Subgroups — #25011897

Order 11008.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#19136345	C₂	1	1	yes	maximal, characteristic
3	#19136324	C₂	1	1	yes	maximal, characteristic
4	#19134501	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132707	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	C₂×#13063331	—	2	1	no	—
10	#14583022	—	2	2	no	—
11	#14583022	C₄	1	4	yes	—
12	C₂×#13063261	C₂²	1	1	yes	characteristic
13	C₂×#13063261	C₂²	1	1	yes	characteristic
14	C₂×#13063261	C₂²	1	1	yes	characteristic
15	C₂×#13063253	—	2	1	no	—
16	C₄×#10621877	—	2	1	no	—
17	#14582716	C₄	1	2	yes	complemented
18	#14582713	C₄	1	2	yes	complemented
19	#14582707	C₄	1	4	yes	complemented
20	#14582664	—	2	1	no	—
21	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#347	—	2	2	no	—
25	C₂×#10621898	—	2	1	no	—
26	#13063347	—	2	1	no	—
27	#13063347	—	2	1	no	—
28	#13063347	C₂×C₄	1	2	yes	—
29	#13063347	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#10621873	—	2	1	no	—
32	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
33	#13063331	—	2	4	no	—
34	#13063331	—	4	2	no	—
35	C₂×#10621877	—	2	1	no	—
36	C₂×#10621877	—	2	1	no	—
37	C₂×#10621877	—	2	1	no	—
38	C₂×#10621872	—	4	1	no	—
39	C₂×C₈×D₄₃	—	4	1	no	—
40	#13063261	—	2	1	no	—
41	#13063261	—	2	1	no	—
42	#13063261	—	2	1	no	—
43	#13063261	C₂×C₄	1	2	yes	—
44	#13063261	C₂×C₄	1	2	yes	—
45	#13063261	C₂×C₄	1	2	yes	—
46	#13063253	—	2	4	no	—
47	#13063236	—	4	1	no	—
48	C₄×C₄₃⋊C₈	—	4	1	no	—
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	M₃₂×C₄₃	—	2	1	no	—
53	M₃₂×C₄₃	—	2	1	no	—
54	M₃₂×C₄₃	—	2	1	no	—
55	M₃₂×C₄₃	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₄₃	—	2	1	no	—
57	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
58	#97×C₄₃	C₂×C₄	1	2	yes	—
59	#97×C₄₃	D₄	1	2	yes	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	4	1	no	—
63	#10621898	—	2	2	no	—
64	#10621898	—	2	2	no	—
65	#10621898	—	4	1	no	—
66	#10621898	—	4	1	no	—
67	C₂×D₁₇₂	—	2	1	no	—
68	C₂×C₄×D₄₃	—	4	1	no	—
69	C₂×Dic₈₆	—	2	1	no	—
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
79	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
81	#10621877	—	2	2	no	—
82	#10621877	—	2	2	no	—
83	#10621877	—	2	2	no	—
84	#10621877	—	4	1	no	—
85	#10621877	—	4	1	no	—
86	#10621877	—	4	1	no	—
87	C₂×C₄₃⋊C₈	—	4	1	no	—
88	C₂×C₄₃⋊C₈	—	4	1	no	—
89	C₂×C₄₃⋊C₈	—	4	1	no	—
90	Dic₁₇₂	—	2	1	no	—
91	Dic₁₇₂	C₄²	1	2	yes	—
92	D₃₄₄	—	2	1	no	—
93	D₃₄₄	C₄²	1	2	yes	—
94	#10621873	—	2	2	no	—
95	#10621873	—	2	2	no	—
96	#10621872	—	4	4	no	—
97	C₈×D₄₃	—	4	4	no	—
98	C₁₆×C₄₃	—	2	2	no	—
99	C₁₆×C₄₃	—	2	2	no	—
100	C₂³×C₄₃	#137	1	1	yes	characteristic
101	C₂²×D₄₃	—	4	1	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	8	1	no	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂²⋊D₄₃	—	4	2	no	—
111	C₂²⋊D₄₃	—	4	2	no	—
112	C₂×Dic₄₃	—	4	1	no	—
113	D₁₇₂	—	2	2	no	—
114	D₁₇₂	—	4	1	no	—
115	C₄×D₄₃	—	4	2	no	—
116	C₄×D₄₃	—	4	2	no	—
117	Dic₈₆	—	2	2	no	—
118	Dic₈₆	—	4	1	no	—
119	C₈×C₄₃	—	2	2	no	—
120	C₈×C₄₃	C₂×C₄²	1	1	yes	characteristic
121	C₈×C₄₃	#117	1	1	yes	characteristic
122	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
123	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₄₃⋊C₈	—	4	2	no	—
125	C₄₃⋊C₈	—	4	2	no	—
126	C₄₃⋊C₈	—	4	2	no	—
127	#12651	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	4	1	no	—
129	C₂²×C₄₃	#417	1	1	yes	characteristic
130	C₂²×C₄₃	#440	1	1	yes	characteristic
131	C₂²×C₄₃	#440	1	1	yes	characteristic
132	C₂×D₄₃	—	4	2	no	—
133	C₂×D₄₃	—	8	2	no	—
134	C₄×C₄₃	—	4	2	no	—
135	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₄₃	#418	1	1	yes	characteristic
137	C₄×C₄₃	#442	1	1	yes	characteristic
138	C₄×C₄₃	#442	1	1	yes	characteristic
139	Dic₄₃	—	4	2	no	—
140	C₂×#444	—	43	1	no	—
141	#2148	—	43	1	no	—
142	#2143	—	43	1	no	—
143	C₂×#347	D₄₃	1	1	yes	characteristic, complemented
144	#1778	—	43	1	no	—
145	#1388	—	43	1	no	—
146	#1385	—	43	1	no	—
147	C₂×C₄₃	—	2	1	no	—
148	C₂×C₄₃	—	2	1	no	—
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	#1762	1	1	yes	characteristic
151	D₄₃	—	8	2	no	—
152	C₂×#135	—	43	1	no	—
153	C₂×#131	—	86	1	no	—
