Subgroups — #25966142

Order 11008.

3386 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583704	C₂	1	4	yes	maximal, complemented
2	#19148373	C₂	1	2	yes	maximal, complemented
3	#19145081	C₂	1	12	yes	maximal, complemented
4	#19145047	C₂	1	6	yes	maximal, complemented
5	#19145023	C₂	1	3	yes	maximal, complemented
6	C₂×#14583276	C₂	1	3	yes	maximal, complemented
7	C₄×#131×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063371	C₂²	1	4	yes	—
9	#14583704	—	2	32	no	—
10	C₂×#13063353	C₂²	1	24	yes	complemented
11	C₂×#13063347	C₂²	1	12	yes	complemented
12	C₂²×D₃₄₄	C₂²	1	12	yes	complemented
13	#14583571	—	2	4	no	—
14	D₄×D₁₇₂	—	2	12	no	—
15	#14583468	C₂²	1	12	yes	complemented
16	#14583449	C₂²	1	6	yes	complemented
17	C₂×#13063322	C₂²	1	6	yes	complemented
18	#14583288	C₂²	1	24	yes	complemented
19	#14583283	C₂²	1	12	yes	complemented
20	#14583279	C₂²	1	1	yes	characteristic, complemented
21	#14583276	C₂²	1	3	yes	complemented
22	#14583276	C₂²	1	6	yes	complemented
23	#14583276	C₂²	1	12	yes	complemented
24	#14583272	C₂²	1	6	yes	complemented
25	C₂×#131×C₄₃	C₂²	1	2	yes	—
26	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#405	C₂²	1	6	yes	complemented
28	C₄×M₁₆×C₄₃	C₂²	1	3	yes	complemented
29	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	complemented
30	#13063371	—	2	32	no	—
31	C₂×#10621902	—	2	4	no	—
32	C₂×#10621898	C₂³	1	12	yes	—
33	C₂²×D₁₇₂	—	2	12	no	—
34	C₂²×D₁₇₂	C₂³	1	12	yes	—
35	C₂×D₄×D₄₃	—	2	12	no	—
36	#13063353	—	2	96	no	—
37	#13063347	—	2	48	no	—
38	C₂×Dic₁₇₂	C₂³	1	4	yes	—
39	C₂×#10621873	C₂³	1	24	yes	—
40	C₂×D₃₄₄	—	2	48	no	—
41	C₂×D₃₄₄	C₂³	1	12	yes	—
42	C₂×D₃₄₄	C₂³	1	24	yes	—
43	#13063325	C₂³	1	2	yes	—
44	#13063322	—	2	12	no	—
45	#13063322	C₂³	1	6	yes	—
46	#13063322	C₂³	1	12	yes	—
47	#13063313	C₂³	1	12	yes	—
48	#13063300	—	2	24	no	—
49	#13063299	—	2	12	no	—
50	#13063295	—	2	24	no	—
51	C₄×D₁₇₂	—	2	12	no	—
52	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
53	#131×C₄₃	D₄	1	16	yes	complemented
54	C₂×M₁₆×C₄₃	C₂³	1	6	yes	—
55	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
56	C₄×Q₈×C₄₃	D₄	1	2	yes	—
57	C₄×D₄×C₄₃	D₄	1	6	yes	—
58	#117×C₄₃	C₂³	1	3	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
60	#97×C₄₃	C₂³	1	6	yes	—
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₄₃	C₂³	1	3	yes	—
63	C₄×C₈×C₄₃	C₂³	1	6	yes	—
64	C₂³×D₄₃	—	2	12	no	—
65	#41×C₄₃	C₂×D₄	1	8	yes	—
