Subgroups — #25250885

Order 11008.

806 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	C₂×#14583335	C₂	1	2	yes	maximal, complemented
2	C₄×#13063347	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136632	C₂	1	4	yes	maximal, complemented
4	#19136627	C₂	1	4	yes	maximal, complemented
5	#19136351	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136311	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	1	yes	characteristic
9	C₄×#10621898	—	2	1	no	—
10	#14583335	—	2	8	no	—
11	C₂×#13063269	C₂²	1	4	yes	complemented
12	C₂×#13063266	C₂²	1	4	yes	complemented
13	C₂×#13063265	C₂²	1	2	yes	—
14	C₄×Dic₁₇₂	C₂²	1	2	yes	complemented
15	C₄×#10621873	—	2	2	no	—
16	C₄×D₃₄₄	C₂²	1	2	yes	complemented
17	#14582937	C₂²	1	1	yes	characteristic
18	#14582896	—	2	1	no	—
19	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	4	yes	complemented
20	#14582700	C₂²	1	8	yes	complemented
21	C₂²×D₈×C₄₃	—	2	1	no	—
22	C₂²×D₈×C₄₃	C₂²	1	2	yes	—
23	C₂×#127×C₄₃	—	2	1	no	—
24	C₄₃×#493	—	2	2	no	—
25	C₄₃×#493	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	—	2	1	no	—
28	#13063347	—	2	2	no	—
29	#13063347	D₄	1	4	yes	complemented
30	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
31	C₂×#10621873	—	2	1	no	—
32	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
33	C₄×Dic₈₆	—	2	2	no	—
34	C₄×D₁₇₂	—	2	2	no	—
35	C₄×C₂²⋊D₄₃	—	4	2	no	—
36	#13063284	—	2	1	no	—
37	#13063282	—	4	1	no	—
38	C₄²×D₄₃	—	4	1	no	—
39	#13063269	—	2	8	no	—
40	#13063266	—	2	8	no	—
41	#13063265	—	2	4	no	—
42	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
43	#13063259	C₂³	1	2	yes	—
44	#13063257	—	2	1	no	—
45	#13063250	—	2	2	no	—
46	#13063247	—	2	2	no	—
47	C₂²×D₄×C₄₃	—	2	2	no	—
48	C₂²×D₄×C₄₃	—	4	2	no	—
49	C₂×D₈×C₄₃	—	2	4	no	—
50	C₂×D₈×C₄₃	—	2	4	no	—
51	C₂×D₈×C₄₃	—	2	2	no	—
52	C₂×D₈×C₄₃	—	2	4	no	—
53	C₂×D₈×C₄₃	—	4	4	no	—
54	C₂×D₈×C₄₃	C₂³	1	4	yes	—
55	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	#127×C₄₃	—	2	4	no	—
58	#127×C₄₃	—	4	1	no	—
59	#127×C₄₃	—	4	1	no	—
60	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₄₃	D₄	1	2	yes	—
62	C₄×C₈×C₄₃	C₂³	1	2	yes	—
63	C₂⁴×C₄₃	—	4	2	no	—
64	C₂×D₄×C₄₃	—	2	4	no	—
65	C₂×D₄×C₄₃	—	4	4	no	—
66	C₂×D₄×C₄₃	—	4	4	no	—
67	C₂×D₄×C₄₃	—	4	2	no	—
68	C₂×D₄×C₄₃	—	4	4	no	—
69	C₂×D₄×C₄₃	—	4	4	no	—
70	C₂×D₄×C₄₃	—	8	8	no	—
71	C₂²×C₄×C₄₃	D₈	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₃	—	4	1	no	—
74	#10621898	—	2	4	no	—
75	#10621898	—	4	2	no	—
76	C₂×D₁₇₂	—	2	1	no	—
77	C₂×C₄×D₄₃	—	4	1	no	—
78	C₂×C₄×D₄₃	—	4	2	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	D₈×C₄₃	—	2	4	no	—
81	D₈×C₄₃	—	2	4	no	—
82	D₈×C₄₃	—	4	8	no	—
83	D₈×C₄₃	—	4	4	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	2	yes	—
89	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₄₃	D₈	1	1	yes	characteristic
91	C₄²×C₄₃	D₈	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	2	yes	—
93	C₂²⋊Dic₄₃	—	4	1	no	—
94	(C₂×D₄₃)⋊C₄	—	4	2	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	Dic₄₃⋊C₄	—	4	2	no	—
97	C₄×Dic₄₃	—	4	1	no	—
98	C₄×Dic₄₃	—	4	2	no	—
99	Dic₁₇₂	—	2	1	no	—
100	Dic₁₇₂	C₂×D₄	1	2	yes	complemented
101	D₃₄₄	—	2	1	no	—
