Subgroups — #25095109

Order 11008.

1206 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583012	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
3	#19136957	C₂	1	2	yes	maximal, complemented
4	#19134492	C₂	1	2	yes	maximal, complemented
5	C₄×#13063257	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134421	C₂	1	2	yes	maximal, complemented
7	C₄×#13063247	C₂	1	4	yes	maximal, complemented
8	C₄₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₂×C₄×D₁₇₂	C₂²	1	4	yes	—
12	#14583012	C₄	1	16	yes	complemented
13	C₄×D₃₄₄	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063257	C₂²	1	2	yes	—
16	C₂×#13063247	C₂²	1	2	yes	—
17	C₂×#13063247	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
20	#14582746	—	2	2	no	—
21	#14582746	—	2	4	no	—
22	#95×D₄₃	—	2	2	no	—
23	#14582662	C₂²	1	2	yes	—
24	#14582662	C₂²	1	4	yes	—
25	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#97×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#376	C₂²	1	1	yes	characteristic
30	C₂²×D₁₇₂	C₂³	1	2	yes	—
31	C₂²×C₄×D₄₃	—	2	2	no	—
32	C₂²×C₄×D₄₃	—	2	4	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₂×D₃₄₄	C₂×C₄	1	8	yes	—
35	C₄×D₁₇₂	—	2	8	no	—
36	C₄×D₁₇₂	—	2	4	no	—
37	C₄×D₁₇₂	C₂×C₄	1	8	yes	complemented
38	C₄×D₁₇₂	C₂×C₄	1	16	yes	complemented
39	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
42	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
43	C₂×C₄×Dic₄₃	—	2	4	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	#13063259	C₂×C₄	1	4	yes	complemented
46	#13063257	C₂×C₄	1	8	yes	complemented
47	#13063247	C₂×C₄	1	8	yes	complemented
48	#13063247	C₂×C₄	1	16	yes	complemented
49	C₄₃⋊#95	—	2	2	no	—
50	C₄₃⋊#95	—	2	4	no	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
56	#97×C₄₃	—	2	2	no	—
57	#97×C₄₃	C₂×C₄	1	8	yes	complemented
58	C₄×C₈×C₄₃	—	2	4	no	—
59	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
60	#95×C₄₃	D₄	1	2	yes	—
61	#95×C₄₃	D₄	1	2	yes	—
62	C₂³×D₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	4	no	—
69	C₂×D₁₇₂	—	2	4	no	—
70	C₂×D₁₇₂	—	2	2	no	—
71	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
72	C₂×D₁₇₂	C₂²×C₄	1	8	yes	—
73	C₂×C₄×D₄₃	—	2	16	no	—
74	C₂×C₄×D₄₃	—	2	8	no	—
75	C₂×C₄×D₄₃	—	8	4	no	—
76	C₂×C₄×D₄₃	—	8	4	no	—
77	C₂×C₄×D₄₃	—	8	4	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
85	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
87	C₄²×C₄₃	#41	1	4	yes	—
88	C₄²×C₄₃	#41	1	2	yes	—
89	(C₂×D₄₃)⋊C₄	—	2	8	no	—
90	(C₂×D₄₃)⋊C₄	—	2	16	no	—
91	C₄⋊Dic₄₃	—	2	4	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
94	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
95	C₄×Dic₄₃	—	8	4	no	—
96	C₄×Dic₄₃	—	8	2	no	—
97	D₃₄₄	C₄²	1	16	yes	complemented
98	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₂²×D₄₃	—	2	8	no	—
100	C₂²×D₄₃	—	2	4	no	—
101	C₂²×D₄₃	—	4	4	no	—
102	C₂²×D₄₃	—	4	2	no	—