66	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₄₃	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
70	C₂×C₂²⋊D₄₃	—	2	12	no	—
71	#10621902	—	4	16	no	—
72	D₄×D₄₃	—	4	48	no	—
73	#10621898	—	2	48	no	—
74	C₂×D₁₇₂	—	2	12	no	—
75	C₂×D₁₇₂	—	2	12	no	—
76	C₂×D₁₇₂	—	2	24	no	—
77	C₂×D₁₇₂	—	2	48	no	—
78	C₂×D₁₇₂	—	8	24	no	—
79	C₂×D₁₇₂	C₂⁴	1	12	yes	—
80	C₂×C₄×D₄₃	—	2	12	no	—
81	C₂×Dic₈₆	C₂⁴	1	4	yes	—
82	M₁₆×C₄₃	C₂×D₄	1	24	yes	complemented
83	C₂×C₈×C₄₃	C₂×D₄	1	24	yes	complemented
84	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
85	C₂×C₈×C₄₃	C₂⁴	1	6	yes	—
86	C₄⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
87	C₂²⋊C₄×C₄₃	C₂×D₄	1	6	yes	—
88	C₄²×C₄₃	C₂×D₄	1	6	yes	—
89	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₄₃	C₂⁴	1	3	yes	—
91	(C₂×D₄₃)⋊C₄	—	2	24	no	—
92	C₄⋊Dic₄₃	—	2	4	no	—
93	Dic₁₇₂	—	2	8	no	—
94	D₃₄₄	—	2	24	no	—
95	D₃₄₄	—	2	48	no	—
96	#10621873	—	2	48	no	—
97	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
98	C₂²×D₄₃	—	2	12	no	—
99	C₂²×D₄₃	—	4	24	no	—
100	C₂²×D₄₃	—	8	24	no	—
101	Q₈×C₄₃	#127	1	4	yes	—
102	D₄×C₄₃	#127	1	12	yes	—
103	C₂×C₄×C₄₃	—	2	12	no	—
104	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
105	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
106	C₂×C₄×C₄₃	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₄₃	—	4	24	no	—
110	C₂×Dic₄₃	—	2	4	no	—
111	D₁₇₂	—	2	24	no	—
112	D₁₇₂	—	4	24	no	—
113	D₁₇₂	—	4	24	no	—
114	D₁₇₂	—	8	24	no	—
115	C₄×D₄₃	—	4	24	no	—
116	Dic₈₆	—	2	8	no	—
117	C₈×C₄₃	C₂²×D₄	1	12	yes	—
118	C₈×C₄₃	C₂²×D₄	1	4	yes	—
119	#60647	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	6	no	—
121	C₂²×C₄₃	C₂×#127	1	6	yes	—
122	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
123	C₂×D₄₃	—	4	12	no	—
124	C₂×D₄₃	—	4	24	no	—
125	C₂×D₄₃	—	16	24	no	—
126	C₄×C₄₃	—	2	6	no	—
127	C₄×C₄₃	C₂×#127	1	6	yes	—
128	C₄×C₄₃	#576	1	4	yes	complemented
129	C₄×C₄₃	C₂³×D₄	1	2	yes	—
130	Dic₄₃	—	4	4	no	—
131	C₂×#576	—	43	4	no	—
132	#3498	—	43	2	no	—
133	#3155	—	43	12	no	—
134	#3151	—	43	6	no	—
135	#3147	—	43	3	no	—
136	C₂×#493	—	43	3	no	—
137	C₄×#131	D₄₃	1	1	yes	characteristic, complemented
138	C₂×C₄₃	—	2	6	no	—
139	C₂×C₄₃	C₂²×#127	1	1	yes	characteristic
140	C₂×C₄₃	C₂×#576	1	2	yes	—
141	D₄₃	—	8	12	no	—
142	C₂×H(5,𝔽₂)	—	43	4	no	—
143	#576	—	86	32	no	—
144	C₂×AGL(1,ℤ/8)	—	43	24	no	—
145	C₂×#135	—	43	12	no	—
146	C₂²×D₈	—	43	12	no	—
147	C₂×#131	C₂×D₄₃	1	2	yes	—
148	#550	—	86	4	no	—
149	D₄²	—	86	12	no	—
150	#535	—	43	12	no	—
151	#532	—	43	6	no	—
