Subgroups — #25015012

Order 11008.

410 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136331	C₂	1	2	yes	maximal, complemented
3	#19136191	C₂	1	1	yes	maximal, characteristic, complemented
4	#19135494	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#508	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	C₂²	1	1	yes	characteristic
7	#14582937	C₂²	1	2	yes	—
8	#14582916	—	2	2	no	—
9	#14582873	—	2	2	no	—
10	#14582854	C₂²	1	1	yes	characteristic
11	#14582727	—	2	1	no	—
12	#14582725	—	2	1	no	—
13	#14582709	—	2	4	no	—
14	#14582674	—	2	1	no	—
15	C₁₆×Dic₄₃	—	2	1	no	—
16	C₂×#135×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#508	—	2	1	no	—
18	C₄₃×#508	—	2	1	no	—
19	C₄₃×#508	—	2	1	no	—
20	C₄₃×#508	—	2	1	no	—
21	C₂×Q₃₂×C₄₃	C₄	1	1	yes	characteristic, complemented
22	C₂×SD₃₂×C₄₃	C₄	1	2	yes	complemented
23	C₂×D₁₆×C₄₃	C₄	1	1	yes	characteristic, complemented
24	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#10621877	D₄	1	1	yes	characteristic
26	#13063284	—	2	2	no	—
27	#13063284	D₄	1	1	yes	characteristic
28	#13063265	—	2	1	no	—
29	#13063265	D₄	1	2	yes	—
30	C₂×C₄₃⋊C₁₆	—	2	1	no	—
31	#13063259	—	2	2	no	—
32	#13063257	—	2	2	no	—
33	#13063252	—	4	4	no	—
34	#13063240	—	4	4	no	—
35	#13063235	—	2	1	no	—
36	C₈×Dic₄₃	—	2	1	no	—
37	C₂×#41×C₄₃	—	2	2	no	—
38	#135×C₄₃	—	2	2	no	—
39	#135×C₄₃	—	2	2	no	—
40	#135×C₄₃	—	2	2	no	—
41	#135×C₄₃	—	2	2	no	—
42	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×SD₁₆×C₄₃	—	2	2	no	—
44	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
46	Q₃₂×C₄₃	—	2	1	no	—
47	Q₃₂×C₄₃	—	2	1	no	—
48	SD₃₂×C₄₃	—	2	2	no	—
49	SD₃₂×C₄₃	—	2	2	no	—
50	D₁₆×C₄₃	—	2	1	no	—
51	D₁₆×C₄₃	—	2	1	no	—
52	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
55	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
58	#41×C₄₃	—	4	2	no	—
59	#41×C₄₃	—	4	2	no	—
60	#41×C₄₃	—	4	2	no	—
61	#41×C₄₃	—	4	2	no	—
62	C₂×Q₈×C₄₃	—	2	2	no	—
63	C₂×D₄×C₄₃	—	2	2	no	—
64	C₂×D₄×C₄₃	—	4	2	no	—
65	C₂²×C₄×C₄₃	—	4	2	no	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₄₃	—	2	2	no	—
68	Q₁₆×C₄₃	—	2	2	no	—
69	SD₁₆×C₄₃	—	4	2	no	—
70	SD₁₆×C₄₃	—	4	2	no	—
71	D₈×C₄₃	—	2	2	no	—
72	D₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
78	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
79	C₂²⋊Dic₄₃	—	4	2	no	—
80	C₂²⋊Dic₄₃	D₈	1	2	yes	—
81	C₄⋊Dic₄₃	—	2	1	no	—
82	C₄⋊Dic₄₃	—	4	2	no	—
83	C₄×Dic₄₃	—	2	1	no	—
84	C₄×Dic₄₃	—	4	2	no	—
85	#10621877	—	2	1	no	—
86	#10621877	D₈	1	2	yes	—
87	C₂×C₄₃⋊C₈	—	2	1	no	—
88	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄₃⋊C₁₆	—	2	2	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₄₃	—	4	2	no	—
96	Q₈×C₄₃	—	4	2	no	—
97	D₄×C₄₃	—	4	2	no	—
98	D₄×C₄₃	—	4	2	no	—
99	D₄×C₄₃	—	8	2	no	—
