Subgroups — #25079807

Order 11008.

646 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	#19145739	C₂	1	2	yes	maximal
2	#19143529	C₂	1	1	yes	maximal, characteristic
3	#19134086	C₂	1	4	yes	maximal, complemented
4	#19133962	C₂	1	4	yes	maximal, complemented
5	#19133343	C₂	1	1	yes	maximal, characteristic
6	#19133267	C₂	1	2	yes	maximal
7	C₂×C₄₃×#494	C₂	1	1	yes	maximal, characteristic
8	#14583456	C₂²	1	1	yes	characteristic
9	#14583295	—	2	2	no	—
10	#14583251	—	2	2	no	—
11	#14583229	—	2	4	no	—
12	#14583181	C₂²	1	8	yes	—
13	#14583029	—	2	1	no	—
14	#14583029	C₂²	1	2	yes	—
15	C₈×Dic₈₆	—	2	1	no	—
16	C₈×Dic₈₆	C₂²	1	2	yes	—
17	#14582952	C₂²	1	2	yes	—
18	#14582925	—	2	1	no	—
19	#14582923	C₂²	1	1	yes	characteristic
20	#14582854	—	2	1	no	—
21	#14582652	C₂²	1	2	yes	—
22	#14582650	C₂²	1	2	yes	—
23	#14582645	C₂²	1	8	yes	—
24	C₂²×Q₁₆×C₄₃	—	2	1	no	—
25	C₂²×Q₁₆×C₄₃	—	2	1	no	—
26	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#494	—	2	1	no	—
28	C₄₃×#494	—	2	1	no	—
29	C₄₃×#494	C₂²	1	2	yes	—
30	C₄₃×#494	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×#10621877	—	2	1	no	—
33	#13063325	D₄	1	1	yes	characteristic
34	#13063325	D₄	1	1	yes	characteristic
35	#13063317	—	2	1	no	—
36	#13063304	—	2	1	no	—
37	#13063304	D₄	1	2	yes	—
38	C₄×Dic₈₆	—	2	1	no	—
39	C₄×Dic₈₆	C₂³	1	2	yes	—
40	#13063284	—	2	1	no	—
41	#13063256	—	2	8	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
44	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
45	Dic₄₃⋊C₈	—	2	2	no	—
46	Dic₄₃⋊C₈	—	2	2	no	—
47	#13063248	—	2	8	no	—
48	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
49	#13063235	—	2	2	no	—
50	C₄×C₄₃⋊C₈	—	2	2	no	—
51	C₈×Dic₄₃	—	2	2	no	—
52	C₂²×Q₈×C₄₃	—	2	2	no	—
53	C₂×Q₁₆×C₄₃	—	2	2	no	—
54	C₂×Q₁₆×C₄₃	—	2	2	no	—
55	C₂×Q₁₆×C₄₃	—	2	2	no	—
56	C₂×Q₁₆×C₄₃	—	2	2	no	—
57	C₂×Q₁₆×C₄₃	—	4	4	no	—
58	C₂×Q₁₆×C₄₃	—	4	4	no	—
59	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₄₃	—	2	2	no	—
62	C₄⋊Q₈×C₄₃	—	2	2	no	—
63	C₄⋊Q₈×C₄₃	—	2	2	no	—
64	C₄⋊Q₈×C₄₃	C₂³	1	4	yes	—
65	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₄₃	—	2	4	no	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂×Q₈×C₄₃	—	4	4	no	—
75	C₂×Q₈×C₄₃	—	4	4	no	—
76	C₂²×C₄×C₄₃	—	4	2	no	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
80	C₂×Dic₈₆	—	2	1	no	—
81	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
82	Q₁₆×C₄₃	—	4	4	no	—
83	Q₁₆×C₄₃	—	4	4	no	—
84	Q₁₆×C₄₃	—	4	4	no	—
85	Q₁₆×C₄₃	—	4	4	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
88	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	4	no	—
93	C₄⋊C₄×C₄₃	—	2	4	no	—
94	C₄⋊C₄×C₄₃	—	4	2	no	—
95	C₄⋊C₄×C₄₃	—	4	2	no	—
96	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₃	#41	1	1	yes	characteristic
99	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊Dic₄₃	—	2	2	no	—
101	C₄⋊Dic₄₃	—	2	1	no	—
102	C₄⋊Dic₄₃	—	2	1	no	—
103	C₄⋊Dic₄₃	—	2	1	no	—
104	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
105	Dic₄₃⋊C₄	—	2	2	no	—
106	Dic₄₃⋊C₄	—	2	2	no	—
107	C₄×Dic₄₃	—	2	2	no	—
108	#10621877	—	4	2	no	—
109	C₂×C₄₃⋊C₈	—	2	2	no	—
110	C₂×C₄₃⋊C₈	—	2	4	no	—
111	C₂³×C₄₃	#121	1	1	yes	characteristic
112	Q₈×C₄₃	—	4	4	no	—
113	Q₈×C₄₃	—	4	4	no	—
114	Q₈×C₄₃	—	8	4	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	2	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	8	4	no	—
119	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	2	2	no	—
129	Dic₈₆	—	2	4	no	—
130	Dic₈₆	—	4	2	no	—
131	C₈×C₄₃	—	2	1	no	—
132	C₈×C₄₃	—	2	1	no	—
133	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
134	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
135	C₄₃⋊C₈	—	4	2	no	—
136	C₄₃⋊C₈	—	4	4	no	—
137	#19204	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	1	no	—
139	C₂²×C₄₃	—	2	1	no	—
140	C₂²×C₄₃	#460	1	2	yes	—
141	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	8	4	no	—
145	C₄×C₄₃	#460	1	2	yes	—
146	C₄×C₄₃	#473	1	2	yes	—
147	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
148	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₄×C₄₃	#578	1	2	yes	—
150	Dic₄₃	—	4	2	no	—
151	Dic₄₃	—	4	2	no	—
152	#3229	—	43	2	no	—
153	C₂×#494	D₄₃	1	1	yes	characteristic
154	#2960	—	43	1	no	—
155	#1690	—	43	4	no	—