152	C₂×#127	—	43	6	no	—
153	C₄×#41	C₂×D₄₃	1	1	yes	characteristic
154	#496	—	43	24	no	—
155	#495	—	43	12	no	—
156	#494	—	43	1	no	—
157	#493	—	43	3	no	—
158	#493	—	43	6	no	—
159	#493	—	43	12	no	—
160	#492	—	43	6	no	—
161	#405	C₂×D₄₃	1	6	yes	complemented
162	C₄×M₁₆	C₂×D₄₃	1	3	yes	complemented
163	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	complemented
164	C₄₃	#60647	1	1	yes	characteristic, complemented, Sylow
165	H(5,𝔽₂)	—	86	32	no	—
166	C₂×#41	—	43	12	no	—
167	C₂×#41	—	86	4	no	—
168	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
169	C₂²×D₄	—	43	12	no	—
170	C₂²×D₄	—	86	12	no	—
171	C₂²×D₄	—	86	12	no	—
172	AGL(1,ℤ/8)	—	86	96	no	—
173	#135	—	86	48	no	—
174	C₂×Q₁₆	—	43	4	no	—
175	C₂×SD₁₆	—	43	24	no	—
176	C₂×D₈	—	43	12	no	—
177	C₂×D₈	—	43	24	no	—
178	C₂×D₈	—	86	48	no	—
179	#131	D₁₇₂	1	16	yes	complemented
180	C₂×M₁₆	C₂²×D₄₃	1	6	yes	—
181	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
182	C₄⋊Q₈	—	43	2	no	—
183	#127	—	43	6	no	—
184	#127	—	43	12	no	—
185	#127	—	86	12	no	—
186	#124	—	43	12	no	—
187	#121	—	86	12	no	—
188	#121	—	86	24	no	—
189	#120	—	86	24	no	—
190	C₄×Q₈	D₁₇₂	1	2	yes	—
191	C₄×D₄	—	86	12	no	—
192	C₄×D₄	D₁₇₂	1	6	yes	—
193	#117	C₂²×D₄₃	1	3	yes	—
194	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
195	#97	C₂²×D₄₃	1	6	yes	—
196	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₄×C₈	C₂²×D₄₃	1	3	yes	—
198	C₄×C₈	C₂²×D₄₃	1	6	yes	—
199	C₂⁴	—	86	12	no	—
200	#41	—	86	48	no	—
201	#41	—	172	16	no	—
202	#41	C₂×D₁₇₂	1	8	yes	—
203	C₂×Q₈	—	43	4	no	—
204	C₂×Q₈	C₂×D₁₇₂	1	1	yes	characteristic
205	C₂×D₄	—	43	12	no	—
206	C₂×D₄	—	86	12	no	—
207	C₂×D₄	—	86	48	no	—
208	C₂×D₄	—	86	24	no	—
209	C₂×D₄	—	86	12	no	—
210	C₂×D₄	—	86	12	no	—
211	C₂×D₄	—	344	24	no	—
212	C₂×D₄	—	172	48	no	—
213	C₂×D₄	C₂×D₁₇₂	1	3	yes	—
214	C₂²×C₄	—	86	12	no	—
215	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
216	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
217	Q₁₆	—	86	8	no	—
218	SD₁₆	—	86	48	no	—
219	D₈	—	86	24	no	—
220	D₈	—	86	48	no	—
221	M₁₆	C₂×D₁₇₂	1	24	yes	complemented
222	C₂×C₈	C₂×D₁₇₂	1	24	yes	complemented
223	C₂×C₈	C₂³×D₄₃	1	2	yes	—
224	C₂×C₈	C₂³×D₄₃	1	6	yes	—
225	C₄⋊C₄	—	86	4	no	—
226	C₄⋊C₄	C₂×D₁₇₂	1	6	yes	—
227	C₂²⋊C₄	—	86	24	no	—
228	C₂²⋊C₄	C₂×D₁₇₂	1	6	yes	—
229	C₄²	C₂×D₁₇₂	1	6	yes	—
230	C₄²	C₂³×D₄₃	1	1	yes	characteristic
231	C₄²	C₂³×D₄₃	1	3	yes	—
232	C₂³	—	86	12	no	—
233	C₂³	—	344	24	no	—
234	C₂³	—	172	24	no	—