100	D₄×C₄₃	—	8	2	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	—	8	2	no	—
103	C₂×C₄×C₄₃	—	8	2	no	—
104	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
110	C₂×Dic₄₃	—	2	2	no	—
111	C₂×Dic₄₃	—	8	2	no	—
112	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₈×C₄₃	#121	1	1	yes	characteristic
115	C₈×C₄₃	#124	1	1	yes	characteristic
116	C₄₃⋊C₈	—	2	2	no	—
117	#13243	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	4	1	no	—
119	C₂²×C₄₃	—	8	2	no	—
120	C₂²×C₄₃	—	16	2	no	—
121	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
122	C₂²×C₄₃	#466	1	1	yes	characteristic
123	C₂²×C₄₃	#493	1	1	yes	characteristic
124	C₄×C₄₃	—	8	2	no	—
125	C₄×C₄₃	#390	1	1	yes	characteristic
126	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
127	C₄×C₄₃	#466	1	1	yes	characteristic
128	C₄×C₄₃	#495	1	1	yes	characteristic
129	Dic₄₃	—	4	2	no	—
130	Dic₄₃	—	16	2	no	—
131	C₂×#508	D₄₃	1	1	yes	characteristic
132	#2158	—	43	1	no	—
133	#2145	—	43	2	no	—
134	#2108	—	43	1	no	—
135	#1976	—	43	2	no	—
136	C₂×C₄₃	—	2	1	no	—
137	C₂×C₄₃	—	2	1	no	—
138	C₂×C₄₃	—	2	1	no	—
139	C₂×C₄₃	—	16	2	no	—
140	C₂×C₄₃	#1970	1	1	yes	characteristic
141	C₂×#135	C₂×D₄₃	1	2	yes	—
142	#508	—	2	1	no	—
143	#508	—	2	1	no	—
144	#508	—	2	1	no	—
145	#508	—	2	1	no	—
146	C₂×Q₃₂	Dic₄₃	1	1	yes	characteristic, complemented
147	C₂×SD₃₂	Dic₄₃	1	2	yes	complemented
148	C₂×D₁₆	Dic₄₃	1	1	yes	characteristic, complemented
149	C₂×M₃₂	—	43	1	no	—
150	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
151	#427	—	43	2	no	—
152	#421	—	86	2	no	—
153	#410	—	86	2	no	—
154	#405	—	43	1	no	—
155	#367	—	86	1	no	—
156	#366	—	86	1	no	—
157	#360	—	86	4	no	—
158	#346	—	86	1	no	—
159	C₄×C₁₆	—	86	1	no	—
160	C₄₃	#13243	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	2	2	no	—
162	#135	—	2	2	no	—
163	#135	—	2	2	no	—
164	#135	—	2	2	no	—
165	#135	—	2	2	no	—
166	C₂×Q₁₆	C₂×Dic₄₃	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×D₈	C₂×Dic₄₃	1	2	yes	—
169	C₂×M₁₆	—	43	1	no	—
170	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
171	#117	—	43	1	no	—
172	#117	—	86	2	no	—
173	Q₃₂	—	2	1	no	—
174	Q₃₂	—	2	1	no	—
175	SD₃₂	—	2	2	no	—
176	SD₃₂	—	2	2	no	—
177	D₁₆	—	2	1	no	—
178	D₁₆	—	2	1	no	—
179	M₃₂	—	43	2	no	—
180	M₃₂	—	86	1	no	—
181	C₂×C₁₆	—	86	1	no	—
182	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
183	C₂×C₁₆	C₂×Dic₄₃	1	1	yes	characteristic
184	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
185	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
186	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
187	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
188	#107	—	86	2	no	—
189	#106	—	86	2	no	—
190	#104	—	172	4	no	—
191	#99	—	172	4	no	—
192	#97	—	86	1	no	—
193	C₄×C₈	—	86	1	no	—
