Subgroups — #25125719

Order 11008.

766 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143185	C₂	1	2	yes	maximal
2	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582925	C₂	1	1	yes	maximal, characteristic
4	#19136107	C₂	1	4	yes	maximal, complemented
5	#19136102	C₂	1	4	yes	maximal, complemented
6	#19134333	C₂	1	2	yes	maximal
7	C₂×C₄₃×#341	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10621877	C₂²	1	1	yes	characteristic
9	C₂×#13063284	C₂²	1	1	yes	characteristic
10	C₂×#13063284	C₂²	1	2	yes	—
11	#14582941	—	2	4	no	—
12	#14582941	C₂²	1	8	yes	—
13	#14582937	—	2	4	no	—
14	#14582937	—	2	2	no	—
15	#14582937	—	2	2	no	—
16	C₂×#13063259	C₂²	1	2	yes	—
17	C₂×#13063257	C₂²	1	2	yes	—
18	#14582925	—	2	2	no	—
19	#14582925	C₂²	1	2	yes	—
20	#14582925	C₂²	1	2	yes	—
21	C₂×Dic₄₃⋊C₈	—	2	2	no	—
22	#14582667	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
24	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#341	C₂²	1	2	yes	—
26	C₄₃×#341	C₂²	1	2	yes	—
27	C₂×#10621877	—	2	1	no	—
28	C₂×#10621877	—	2	4	no	—
29	C₂×#10621877	C₂³	1	2	yes	—
30	C₂²×C₄₃⋊C₈	—	2	1	no	—
31	#13063284	—	2	1	no	—
32	#13063284	—	2	1	no	—
33	#13063284	—	2	2	no	—
34	#13063284	—	2	2	no	—
35	#13063284	—	2	2	no	—
36	#13063284	—	2	2	no	—
37	#13063284	D₄	1	2	yes	—
38	#13063284	C₂³	1	2	yes	—
39	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
41	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
42	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
43	C₂×C₄×Dic₄₃	—	2	1	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	#13063259	—	2	4	no	—
46	#13063259	—	2	8	no	—
47	#13063257	—	2	4	no	—
48	#13063257	—	2	8	no	—
49	Dic₄₃⋊C₈	—	2	4	no	—
50	Dic₄₃⋊C₈	—	2	4	no	—
51	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₄₃	—	2	2	no	—
53	C₂×M₁₆×C₄₃	—	2	4	no	—
54	C₂×M₁₆×C₄₃	Q₈	1	8	yes	—
55	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₈×C₄₃	—	2	2	no	—
57	C₂²×C₈×C₄₃	—	2	4	no	—
58	C₂²×C₈×C₄₃	D₄	1	2	yes	—
59	C₂²×C₈×C₄₃	D₄	1	2	yes	—
60	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂⁴×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₄₃	#41	1	4	yes	—
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	2	2	no	—
72	C₂²×Dic₄₃	—	2	4	no	—
73	M₁₆×C₄₃	—	2	8	no	—
74	M₁₆×C₄₃	—	2	4	no	—
75	M₁₆×C₄₃	C₂×Q₈	1	8	yes	—
76	C₂×C₈×C₄₃	—	2	8	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	2	4	no	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²⋊Dic₄₃	—	2	2	no	—
85	C₂²⋊Dic₄₃	—	2	2	no	—
86	C₂²⋊Dic₄₃	—	4	2	no	—
87	C₂²⋊Dic₄₃	—	4	2	no	—
88	C₄⋊Dic₄₃	—	2	2	no	—
89	C₄⋊Dic₄₃	—	2	4	no	—
90	C₄⋊Dic₄₃	—	2	4	no	—
91	C₄⋊Dic₄₃	C₂×D₄	1	4	yes	—
92	C₄×Dic₄₃	—	2	2	no	—
93	C₄×Dic₄₃	—	2	2	no	—
94	C₄×Dic₄₃	—	4	2	no	—
95	C₄×Dic₄₃	—	4	2	no	—
96	#10621877	—	2	4	no	—
