Subgroups — #25097026

Order 11008.

1190 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582941	C₂	1	4	yes	maximal
2	#19137002	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582769	C₂	1	1	yes	maximal, characteristic
4	#19134763	C₂	1	4	yes	maximal, complemented
5	#19134755	C₂	1	2	yes	maximal, complemented
6	#19134752	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063284	C₂²	1	2	yes	—
9	C₂²×C₄⋊Dic₄₃	—	2	2	no	—
10	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
11	#14582941	—	2	16	no	—
12	C₂×#13063259	C₂²	1	8	yes	—
13	C₂×#13063257	C₂²	1	8	yes	—
14	#14582769	—	2	2	no	—
15	#14582769	C₂²	1	1	yes	characteristic
16	#14582769	C₂²	1	2	yes	—
17	#14582769	C₂²	1	1	yes	characteristic
18	#14582769	C₂²	1	2	yes	—
19	#14582765	C₂²	1	2	yes	—
20	#14582759	—	2	2	no	—
21	C₄×C₄⋊Dic₄₃	—	2	2	no	—
22	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₄₃	C₄	1	8	yes	complemented
25	C₂×#97×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
27	C₂³×Dic₄₃	—	2	2	no	—
28	#13063284	—	2	8	no	—
29	C₂×C₄⋊Dic₄₃	—	2	8	no	—
30	C₂×C₄⋊Dic₄₃	—	2	4	no	—
31	C₂×C₄⋊Dic₄₃	—	2	4	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	—	4	8	no	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
40	C₂×C₄×Dic₄₃	—	2	2	no	—
41	#13063259	—	2	16	no	—
42	#13063257	—	2	16	no	—
43	C₄₃⋊#95	—	2	4	no	—
44	C₂³×C₄×C₄₃	D₄	1	2	yes	—
45	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
47	C₂×M₁₆×C₄₃	D₄	1	8	yes	—
48	C₂×M₁₆×C₄₃	Q₈	1	8	yes	—
49	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#97×C₄₃	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₄₃	C₂×C₄	1	8	yes	complemented
57	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	2	no	—
70	C₂²×Dic₄₃	—	2	2	no	—
71	C₂²×Dic₄₃	—	2	4	no	—
72	C₂²×Dic₄₃	—	4	8	no	—
73	C₂²×Dic₄₃	—	4	4	no	—
74	M₁₆×C₄₃	C₄⋊C₄	1	32	yes	complemented
75	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₄₃	C₂×D₄	1	8	yes	—
77	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
78	C₄²×C₄₃	C₄⋊C₄	1	8	yes	—
79	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
83	C₂²⋊Dic₄₃	—	4	4	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	—	2	8	no	—
86	C₄⋊Dic₄₃	—	2	4	no	—
87	C₄⋊Dic₄₃	—	4	8	no	—
88	C₄⋊Dic₄₃	—	4	4	no	—
89	C₄⋊Dic₄₃	—	4	4	no	—
90	C₄⋊Dic₄₃	—	4	8	no	—
91	C₄⋊Dic₄₃	—	4	4	no	—
92	C₄×Dic₄₃	—	4	4	no	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	—	2	2	no	—
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₄₃	#127	1	2	yes	—
97	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
98	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	16	no	—
102	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	8	yes	—
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₄₃	#127	1	2	yes	—
107	C₂×C₄×C₄₃	C₄⋊Q₈	1	8	yes	—
108	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
110	C₂×C₄×C₄₃	#137	1	4	yes	—
111	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×Dic₄₃	—	4	2	no	—