102	D₃₄₄	C₂×D₄	1	2	yes	complemented
103	#10621873	—	2	4	no	—
104	C₄₃⋊C₁₆	—	2	4	no	—
105	C₂³×C₄₃	—	4	4	no	—
106	C₂³×C₄₃	—	8	8	no	—
107	C₂³×C₄₃	—	8	4	no	—
108	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
109	C₂²×D₄₃	—	4	1	no	—
110	D₄×C₄₃	—	4	4	no	—
111	D₄×C₄₃	—	4	4	no	—
112	D₄×C₄₃	—	8	8	no	—
113	D₄×C₄₃	—	8	4	no	—
114	D₄×C₄₃	—	8	8	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	#121	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²⋊D₄₃	—	4	4	no	—
123	C₂×Dic₄₃	—	4	1	no	—
124	C₂×Dic₄₃	—	4	2	no	—
125	D₁₇₂	—	2	2	no	—
126	D₁₇₂	—	4	1	no	—
127	C₄×D₄₃	—	4	4	no	—
128	C₄×D₄₃	—	8	4	no	—
129	Dic₈₆	—	2	2	no	—
130	Dic₈₆	—	4	1	no	—
131	C₈×C₄₃	—	2	2	no	—
132	C₈×C₄₃	#121	1	2	yes	—
133	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
135	#33582	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	8	8	no	—
138	C₂²×C₄₃	—	8	4	no	—
139	C₂²×C₄₃	—	16	8	no	—
140	C₂²×C₄₃	#459	1	2	yes	—
141	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₄₃	—	4	2	no	—
143	C₂×D₄₃	—	8	2	no	—
144	C₄×C₄₃	—	2	2	no	—
145	C₄×C₄₃	#459	1	2	yes	—
146	C₄×C₄₃	#509	1	4	yes	complemented
147	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
148	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	Dic₄₃	—	4	2	no	—
150	Dic₄₃	—	8	2	no	—
151	C₂×#509	—	43	2	no	—
152	C₂×#493	D₄₃	1	1	yes	characteristic, complemented
153	C₄×#135	—	43	1	no	—
154	#2207	—	43	4	no	—
155	#2206	—	43	4	no	—
156	#2150	—	43	1	no	—
157	#2140	—	43	2	no	—
158	C₂×C₄₃	—	2	2	no	—
159	C₂×C₄₃	—	16	4	no	—
160	C₂×C₄₃	C₂×#459	1	1	yes	characteristic
161	C₂×C₄₃	C₂×#509	1	2	yes	—
162	D₄₃	—	8	2	no	—
163	C₂×#135	—	43	1	no	—
164	C₂²×D₈	—	2	1	no	—
165	C₂²×D₈	C₂×D₄₃	1	2	yes	—
166	C₂×#127	—	2	1	no	—
167	C₄×#41	—	86	1	no	—
168	#509	—	86	8	no	—
169	C₂×SD₃₂	—	43	4	no	—
170	C₂×D₁₆	—	43	4	no	—
171	C₂×M₃₂	—	43	2	no	—
172	#493	—	2	2	no	—
173	#493	C₂×D₄₃	1	4	yes	—
174	C₄×Q₁₆	—	43	2	no	—
175	C₄×SD₁₆	—	86	2	no	—
176	C₄×D₈	—	43	2	no	—
177	#427	—	43	1	no	—
178	#416	—	86	1	no	—
179	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
180	C₄⋊C₁₆	—	43	4	no	—
181	#357	—	43	8	no	—
182	C₄₃	#33582	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	86	1	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	4	2	no	—
186	#135	—	43	4	no	—
187	#135	—	86	2	no	—
188	C₂×Q₁₆	—	43	1	no	—
189	C₂×SD₁₆	—	86	1	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	—	2	4	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	4	4	no	—
195	C₂×D₈	—	43	1	no	—
196	C₂×D₈	C₂²×D₄₃	1	4	yes	—
197	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
199	#127	—	2	4	no	—
200	#127	—	4	1	no	—
201	#127	—	4	1	no	—
202	C₄×Q₈	—	86	2	no	—
203	C₄×D₄	—	86	2	no	—
204	C₄×D₄	—	172	2	no	—
205	#117	—	86	1	no	—
206	#117	—	172	1	no	—
207	C₂×C₄²	—	172	1	no	—
208	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
209	SD₃₂	—	86	8	no	—
210	D₁₆	—	86	8	no	—
211	M₃₂	—	86	4	no	—
212	C₂×C₁₆	—	43	4	no	—
213	#107	—	43	2	no	—
214	#106	—	86	1	no	—
215	#103	—	86	2	no	—
216	#102	—	86	2	no	—
217	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
218	C₄×C₈	C₂²×D₄₃	1	2	yes	—
219	C₂⁴	—	4	2	no	—
220	#41	—	86	4	no	—
221	#41	—	172	2	no	—
