Subgroups — #25252051

Order 11008.

858 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583334	C₂	1	2	yes	maximal, complemented
2	C₂²×D₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19145082	C₂	1	2	yes	maximal, complemented
4	#19136860	C₂	1	4	yes	maximal, complemented
5	#19136857	C₂	1	2	yes	maximal, complemented
6	#19136843	C₂	1	2	yes	maximal, complemented
7	#19136191	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063350	—	2	2	no	—
10	C₂²×D₃₄₄	C₂²	1	2	yes	—
11	#14583463	—	2	2	no	—
12	#14583334	—	2	8	no	—
13	C₂×#13063268	C₂²	1	4	yes	complemented
14	C₂×D₆₈₈	—	2	4	no	—
15	C₂×D₆₈₈	C₂²	1	2	yes	—
16	C₂×D₆₈₈	C₂²	1	2	yes	—
17	C₂×#13063266	C₂²	1	4	yes	complemented
18	C₂×#13063265	C₂²	1	1	yes	characteristic
19	#14583285	—	2	4	no	—
20	#14583284	C₂²	1	4	yes	complemented
21	#14583274	C₂²	1	4	yes	complemented
22	#14582854	C₂²	1	1	yes	characteristic
23	#14582674	C₂²	1	2	yes	complemented
24	C₁₆×Dic₄₃	C₂²	1	2	yes	complemented
25	C₂×#135×C₄₃	C₂²	1	2	yes	—
26	C₄₃×#508	—	2	4	no	—
27	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic, complemented
28	C₂×SD₃₂×C₄₃	C₂²	1	2	yes	complemented
29	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
30	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×D₁₇₂	—	2	2	no	—
32	#13063350	—	4	8	no	—
33	C₂×#10621884	—	2	4	no	—
34	C₂×D₃₄₄	—	2	4	no	—
35	C₂×D₃₄₄	—	2	4	no	—
36	C₂×D₃₄₄	—	2	2	no	—
37	C₂×D₃₄₄	C₂³	1	4	yes	—
38	C₂×#10621882	—	2	4	no	—
39	C₂×#10621877	D₄	1	1	yes	characteristic
40	#13063321	—	2	4	no	—
41	#13063314	—	2	4	no	—
42	#13063299	—	4	4	no	—
43	#13063293	—	4	4	no	—
44	#13063284	D₄	1	1	yes	characteristic
45	#13063268	—	2	8	no	—
46	D₆₈₈	—	2	4	no	—
47	D₆₈₈	—	2	4	no	—
48	#13063266	—	2	8	no	—
49	#13063265	D₄	1	4	yes	complemented
50	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
51	#13063235	D₄	1	2	yes	—
52	C₈×Dic₄₃	C₂³	1	2	yes	—
53	C₂×#41×C₄₃	—	2	2	no	—
54	#135×C₄₃	—	2	4	no	—
55	#135×C₄₃	—	2	4	no	—
56	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
57	C₂×SD₁₆×C₄₃	—	2	2	no	—
58	C₂×D₈×C₄₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	Q₃₂×C₄₃	—	2	2	no	—
61	SD₃₂×C₄₃	—	2	4	no	—
62	D₁₆×C₄₃	—	2	2	no	—
63	C₂×C₁₆×C₄₃	D₄	1	4	yes	complemented
64	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
66	C₂³×D₄₃	—	4	2	no	—
67	#41×C₄₃	—	4	4	no	—
68	#41×C₄₃	—	4	4	no	—
69	C₂×Q₈×C₄₃	—	2	2	no	—
70	C₂×D₄×C₄₃	—	2	2	no	—
71	C₂×D₄×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	—	4	2	no	—
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₃	—	4	8	no	—
75	C₂×D₁₇₂	—	2	4	no	—
76	C₂×D₁₇₂	—	4	2	no	—
77	C₂×D₁₇₂	—	4	4	no	—
78	C₂×D₁₇₂	—	4	4	no	—
79	Q₁₆×C₄₃	—	2	2	no	—
80	Q₁₆×C₄₃	—	2	2	no	—
81	SD₁₆×C₄₃	—	4	4	no	—
82	D₈×C₄₃	—	2	2	no	—
83	D₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₄₃	D₈	1	2	yes	—
89	#10621884	—	4	8	no	—
90	#10621882	—	4	8	no	—
91	(C₂×D₄₃)⋊C₄	—	4	8	no	—
92	C₄⋊Dic₄₃	D₈	1	2	yes	—
93	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
94	#10621877	D₈	1	4	yes	—
95	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
96	D₃₄₄	—	2	4	no	—