194	#41	—	4	2	no	—
195	#41	—	4	2	no	—
196	#41	—	4	2	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
203	Q₁₆	—	2	2	no	—
204	Q₁₆	—	2	2	no	—
205	SD₁₆	—	4	2	no	—
206	SD₁₆	—	4	2	no	—
207	D₈	—	2	2	no	—
208	D₈	—	2	2	no	—
209	M₁₆	—	43	2	no	—
210	M₁₆	—	86	1	no	—
211	C₂×C₈	—	86	1	no	—
212	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
213	C₂×C₈	#10621900	1	1	yes	characteristic
214	C₂×C₈	#10621900	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
218	C₄⋊C₄	—	86	1	no	—
219	C₄⋊C₄	—	172	2	no	—
220	C₂²⋊C₄	—	43	2	no	—
221	C₂²⋊C₄	—	172	2	no	—
222	C₄²	—	86	1	no	—
223	C₄²	—	172	2	no	—
224	C₁₆	—	86	2	no	—
225	C₁₆	C₂²⋊Dic₄₃	1	1	yes	characteristic
226	C₁₆	C₂²⋊Dic₄₃	1	1	yes	characteristic
227	C₁₆	C₂²⋊Dic₄₃	1	1	yes	characteristic
228	C₁₆	C₂²⋊Dic₄₃	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	D₈×D₄₃	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	86	2	no	—
241	C₂×C₄	—	344	2	no	—
242	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
243	C₂×C₄	#13063296	1	1	yes	characteristic
244	C₂×C₄	#13063321	1	1	yes	characteristic
245	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
246	C₂×C₄	#13063344	1	1	yes	characteristic
247	C₂×C₄	#13063344	1	1	yes	characteristic
248	C₈	—	86	2	no	—
249	C₈	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
250	C₈	D₄×Dic₄₃	1	1	yes	characteristic
251	C₈	#13063296	1	1	yes	characteristic
252	C₈	#13063312	1	1	yes	characteristic
253	C₂²	—	4	1	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	16	2	no	—
256	C₂²	D₈×Dic₄₃	1	1	yes	characteristic
257	C₂²	#14583132	1	1	yes	characteristic
258	C₂²	#14583274	1	1	yes	characteristic
259	C₄	—	8	2	no	—
260	C₄	—	172	2	no	—
261	C₄	—	688	2	no	—
262	C₄	#14582802	1	1	yes	characteristic
263	C₄	D₈×Dic₄₃	1	1	yes	characteristic
264	C₄	#14583132	1	1	yes	characteristic
265	C₄	#14583282	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	1	no	—
269	C₂	—	16	2	no	—
270	C₂	#19135462	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583132
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting, prime radical	C₂×C₄₃×#508	C₂
socle	C₂×C₄₃	#1970
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×#508	D₄₃
hypocenter	C₄₃	#13243
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #19136379	C₂	semidirect
2. #19136331	C₂	semidirect
3. #19136191	C₂	semidirect
4. #19135494	C₂	semidirect
21. C₂×Q₃₂×C₄₃	C₄	semidirect
22. C₂×SD₃₂×C₄₃	C₄	semidirect
23. C₂×D₁₆×C₄₃	C₄	semidirect
146. C₂×Q₃₂	Dic₄₃	semidirect
147. C₂×SD₃₂	Dic₄₃	semidirect
148. C₂×D₁₆	Dic₄₃	semidirect
160. C₄₃	#13243	semidirect

Maximal subgroups

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

subgroup	index
#19136379	2
#19136331	2
#19136191	2
#19135494	2
C₂×C₄₃×#508	2
#13243	43