116	C₂×Dic₄₃	—	4	2	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	8	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	8	8	no	—
122	C₈×C₄₃	C₂×C₄⋊C₄	1	16	yes	complemented
123	#20800	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	#384	1	4	yes	—
129	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
131	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
133	C₂²×C₄₃	C₂×#137	1	2	yes	—
134	C₄×C₄₃	—	2	4	no	—
135	C₄×C₄₃	#384	1	4	yes	—
136	C₄×C₄₃	#428	1	8	yes	complemented
137	C₄×C₄₃	C₂²×C₄⋊C₄	1	2	yes	—
138	C₄×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
139	Dic₄₃	—	8	2	no	—
140	Dic₄₃	—	8	4	no	—
141	Dic₄₃	—	8	2	no	—
142	C₂×#428	—	43	4	no	—
143	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
144	#2285	—	43	1	no	—
145	C₂×#384	—	43	1	no	—
146	#1847	—	43	4	no	—
147	#1845	—	43	2	no	—
148	#1844	—	43	2	no	—
149	C₂×C₄₃	—	2	4	no	—
150	C₂×C₄₃	C₂×#384	1	1	yes	characteristic
151	C₂×C₄₃	C₂×#428	1	2	yes	—
152	C₂×C₄₃	#3151	1	1	yes	characteristic
153	C₂×C₄₃	#3152	1	1	yes	characteristic
154	C₂×C₄₃	#3165	1	2	yes	—
155	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
156	C₂×#117	—	43	2	no	—
157	C₂²×C₄⋊C₄	—	43	2	no	—
158	C₂²×C₄⋊C₄	—	86	2	no	—
159	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
160	#428	—	86	16	no	—
161	C₂×#107	—	43	8	no	—
162	C₂×#106	—	43	8	no	—
163	C₄×M₁₆	Dic₄₃	1	8	yes	complemented
164	C₂×#97	C₂×D₄₃	1	2	yes	—
165	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
166	#384	—	43	1	no	—
167	#384	—	43	2	no	—
168	#384	—	43	1	no	—
169	#384	—	43	2	no	—
170	#384	—	86	2	no	—
171	#382	—	43	2	no	—
172	#380	—	86	2	no	—
173	C₄×C₄⋊C₄	—	86	2	no	—
174	C₄₃	#20800	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	86	2	no	—
176	C₂³×C₄	D₁₇₂	1	2	yes	—
177	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×M₁₆	Dic₈₆	1	8	yes	—
179	C₂×M₁₆	D₁₇₂	1	8	yes	—
180	C₂×M₁₆	C₂×Dic₄₃	1	8	yes	—
181	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
182	#117	—	86	8	no	—
183	C₂×C₄⋊C₄	—	43	4	no	—
184	C₂×C₄⋊C₄	—	43	2	no	—
185	C₂×C₄⋊C₄	—	43	2	no	—
186	C₂×C₄⋊C₄	—	86	8	no	—
187	C₂×C₄⋊C₄	—	86	2	no	—
188	C₂×C₄⋊C₄	—	86	4	no	—
189	C₂×C₄⋊C₄	—	86	4	no	—
190	C₂×C₄⋊C₄	—	86	2	no	—
191	C₂×C₄⋊C₄	—	86	2	no	—
192	C₂×C₄⋊C₄	—	172	8	no	—
193	C₂×C₂²⋊C₄	—	86	2	no	—
194	C₂×C₄²	—	86	2	no	—
195	C₂×C₄²	Dic₈₆	1	4	yes	—
196	C₂×C₄²	D₁₇₂	1	2	yes	—
197	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
198	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
200	#107	—	86	16	no	—
201	#106	—	86	16	no	—
202	#97	C₂×Dic₄₃	1	8	yes	complemented
203	C₄×C₈	C₂×Dic₄₃	1	8	yes	complemented
204	#95	—	86	4	no	—
205	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	86	2	no	—
209	C₂²×C₄	—	86	2	no	—
210	C₂²×C₄	—	86	4	no	—
211	C₂²×C₄	—	172	8	no	—
212	C₂²×C₄	—	172	4	no	—
213	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
214	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
215	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
216	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