156	#1670	—	43	4	no	—
157	#1564	—	43	1	no	—
158	#1546	—	43	2	no	—
159	C₂×C₄₃	—	2	2	no	—
160	C₂×C₄₃	C₂×#460	1	1	yes	characteristic
161	C₂×C₄₃	#3059	1	1	yes	characteristic
162	C₂×C₄₃	C₂×#473	1	1	yes	characteristic
163	C₂²×Q₁₆	—	2	1	no	—
164	C₂²×Q₁₆	—	2	1	no	—
165	#533	—	43	1	no	—
166	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
167	#498	—	86	2	no	—
168	#494	—	2	1	no	—
169	#494	—	2	1	no	—
170	#494	C₂×D₄₃	1	2	yes	—
171	#494	C₂×D₄₃	1	2	yes	—
172	#487	—	86	2	no	—
173	#483	—	86	4	no	—
174	#475	—	43	8	no	—
175	#446	—	43	2	no	—
176	#446	—	86	1	no	—
177	C₈×Q₈	—	43	2	no	—
178	C₈×Q₈	—	86	1	no	—
179	#431	—	43	2	no	—
180	#424	—	86	1	no	—
181	#423	—	43	1	no	—
182	#405	—	86	1	no	—
183	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
184	#335	—	43	2	no	—
185	#334	—	43	2	no	—
186	#332	—	43	8	no	—
187	C₄₃	#19204	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	2	2	no	—
189	C₂×Q₁₆	—	2	2	no	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	—	4	4	no	—
194	C₂×Q₁₆	—	4	4	no	—
195	C₂×M₁₆	—	86	1	no	—
196	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	43	1	no	—
202	C₄⋊Q₈	—	43	1	no	—
203	C₄⋊Q₈	C₂²×D₄₃	1	4	yes	—
204	#125	—	86	1	no	—
205	C₂²⋊Q₈	—	43	2	no	—
206	C₂²⋊Q₈	—	86	1	no	—
207	C₄×Q₈	—	43	2	no	—
208	C₄×Q₈	—	86	1	no	—
209	#117	—	86	1	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	#106	—	86	8	no	—
214	C₄⋊C₈	—	43	2	no	—
215	C₄⋊C₈	—	43	4	no	—
216	C₄⋊C₈	—	86	2	no	—
217	C₄⋊C₈	—	86	2	no	—
218	C₄⋊C₈	—	86	1	no	—
219	#103	—	86	8	no	—
220	C₂²⋊C₈	—	86	2	no	—
221	#97	—	86	2	no	—
222	C₄×C₈	—	86	2	no	—
223	C₄×C₈	—	86	2	no	—
224	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	4	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	43	2	no	—
233	C₂×Q₈	—	86	1	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
236	C₂²×C₄	#10621900	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
238	Q₁₆	—	4	4	no	—
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	4	no	—
241	Q₁₆	—	4	4	no	—
242	M₁₆	—	172	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	86	4	no	—
245	C₂×C₈	—	86	2	no	—
246	C₂×C₈	#10621883	1	4	yes	—
247	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	43	2	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	2	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	2	no	—
261	C₂²⋊C₄	—	86	2	no	—
262	C₄²	—	86	2	no	—
263	C₄²	D₄×D₄₃	1	1	yes	characteristic
264	C₄²	#10621900	1	1	yes	characteristic
265	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
266	C₄²	C₂³×D₄₃	1	1	yes	characteristic
267	C₂³	#13063296	1	1	yes	characteristic
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	8	4	no	—
271	Q₈	—	86	4	no	—
272	Q₈	—	172	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	86	2	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	#13063296	1	1	yes	characteristic
280	C₂×C₄	#13063296	1	1	yes	characteristic
281	C₂×C₄	#13063296	1	1	yes	characteristic
282	C₂×C₄	C₂×#10621883	1	2	yes	—
283	C₂×C₄	#13063351	1	2	yes	—
284	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	172	2	no	—
290	C₈	—	172	4	no	—
291	C₈	C₂×#10621883	1	2	yes	—
292	C₈	C₂×#10621883	1	2	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	#14583096	1	2	yes	—
296	C₂²	C₂×#13063296	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	4	no	—
300	C₄	—	172	2	no	—
301	C₄	—	172	2	no	—
302	C₄	#14583096	1	2	yes	—
303	C₄	#14583171	1	2	yes	—
304	C₄	C₂²×#10621883	1	1	yes	characteristic
305	C₄	C₂×#13063351	1	1	yes	characteristic
306	C₄	#14583712	1	2	yes	—
307	C₂	—	2	2	no	—
308	C₂	C₂×#14583096	1	1	yes	characteristic
309	C₂	#19144246	1	1	yes	characteristic
310	C₂	C₂×#14583171	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063296
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#494	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂³×C₄₃	#121
hypercenter	C₂×#494	D₄₃
hypocenter	C₄₃	#19204
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #19134086	C₂	semidirect
4. #19133962	C₂	semidirect
187. C₄₃	#19204	semidirect

Maximal subgroups

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

subgroup	index
#19145739	2
#19143529	2
#19134086	2
#19133962	2
#19133343	2
#19133267	2
C₂×C₄₃×#494	2
#19204	43