103	C₂²×D₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₄×D₄	1	8	yes	—
110	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₄₃	—	2	8	no	—
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	D₁₇₂	—	2	4	no	—
121	D₁₇₂	—	2	8	no	—
122	D₁₇₂	C₂×C₄²	1	8	yes	—
123	C₄×D₄₃	—	8	16	no	—
124	C₄×D₄₃	—	8	8	no	—
125	C₈×C₄₃	—	2	2	no	—
126	C₈×C₄₃	—	2	4	no	—
127	C₈×C₄₃	C₂×C₄²	1	4	yes	—
128	#20503	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	C₂×C₄×D₄	1	2	yes	—
130	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₃	#576	1	2	yes	—
132	C₂²×C₄₃	#577	1	2	yes	—
133	C₂×D₄₃	—	2	8	no	—
134	C₂×D₄₃	—	4	8	no	—
135	C₂×D₄₃	—	4	8	no	—
136	C₂×D₄₃	—	4	4	no	—
137	C₂×D₄₃	—	4	4	no	—
138	C₄×C₄₃	—	2	8	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
143	C₄×C₄₃	C₄×#41	1	2	yes	—
144	Dic₄₃	—	4	8	no	—
145	Dic₄₃	—	4	4	no	—
146	C₂×#442	—	43	2	no	—
147	C₂×C₄×D₈	—	43	1	no	—
148	#2275	—	43	2	no	—
149	#1776	—	43	2	no	—
150	C₄×#106	—	43	1	no	—
151	#1762	—	43	2	no	—
152	C₄×#102	—	43	4	no	—
153	#1726	D₄₃	1	1	yes	characteristic, complemented
154	C₂×C₄₃	C₄²×D₄	1	1	yes	characteristic
155	C₂×C₄₃	#2946	1	4	yes	—
156	C₂×C₄₃	#2948	1	1	yes	characteristic
157	C₂×C₄₃	#2952	1	1	yes	characteristic
158	D₄₃	—	4	8	no	—
159	C₂²×D₈	—	43	1	no	—
160	C₂×C₄×D₄	—	43	2	no	—
161	C₂×C₄×D₄	—	43	4	no	—
162	#442	—	43	16	no	—
163	C₄×D₈	—	43	8	no	—
164	C₂×#107	—	43	1	no	—
165	C₂×#106	—	43	2	no	—
166	C₂×#102	—	43	2	no	—
167	C₂×#102	—	43	4	no	—
168	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
169	C₂×#97	C₂×D₄₃	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
171	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	43	4	no	—
173	C₄×C₄⋊C₄	—	43	2	no	—
174	C₄×C₂²⋊C₄	—	86	2	no	—
175	C₄×C₂²⋊C₄	—	86	4	no	—
176	#376	C₂×D₄₃	1	1	yes	characteristic
177	C₂×#95	—	86	2	no	—
178	#340	—	43	2	no	—
179	#340	—	43	4	no	—
180	C₄₃	#20503	1	1	yes	characteristic, complemented, Sylow
181	C₂²×D₄	—	43	2	no	—
182	C₂³×C₄	—	86	2	no	—
183	C₂³×C₄	—	86	4	no	—
184	C₂×D₈	—	43	4	no	—
185	C₂×D₈	—	43	8	no	—
186	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₄×D₄	—	43	8	no	—
190	C₄×D₄	—	43	16	no	—
191	C₄×D₄	—	86	8	no	—
192	C₄×D₄	—	86	4	no	—
193	C₂×C₄⋊C₄	—	43	4	no	—
194	C₂×C₄⋊C₄	—	43	2	no	—
195	C₂×C₂²⋊C₄	—	86	4	no	—
196	C₂×C₂²⋊C₄	—	86	2	no	—
197	C₂×C₄²	—	86	4	no	—
198	C₂×C₄²	—	86	2	no	—
199	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
201	#107	—	43	4	no	—
202	#106	—	43	8	no	—
203	#102	—	43	8	no	—
204	#102	—	43	16	no	—
205	#97	—	2	2	no	—
206	#97	C₄×D₄₃	1	8	yes	complemented
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
209	#95	—	86	2	no	—
210	#95	—	86	4	no	—
211	#95	D₁₇₂	1	2	yes	—
212	#95	D₁₇₂	1	2	yes	—
213	C₂⁴	—	86	2	no	—
214	C₂×D₄	—	43	4	no	—
215	C₂×D₄	—	43	8	no	—
216	C₂×D₄	—	86	4	no	—
217	C₂×D₄	—	86	2	no	—
218	C₂²×C₄	—	86	2	no	—
219	C₂²×C₄	—	86	4	no	—
220	C₂²×C₄	—	86	16	no	—
221	C₂²×C₄	—	86	8	no	—
222	C₂²×C₄	—	344	4	no	—