217	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
219	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
221	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
222	M₁₆	C₄⋊Dic₄₃	1	32	yes	complemented
223	C₂×C₈	C₂×Dic₈₆	1	8	yes	—
224	C₂×C₈	C₂×D₁₇₂	1	8	yes	—
225	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
226	C₄⋊C₄	—	86	4	no	—
227	C₄⋊C₄	—	86	8	no	—
228	C₄⋊C₄	—	86	4	no	—
229	C₄⋊C₄	—	172	8	no	—
230	C₄⋊C₄	—	172	4	no	—
231	C₄⋊C₄	—	172	8	no	—
232	C₄⋊C₄	—	172	4	no	—
233	C₄⋊C₄	—	172	4	no	—
234	C₂²⋊C₄	—	172	4	no	—
235	C₄²	—	172	4	no	—
236	C₄²	C₄⋊Dic₄₃	1	8	yes	—
237	C₄²	C₂×Dic₈₆	1	2	yes	—
238	C₄²	C₂×Dic₈₆	1	2	yes	—
239	C₄²	C₂²×Dic₄₃	1	2	yes	—
240	C₄²	C₂²×Dic₄₃	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
244	C₂³	#13063322	1	2	yes	—
245	C₂³	#13063325	1	2	yes	—
246	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	344	8	no	—
251	C₂×C₄	—	172	8	no	—
252	C₂×C₄	—	172	4	no	—
253	C₂×C₄	—	172	4	no	—
254	C₂×C₄	—	172	2	no	—
255	C₂×C₄	—	172	2	no	—
256	C₂×C₄	—	172	4	no	—
257	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₄₃	1	8	yes	—
259	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
261	C₂×C₄	#13063322	1	2	yes	—
262	C₂×C₄	#13063325	1	8	yes	—
263	C₂×C₄	#13063325	1	2	yes	—
264	C₂×C₄	#13063353	1	4	yes	—
265	C₂×C₄	#13063358	1	4	yes	—
266	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
268	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
269	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
270	C₈	C₂×C₄⋊Dic₄₃	1	16	yes	complemented
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	#14582769	1	4	yes	—
276	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
277	C₂²	C₂×#13063322	1	1	yes	characteristic
278	C₂²	C₂×#13063325	1	1	yes	characteristic
279	C₂²	C₂×#13063353	1	2	yes	—
280	C₂²	C₂×#13063358	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	344	4	no	—
283	C₄	—	344	2	no	—
284	C₄	—	344	2	no	—
285	C₄	#14582769	1	4	yes	—
286	C₄	#14582941	1	8	yes	complemented
287	C₄	C₂²×C₄⋊Dic₄₃	1	2	yes	—
288	C₄	C₂×#13063325	1	2	yes	—
289	C₂	—	2	4	no	—
290	C₂	C₂×#14582769	1	1	yes	characteristic
291	C₂	C₂×#14582941	1	2	yes	—
292	C₂	#19145047	1	1	yes	characteristic
293	C₂	#19145054	1	1	yes	characteristic
294	C₂	#19145141	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	#20800
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #19134763	C₂	semidirect
5. #19134755	C₂	semidirect
6. #19134752	C₂	semidirect
24. C₄×M₁₆×C₄₃	C₄	semidirect
55. #97×C₄₃	C₂×C₄	semidirect
56. C₄×C₈×C₄₃	C₂×C₄	semidirect
74. M₁₆×C₄₃	C₄⋊C₄	semidirect
122. C₈×C₄₃	C₂×C₄⋊C₄	semidirect
136. C₄×C₄₃	#428	semidirect
163. C₄×M₁₆	Dic₄₃	semidirect
174. C₄₃	#20800	semidirect
202. #97	C₂×Dic₄₃	semidirect
203. C₄×C₈	C₂×Dic₄₃	semidirect
222. M₁₆	C₄⋊Dic₄₃	semidirect
270. C₈	C₂×C₄⋊Dic₄₃	semidirect
286. C₄	#14582941	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582941	2
#19137002	2
C₂×#14582769	2
#19134763	2
#19134755	2
#19134752	2
C₂×C₄×M₁₆×C₄₃	2
#20800	43