97	#10621877	—	4	2	no	—
98	C₂×C₄₃⋊C₈	—	2	2	no	—
99	C₂×C₄₃⋊C₈	—	4	2	no	—
100	C₂³×C₄₃	—	4	2	no	—
101	C₂³×C₄₃	#121	1	1	yes	characteristic
102	C₂³×C₄₃	#121	1	1	yes	characteristic
103	C₂³×C₄₃	#125	1	4	yes	—
104	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₄₃	#123	1	2	yes	—
110	C₂×C₄×C₄₃	#123	1	2	yes	—
111	C₂×C₄×C₄₃	#125	1	4	yes	—
112	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
118	C₂×Dic₄₃	—	2	4	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	8	4	no	—
122	C₈×C₄₃	—	2	8	no	—
123	C₈×C₄₃	—	2	2	no	—
124	C₈×C₄₃	—	2	2	no	—
125	C₄₃⋊C₈	—	4	2	no	—
126	#23674	—	43	1	no	maximal, Sylow
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	4	4	no	—
131	C₂²×C₄₃	#398	1	2	yes	—
132	C₂²×C₄₃	#398	1	2	yes	—
133	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
134	C₂²×C₄₃	C₂×#125	1	2	yes	—
135	C₄×C₄₃	#398	1	2	yes	—
136	C₄×C₄₃	#398	1	2	yes	—
137	C₄×C₄₃	C₂×C₂²⋊Q₈	1	2	yes	—
138	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
139	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	Dic₄₃	—	4	4	no	—
141	Dic₄₃	—	8	4	no	—
142	#2911	—	43	2	no	—
143	C₂×#427	—	43	1	no	—
144	C₂×#424	—	43	1	no	—
145	#2090	—	43	4	no	—
146	#2089	—	43	4	no	—
147	#1742	—	43	2	no	—
148	C₂×#341	D₄₃	1	1	yes	characteristic
149	C₂×C₄₃	—	2	2	no	—
150	C₂×C₄₃	—	2	2	no	—
151	C₂×C₄₃	—	2	2	no	—
152	C₂×C₄₃	C₂×#398	1	1	yes	characteristic
153	C₂×C₄₃	#3067	1	2	yes	—
154	C₂²×M₁₆	—	43	1	no	—
155	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
156	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
157	C₂×#117	—	43	1	no	—
158	C₂×#117	—	43	2	no	—
159	#428	—	43	8	no	—
160	#428	—	86	4	no	—
161	#427	—	86	4	no	—
162	#427	—	86	2	no	—
163	#427	—	86	2	no	—
164	C₂×#107	—	43	2	no	—
165	C₂×#106	—	43	2	no	—
166	#424	—	43	2	no	—
167	#424	—	43	2	no	—
168	#424	—	86	2	no	—
169	C₂×C₄⋊C₈	—	86	2	no	—
170	#343	—	43	8	no	—
171	#341	C₂×D₄₃	1	2	yes	—
172	#341	C₂×D₄₃	1	2	yes	—
173	C₄₃	#23674	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	4	no	—
177	C₂×M₁₆	—	43	2	no	—
178	C₂×M₁₆	—	86	1	no	—
179	C₂×M₁₆	—	86	4	no	—
180	C₂×M₁₆	Dic₈₆	1	8	yes	—
181	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	86	1	no	—
185	C₂²×C₈	D₁₇₂	1	2	yes	—
186	C₂²×C₈	D₁₇₂	1	2	yes	—
187	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	#117	—	43	2	no	—
190	#117	—	43	2	no	—
191	#117	—	86	1	no	—
192	#117	—	86	1	no	—
193	#117	—	86	2	no	—
194	#117	—	86	2	no	—
195	#117	—	86	2	no	—
196	#117	—	86	2	no	—
197	C₂×C₄⋊C₄	—	43	2	no	—
198	C₂×C₄⋊C₄	—	43	4	no	—
199	C₂×C₂²⋊C₄	—	86	1	no	—
200	C₂×C₂²⋊C₄	—	86	2	no	—
201	C₂×C₄²	—	86	1	no	—
202	C₂×C₄²	—	86	2	no	—
203	#107	—	86	8	no	—
204	#107	—	86	4	no	—
205	#106	—	86	8	no	—
206	#106	—	86	4	no	—
207	C₄⋊C₈	—	86	4	no	—
208	C₄⋊C₈	—	86	4	no	—
209	C₂⁴	#10621902	1	1	yes	characteristic