97	D₃₄₄	—	2	4	no	—
98	D₃₄₄	—	4	4	no	—
99	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
100	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
101	C₄₃⋊C₁₆	C₂×D₄	1	4	yes	—
102	C₂³×C₄₃	—	4	2	no	—
103	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₄₃	—	4	4	no	—
105	C₂²×D₄₃	—	8	4	no	—
106	C₂²×D₄₃	—	16	4	no	—
107	Q₈×C₄₃	—	4	2	no	—
108	Q₈×C₄₃	—	4	2	no	—
109	D₄×C₄₃	—	4	2	no	—
110	D₄×C₄₃	—	4	2	no	—
111	D₄×C₄₃	—	8	4	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	8	4	no	—
114	C₂×C₄×C₄₃	#127	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₄₃	—	16	8	no	—
119	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
120	D₁₇₂	—	4	4	no	—
121	D₁₇₂	—	4	4	no	—
122	D₁₇₂	—	8	4	no	—
123	C₈×C₄₃	#127	1	2	yes	—
124	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₄₃⋊C₈	C₂×D₈	1	4	yes	—
127	#33720	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	8	2	no	—
130	C₂²×C₄₃	—	8	4	no	—
131	C₂²×C₄₃	#493	1	2	yes	—
132	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
133	C₂×D₄₃	—	8	4	no	—
134	C₂×D₄₃	—	8	8	no	—
135	C₂×D₄₃	—	16	8	no	—
136	C₄×C₄₃	—	8	2	no	—
137	C₄×C₄₃	#493	1	2	yes	—
138	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
140	Dic₄₃	—	2	4	no	—
141	C₂×#509	—	43	2	no	—
142	C₂×#508	D₄₃	1	1	yes	characteristic, complemented
143	C₂²×D₁₆	—	43	1	no	—
144	#3155	—	43	2	no	—
145	#2250	—	43	4	no	—
146	#2249	—	43	2	no	—
147	#2246	—	43	2	no	—
148	#2108	—	43	1	no	—
149	C₂×C₄₃	—	2	2	no	—
150	C₂×C₄₃	—	16	2	no	—
151	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
152	C₂×C₄₃	#3415	1	2	yes	—
153	D₄₃	—	16	4	no	—
154	C₂×AGL(1,ℤ/8)	—	86	2	no	—
155	C₂×#135	C₂×D₄₃	1	2	yes	—
156	C₂²×D₈	—	43	2	no	—
157	#535	—	86	2	no	—
158	#509	—	86	8	no	—
159	#508	—	2	4	no	—
160	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic, complemented
161	C₂×SD₃₂	—	43	4	no	—
162	C₂×SD₃₂	C₂×D₄₃	1	2	yes	complemented
163	C₂×D₁₆	—	43	4	no	—
164	C₂×D₁₆	—	43	2	no	—
165	C₂×D₁₆	—	43	2	no	—
166	C₂×D₁₆	—	86	4	no	—
167	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
168	C₂×M₃₂	—	43	1	no	—
169	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
170	#496	—	86	4	no	—
171	#495	—	43	4	no	—
172	#493	—	43	4	no	—
173	#405	—	43	1	no	—
174	#346	—	43	2	no	—
175	C₄×C₁₆	—	43	2	no	—
176	C₄₃	#33720	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	2	2	no	—
178	C₂²×D₄	—	86	2	no	—
179	AGL(1,ℤ/8)	—	172	8	no	—
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×SD₁₆	—	86	4	no	—
185	C₂×D₈	—	43	4	no	—
186	C₂×D₈	—	86	4	no	—
187	C₂×D₈	—	86	4	no	—
188	C₂×D₈	—	86	2	no	—
189	C₂×D₈	—	86	4	no	—
190	C₂×D₈	C₂²×D₄₃	1	2	yes	—
191	C₂×M₁₆	—	43	1	no	—
192	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	#127	—	86	4	no	—
194	#124	—	86	4	no	—
195	#121	—	172	4	no	—
196	#120	—	172	4	no	—
197	#117	—	43	1	no	—
198	Q₃₂	—	2	2	no	—
199	SD₃₂	—	2	4	no	—
200	SD₃₂	—	86	8	no	—
201	D₁₆	—	2	2	no	—
202	D₁₆	—	86	8	no	—
203	D₁₆	—	86	4	no	—
204	D₁₆	—	86	4	no	—
205	M₃₂	—	43	4	no	—
206	C₂×C₁₆	—	43	2	no	—
207	C₂×C₁₆	C₂²⋊D₄₃	1	4	yes	complemented
208	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