223	C₂²×C₄	—	344	4	no	—
224	C₂²×C₄	—	344	4	no	—
225	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
227	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	D₈	—	43	16	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
235	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
237	C₄⋊C₄	—	43	8	no	—
238	C₄⋊C₄	—	43	4	no	—
239	C₄⋊C₄	—	86	4	no	—
240	C₄⋊C₄	—	86	2	no	—
241	C₂²⋊C₄	—	86	8	no	—
242	C₂²⋊C₄	—	86	16	no	—
243	C₄²	—	344	4	no	—
244	C₄²	—	344	2	no	—
245	C₄²	C₂×C₄×D₄₃	1	2	yes	—
246	C₄²	C₂×C₄×D₄₃	1	4	yes	—
247	C₄²	#10621898	1	4	yes	—
248	C₄²	#10621898	1	2	yes	—
249	C₂³	—	86	8	no	—
250	C₂³	—	86	4	no	—
251	C₂³	—	172	4	no	—
252	C₂³	—	172	2	no	—
253	C₂³	—	172	2	no	—
254	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
255	D₄	—	43	8	no	—
256	D₄	—	86	4	no	—
257	D₄	—	86	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	86	8	no	—
263	C₂×C₄	—	86	4	no	—
264	C₂×C₄	—	344	16	no	—
265	C₂×C₄	—	344	8	no	—
266	C₂×C₄	—	172	4	no	—
267	C₂×C₄	—	172	4	no	—
268	C₂×C₄	—	172	4	no	—
269	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
270	C₂×C₄	C₄×D₁₇₂	1	8	yes	—
271	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
274	C₂×C₄	C₂×#10621898	1	2	yes	—
275	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	2	4	no	—
278	C₈	C₄²×D₄₃	1	4	yes	—
279	C₂²	—	86	8	no	—
280	C₂²	—	172	8	no	—
281	C₂²	—	172	8	no	—
282	C₂²	—	172	4	no	—
283	C₂²	—	172	4	no	—
284	C₂²	C₂×C₄×D₁₇₂	1	2	yes	—
285	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
286	C₂²	#14583704	1	2	yes	—
287	C₂²	#14583711	1	2	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	172	8	no	—
292	C₄	—	172	4	no	—
293	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
294	C₄	C₄×#10621898	1	1	yes	characteristic
295	C₄	C₄×#10621898	1	2	yes	—
296	C₂	—	172	8	no	—
297	C₂	C₄²×D₁₇₂	1	1	yes	characteristic
298	C₂	#19143429	1	4	yes	—
299	C₂	#19143448	1	1	yes	characteristic
300	C₂	#19143480	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1726	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1726	D₄₃
hypocenter	C₄₃	#20503
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
3. #19136957	C₂	semidirect
4. #19134492	C₂	semidirect
5. C₄×#13063257	C₂	semidirect
6. #19134421	C₂	semidirect
7. C₄×#13063247	C₂	semidirect
8. C₄₃×#1726	C₂	semidirect
12. #14583012	C₄	semidirect
13. C₄×D₃₄₄	C₄	semidirect
37. C₄×D₁₇₂	C₂×C₄	semidirect
38. C₄×D₁₇₂	C₂×C₄	semidirect
45. #13063259	C₂×C₄	semidirect
46. #13063257	C₂×C₄	semidirect
47. #13063247	C₂×C₄	semidirect
48. #13063247	C₂×C₄	semidirect
57. #97×C₄₃	C₂×C₄	semidirect
59. C₄×C₈×C₄₃	C₂×C₄	semidirect
97. D₃₄₄	C₄²	semidirect
153. #1726	D₄₃	semidirect
180. C₄₃	#20503	semidirect
206. #97	C₄×D₄₃	semidirect
208. C₄×C₈	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583012	2
C₂×C₄×D₃₄₄	2
#19136957	2
#19134492	2
C₄×#13063257	2
#19134421	2
C₄×#13063247	2
C₄₃×#1726	2
#20503	43