222	C₂×Q₈	—	86	1	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	8	8	no	—
230	C₂×D₄	—	86	1	no	—
231	C₂×D₄	—	172	1	no	—
232	C₂²×C₄	—	172	2	no	—
233	C₂²×C₄	—	172	1	no	—
234	C₂²×C₄	#10621882	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
236	Q₁₆	—	43	2	no	—
237	Q₁₆	—	86	1	no	—
238	SD₁₆	—	86	4	no	—
239	D₈	—	2	4	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	8	no	—
242	D₈	—	4	4	no	—
243	D₈	—	43	2	no	—
244	D₈	—	86	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	D₄×D₄₃	1	2	yes	—
247	C₂×C₈	#10621900	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	172	2	no	—
253	C₂²⋊C₄	—	172	1	no	—
254	C₂²⋊C₄	—	172	2	no	—
255	C₄²	—	172	1	no	—
256	C₄²	—	172	2	no	—
257	C₄²	#10621882	1	1	yes	characteristic
258	C₄²	#10621882	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
260	C₁₆	—	86	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	8	8	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	172	1	no	—
265	C₂³	C₂×#10621882	1	1	yes	characteristic
266	Q₈	—	86	2	no	—
267	Q₈	—	172	1	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	8	no	—
272	D₄	—	8	4	no	—
273	D₄	—	86	2	no	—
274	D₄	—	172	4	no	—
275	D₄	—	172	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	344	4	no	—
278	C₂×C₄	—	172	2	no	—
279	C₂×C₄	—	172	1	no	—
280	C₂×C₄	—	172	4	no	—
281	C₂×C₄	#13063296	1	2	yes	—
282	C₂×C₄	C₂×#10621882	1	2	yes	—
283	C₂×C₄	C₂×#10621882	1	2	yes	—
284	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
285	C₂×C₄	#13063351	1	2	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	#13063296	1	2	yes	—
289	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
290	C₈	C₂×#10621900	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	8	no	—
294	C₂²	—	16	8	no	—
295	C₂²	—	344	2	no	—
296	C₂²	—	172	2	no	—
297	C₂²	#14583091	1	2	yes	—
298	C₂²	C₂²×#10621882	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	344	2	no	—
301	C₄	—	172	2	no	—
302	C₄	#14583091	1	2	yes	—
303	C₄	#14583335	1	4	yes	complemented
304	C₄	C₂×#13063296	1	1	yes	characteristic
305	C₄	C₂×#13063351	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	—	344	2	no	—
309	C₂	C₂×#14583091	1	1	yes	characteristic
310	C₂	C₂×#14583335	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#493	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₂×#493	D₄₃
hypocenter	C₄₃	#33582
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#14583335	C₂	semidirect
2. C₄×#13063347	C₂	semidirect
3. #19136632	C₂	semidirect
4. #19136627	C₂	semidirect
5. #19136351	C₂	semidirect
6. #19136311	C₂	semidirect
7. C₂×C₄₃×#493	C₂	semidirect
11. C₂×#13063269	C₂²	semidirect
12. C₂×#13063266	C₂²	semidirect
14. C₄×Dic₁₇₂	C₂²	semidirect
16. C₄×D₃₄₄	C₂²	semidirect
19. C₄⋊(C₄₃⋊C₁₆)	C₂²	semidirect
20. #14582700	C₂²	semidirect
29. #13063347	D₄	semidirect
100. Dic₁₇₂	C₂×D₄	semidirect
102. D₃₄₄	C₂×D₄	semidirect
146. C₄×C₄₃	#509	semidirect
152. C₂×#493	D₄₃	semidirect
182. C₄₃	#33582	semidirect
303. C₄	#14583335	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583335	2
C₄×#13063347	2
#19136632	2
#19136627	2
#19136351	2
#19136311	2
C₂×C₄₃×#493	2
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