210	C₂²×C₄	—	86	2	no	—
211	C₂²×C₄	—	86	4	no	—
212	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
213	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	#10621900	1	4	yes	—
218	C₂²×C₄	#10621902	1	1	yes	characteristic
219	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
220	M₁₆	—	2	8	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	86	4	no	—
223	M₁₆	—	172	2	no	—
224	M₁₆	C₂×Dic₈₆	1	8	yes	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	86	2	no	—
232	C₂×C₈	—	172	2	no	—
233	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
234	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
235	C₄⋊C₄	—	43	4	no	—
236	C₄⋊C₄	—	86	2	no	—
237	C₄⋊C₄	—	86	4	no	—
238	C₄⋊C₄	—	86	4	no	—
239	C₂²⋊C₄	—	86	2	no	—
240	C₂²⋊C₄	—	86	2	no	—
241	C₂²⋊C₄	—	172	2	no	—
242	C₂²⋊C₄	—	172	2	no	—
243	C₄²	—	86	2	no	—
244	C₄²	—	86	2	no	—
245	C₄²	—	172	2	no	—
246	C₄²	—	172	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	#13063300	1	1	yes	characteristic
249	C₂³	#13063300	1	1	yes	characteristic
250	C₂³	#13063316	1	4	yes	—
251	C₂³	C₂×#10621902	1	1	yes	characteristic
252	C₂×C₄	—	86	4	no	—
253	C₂×C₄	—	344	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	—	172	4	no	—
256	C₂×C₄	#13063300	1	1	yes	characteristic
257	C₂×C₄	#13063300	1	1	yes	characteristic
258	C₂×C₄	#13063302	1	4	yes	—
259	C₂×C₄	#13063302	1	4	yes	—
260	C₂×C₄	#13063309	1	2	yes	—
261	C₂×C₄	#13063309	1	2	yes	—
262	C₂×C₄	#13063316	1	4	yes	—
263	C₂×C₄	#13063325	1	2	yes	—
264	C₂×C₄	#13063325	1	2	yes	—
265	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
268	C₂×C₄	C₂×#10621900	1	2	yes	—
269	C₈	—	2	8	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	172	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	#14582831	1	2	yes	—
278	C₂²	#14582831	1	2	yes	—
279	C₂²	C₂×#13063300	1	1	yes	characteristic
280	C₂²	C₂×#13063316	1	2	yes	—
281	C₄	—	344	4	no	—
282	C₄	—	172	4	no	—
283	C₄	#14582831	1	2	yes	—
284	C₄	#14582831	1	2	yes	—
285	C₄	C₂×#13063302	1	2	yes	—
286	C₄	C₂×#13063309	1	1	yes	characteristic
287	C₄	C₂×#13063325	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₂×#14582831	1	1	yes	characteristic
292	C₂	#19144316	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063300
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#341	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂⁴×C₄₃	#41
hypercenter	C₂×#341	D₄₃
hypocenter	C₄₃	#23674
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
4. #19136107	C₂	semidirect
5. #19136102	C₂	semidirect
173. C₄₃	#23674	semidirect

Maximal subgroups

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

subgroup	index
#19143185	2
C₂×#14582937	2
C₂×#14582925	2
#19136107	2
#19136102	2
#19134333	2
C₂×C₄₃×#341	2
#23674	43