235	C₂³	C₂²×D₁₇₂	1	3	yes	—
236	Q₈	—	86	8	no	—
237	Q₈	#13063322	1	4	yes	—
238	D₄	—	86	24	no	—
239	D₄	—	344	24	no	—
240	D₄	—	172	24	no	—
241	D₄	—	172	24	no	—
242	D₄	—	172	24	no	—
243	D₄	#13063322	1	12	yes	—
244	C₂×C₄	—	2	12	no	—
245	C₂×C₄	—	86	4	no	—
246	C₂×C₄	—	172	24	no	—
247	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
248	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
249	C₂×C₄	C₂²×D₁₇₂	1	12	yes	—
250	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
251	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
252	C₈	C₂²×D₁₇₂	1	4	yes	—
253	C₈	C₂²×D₁₇₂	1	12	yes	—
254	C₂²	—	2	6	no	—
255	C₂²	—	172	24	no	—
256	C₂²	—	172	12	no	—
257	C₂²	—	688	24	no	—
258	C₂²	C₂×#13063322	1	6	yes	—
259	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
260	C₄	—	2	6	no	—
261	C₄	—	172	4	no	—
262	C₄	C₂×#13063322	1	6	yes	—
263	C₄	#14583704	1	4	yes	complemented
264	C₄	C₂³×D₁₇₂	1	2	yes	—
265	C₂	—	2	6	no	—
266	C₂	—	344	12	no	—
267	C₂	C₂²×#13063322	1	1	yes	characteristic
268	C₂	C₂×#14583704	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₄×#131×C₄₃	C₂
socle	C₂²×C₄₃	C₂³×D₄
hypercenter	C₄×#131	D₄₃
hypocenter	C₄₃	#60647
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#14583704	C₂	semidirect
2. #19148373	C₂	semidirect
3. #19145081	C₂	semidirect
4. #19145047	C₂	semidirect
5. #19145023	C₂	semidirect
6. C₂×#14583276	C₂	semidirect
7. C₄×#131×C₄₃	C₂	semidirect
10. C₂×#13063353	C₂²	semidirect
11. C₂×#13063347	C₂²	semidirect
12. C₂²×D₃₄₄	C₂²	semidirect
15. #14583468	C₂²	semidirect
16. #14583449	C₂²	semidirect
17. C₂×#13063322	C₂²	semidirect
18. #14583288	C₂²	semidirect
19. #14583283	C₂²	semidirect
20. #14583279	C₂²	semidirect
21. #14583276	C₂²	semidirect
22. #14583276	C₂²	semidirect
23. #14583276	C₂²	semidirect
24. #14583272	C₂²	semidirect
27. C₄₃×#405	C₂²	semidirect
28. C₄×M₁₆×C₄₃	C₂²	semidirect
29. C₂×C₄×C₈×C₄₃	C₂²	semidirect
53. #131×C₄₃	D₄	semidirect
82. M₁₆×C₄₃	C₂×D₄	semidirect
83. C₂×C₈×C₄₃	C₂×D₄	semidirect
128. C₄×C₄₃	#576	semidirect
137. C₄×#131	D₄₃	semidirect
161. #405	C₂×D₄₃	semidirect
162. C₄×M₁₆	C₂×D₄₃	semidirect
163. C₂×C₄×C₈	C₂×D₄₃	semidirect
164. C₄₃	#60647	semidirect
179. #131	D₁₇₂	semidirect
221. M₁₆	C₂×D₁₇₂	semidirect
222. C₂×C₈	C₂×D₁₇₂	semidirect
263. C₄	#14583704	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583704	2
#19148373	2
#19145081	2
#19145047	2
#19145023	2
C₂×#14583276	2
C₄×#131×C₄₃	2
#60647	43