209	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
210	#97	—	43	2	no	—
211	C₄×C₈	—	43	2	no	—
212	C₂⁴	—	172	2	no	—
213	#41	—	4	4	no	—
214	#41	—	4	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	86	4	no	—
219	C₂×D₄	—	172	8	no	—
220	C₂×D₄	—	172	4	no	—
221	C₂×D₄	—	172	2	no	—
222	C₂×D₄	—	172	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	SD₁₆	—	172	8	no	—
229	D₈	—	2	2	no	—
230	D₈	—	2	2	no	—
231	D₈	—	86	4	no	—
232	D₈	—	86	4	no	—
233	D₈	—	172	8	no	—
234	D₈	—	172	4	no	—
235	M₁₆	—	43	4	no	—
236	C₂×C₈	—	43	2	no	—
237	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
238	C₂×C₈	D₄×D₄₃	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
240	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
241	C₄⋊C₄	—	43	2	no	—
242	C₂²⋊C₄	—	43	2	no	—
243	C₂²⋊C₄	—	172	8	no	—
244	C₄²	—	43	2	no	—
245	C₁₆	—	43	4	no	—
246	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
247	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
248	C₂³	—	4	2	no	—
249	C₂³	—	344	4	no	—
250	C₂³	—	172	4	no	—
251	C₂³	—	688	4	no	—
252	C₂³	D₈×D₄₃	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	344	4	no	—
259	D₄	—	172	4	no	—
260	D₄	—	172	4	no	—
261	D₄	—	688	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	43	4	no	—
265	C₂×C₄	#13063321	1	2	yes	—
266	C₂×C₄	D₈×D₄₃	1	2	yes	—
267	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
269	C₈	—	43	4	no	—
270	C₈	#13063321	1	2	yes	—
271	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
272	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	344	4	no	—
277	C₂²	—	344	8	no	—
278	C₂²	—	688	8	no	—
279	C₂²	#14583274	1	2	yes	—
280	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
281	C₄	—	8	2	no	—
282	C₄	—	86	4	no	—
283	C₄	#14583274	1	2	yes	—
284	C₄	C₂×#13063321	1	1	yes	characteristic
285	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	16	2	no	—
288	C₂	—	688	4	no	—
289	C₂	C₂×#14583274	1	1	yes	characteristic
290	C₂	#19147733	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#508	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#508	D₄₃
hypocenter	C₄₃	#33720
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#14583334	C₂	semidirect
2. C₂²×D₆₈₈	C₂	semidirect
3. #19145082	C₂	semidirect
4. #19136860	C₂	semidirect
5. #19136857	C₂	semidirect
6. #19136843	C₂	semidirect
7. #19136191	C₂	semidirect
8. C₂×C₄₃×#508	C₂	semidirect
13. C₂×#13063268	C₂²	semidirect
17. C₂×#13063266	C₂²	semidirect
20. #14583284	C₂²	semidirect
21. #14583274	C₂²	semidirect
23. #14582674	C₂²	semidirect
24. C₁₆×Dic₄₃	C₂²	semidirect
27. C₂×Q₃₂×C₄₃	C₂²	semidirect
28. C₂×SD₃₂×C₄₃	C₂²	semidirect
29. C₂×D₁₆×C₄₃	C₂²	semidirect
49. #13063265	D₄	semidirect
63. C₂×C₁₆×C₄₃	D₄	semidirect
142. C₂×#508	D₄₃	semidirect
160. C₂×Q₃₂	C₂×D₄₃	semidirect
162. C₂×SD₃₂	C₂×D₄₃	semidirect
167. C₂×D₁₆	C₂×D₄₃	semidirect
176. C₄₃	#33720	semidirect
207. C₂×C₁₆	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583334	2
C₂²×D₆₈₈	2
#19145082	2
#19136860	2
#19136857	2
#19136843	2
#19136191	2
C₂×C₄₃×#508	2
#33720	43