semidirect
30. C₂×Dic₁₇₂	Q₈	semidirect
31. C₂×#10621873	Q₈	semidirect
32. C₂×D₃₄₄	Q₈	semidirect
149. #3162	D₄₃	semidirect
180. C₄₃	#29155	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583022	2
#19135764	2
#19135491	2
#19134718	2
#19134646	2
#19134638	2
#19134479	2
C₄₃×#3162	2
#29155	43