154	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
155	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
156	#444	—	43	4	no	—
157	#444	—	86	2	no	—
158	C₂×#108	—	43	1	no	—
159	C₂×#108	—	43	1	no	—
160	C₂×#108	—	43	1	no	—
161	C₂×#104	—	86	1	no	—
162	C₄×M₁₆	—	86	1	no	—
163	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
164	#362	—	43	2	no	—
165	#361	—	43	2	no	—
166	#359	—	43	4	no	—
167	#347	—	2	2	no	—
168	#341	—	86	1	no	—
169	C₄₃	#12651	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	86	1	no	—
171	#135	—	43	2	no	—
172	#135	—	43	2	no	—
173	#135	—	86	1	no	—
174	#135	—	86	1	no	—
175	C₂×Q₁₆	—	43	1	no	—
176	C₂×SD₁₆	—	86	1	no	—
177	C₂×D₈	—	43	1	no	—
178	#131	—	86	4	no	—
179	#131	—	172	2	no	—
180	C₂×M₁₆	—	86	1	no	—
181	C₂×M₁₆	—	86	1	no	—
182	C₂×M₁₆	—	86	1	no	—
183	C₂×M₁₆	—	172	1	no	—
184	C₂²×C₈	—	172	1	no	—
185	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
187	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	—	2	1	no	—
191	M₃₂	C₄×D₄₃	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
194	#108	—	43	2	no	—
195	#108	—	43	2	no	—
196	#108	—	43	2	no	—
197	#108	—	86	1	no	—
198	#108	—	86	1	no	—
199	#108	—	86	1	no	—
200	#104	—	86	4	no	—
201	#97	—	172	1	no	—
202	#97	C₄×D₄₃	1	2	yes	—
203	#97	C₂²⋊D₄₃	1	2	yes	—
204	C₄×C₈	—	172	1	no	—
205	#41	—	86	2	no	—
206	#41	—	86	2	no	—
207	#41	—	172	1	no	—
208	#41	—	172	1	no	—
209	C₂×Q₈	—	86	1	no	—
210	C₂×D₄	—	86	1	no	—
211	C₂×D₄	—	172	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	172	1	no	—
214	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
215	Q₁₆	—	43	2	no	—
216	Q₁₆	—	86	1	no	—
217	SD₁₆	—	86	2	no	—
218	SD₁₆	—	86	2	no	—
219	D₈	—	43	2	no	—
220	D₈	—	86	1	no	—
221	M₁₆	—	86	2	no	—
222	M₁₆	—	86	2	no	—
223	M₁₆	—	86	2	no	—
224	M₁₆	—	172	4	no	—
225	M₁₆	—	172	1	no	—
226	M₁₆	—	172	1	no	—
227	M₁₆	—	172	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	172	4	no	—
231	C₂×C₈	—	172	1	no	—
232	C₂×C₈	—	172	1	no	—
233	C₂×C₈	—	172	1	no	—
234	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
238	C₂×C₈	#10621898	1	1	yes	characteristic
239	C₂×C₈	#10621898	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₂³	—	172	1	no	—
246	C₂³	#13063358	1	1	yes	characteristic
247	Q₈	—	86	2	no	—
248	Q₈	—	172	1	no	—
249	D₄	—	86	2	no	—
250	D₄	—	172	2	no	—
251	D₄	—	172	2	no	—
252	D₄	—	172	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	172	1	no	—
256	C₂×C₄	—	172	2	no	—
257	C₂×C₄	—	172	2	no	—
258	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
260	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
261	C₂×C₄	#13063353	1	1	yes	characteristic
262	C₂×C₄	#13063353	1	1	yes	characteristic
263	C₂×C₄	#13063353	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	172	2	no	—
266	C₈	—	172	2	no	—
267	C₈	—	172	2	no	—
268	C₈	C₄²×D₄₃	1	1	yes	characteristic
269	C₈	#13063282	1	1	yes	characteristic
270	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
271	C₈	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	344	2	no	—
274	C₂²	—	172	2	no	—
275	C₂²	#14582901	1	1	yes	characteristic
276	C₂²	#14583001	1	1	yes	characteristic
277	C₂²	#14583001	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	172	2	no	—
280	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
281	C₄	#14582905	1	1	yes	characteristic
282	C₄	#14583012	1	1	yes	characteristic
283	C₄	#14583012	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	344	2	no	—
288	C₂	#19134421	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583012
commutator	C₈×C₄₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄₃×#347	C₂
socle	C₂×C₄₃	#1762
prime radical	C₂×#13063347	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#347	D₄₃
hypocenter	C₄₃	#12651
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #19134501	C₂	semidirect
5. #19132707	C₂	semidirect
6. #19132699	C₂	semidirect
7. C₂×C₄₃×#347	C₂	semidirect
17. #14582716	C₄	semidirect
18. #14582713	C₄	semidirect
19. #14582707	C₄	semidirect
143. C₂×#347	D₄₃	semidirect
169. C₄₃	#12651	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
#19136345	2
#19136324	2
#19134501	2
#19132707	2
#19132699	2
C₂×C₄₃×#347	2
#12651	43