Subgroups — #25078967

Order 11008.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19144635	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
4	#19134000	C₂	1	2	yes	maximal, complemented
5	#19133990	C₂	1	2	yes	maximal, complemented
6	#19133978	C₂	1	4	yes	maximal, complemented
7	#19133250	C₂	1	2	yes	maximal
8	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	#14583304	C₂²	1	1	yes	characteristic
12	#14583304	C₂²	1	1	yes	characteristic
13	#14583301	—	2	1	no	—
14	#14583301	—	2	1	no	—
15	#14583297	C₂²	1	2	yes	—
16	#14583235	—	2	2	no	—
17	#14583195	C₂²	1	4	yes	—
18	#14583110	C₂²	1	4	yes	—
19	#14583057	—	2	2	no	—
20	C₈×Dic₈₆	—	2	1	no	—
21	C₈×Dic₈₆	—	2	1	no	—
22	C₈×Dic₈₆	C₂²	1	2	yes	—
23	C₈×Dic₈₆	C₂²	1	2	yes	—
24	C₄×Dic₁₇₂	—	2	4	no	—
25	#14582937	—	2	1	no	—
26	#14582937	—	2	1	no	—
27	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×Dic₄₃⋊C₈	—	2	1	no	—
29	C₂×C₈×Dic₄₃	—	2	1	no	—
30	#14582647	C₂²	1	4	yes	—
31	#14582646	C₂²	1	4	yes	—
32	C₄₃×#431	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₄₃×#335	C₂²	1	2	yes	—
35	C₄₃×#334	C₂²	1	2	yes	—
36	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
37	C₂×Dic₁₇₂	—	2	4	no	—
38	C₂²×C₄₃⋊C₈	—	2	1	no	—
39	#13063325	C₂³	1	2	yes	—
40	#13063325	C₂³	1	2	yes	—
41	#13063317	—	2	2	no	—
42	#13063304	—	2	2	no	—
43	#13063304	—	2	2	no	—
44	C₄×Dic₈₆	—	2	1	no	—
45	C₄×Dic₈₆	—	2	1	no	—
46	C₄×Dic₈₆	—	2	2	no	—
47	C₄×Dic₈₆	—	2	2	no	—
48	C₄×Dic₈₆	D₄	1	2	yes	—
49	C₄×Dic₈₆	C₂³	1	2	yes	—
50	#13063284	—	2	2	no	—
51	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
52	C₂×Dic₄₃⋊C₄	—	2	1	no	—
53	C₂×C₄×Dic₄₃	—	2	1	no	—
54	#13063259	—	2	1	no	—
55	#13063259	—	2	1	no	—
56	#13063259	—	2	1	no	—
57	#13063259	—	2	1	no	—
58	#13063259	—	2	4	no	—
59	#13063257	—	2	2	no	—
60	#13063257	—	2	2	no	—
61	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
62	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₄₃⋊C₈	—	2	2	no	—
64	Dic₄₃⋊C₈	—	2	2	no	—
65	#13063250	—	2	4	no	—
66	#13063250	—	2	4	no	—
67	C₈×Dic₄₃	—	2	2	no	—
68	C₈×Dic₄₃	—	2	2	no	—
69	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₄₃	C₂³	1	4	yes	—
73	C₂²⋊C₈×C₄₃	—	2	2	no	—
74	C₄×C₈×C₄₃	—	2	2	no	—
75	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
82	C₂²×Dic₄₃	—	2	1	no	—
83	C₂²×Dic₄₃	—	2	1	no	—
84	C₂×Dic₈₆	—	2	1	no	—
85	C₂×Dic₈₆	—	2	4	no	—
86	C₂×Dic₈₆	—	2	2	no	—
87	C₂×Dic₈₆	—	2	2	no	—
88	C₂×Dic₈₆	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	—	2	4	no	—
91	C₂×C₈×C₄₃	Q₁₆	1	4	yes	complemented
92	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
101	C₄⋊Dic₄₃	—	2	1	no	—
102	C₄⋊Dic₄₃	—	2	2	no	—
103	C₄⋊Dic₄₃	—	2	2	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	C₄⋊Dic₄₃	—	2	2	no	—
106	C₄⋊Dic₄₃	C₂×D₄	1	2	yes	—
107	Dic₄₃⋊C₄	—	2	2	no	—
108	Dic₄₃⋊C₄	—	2	2	no	—
109	Dic₄₃⋊C₄	—	4	2	no	—
110	Dic₄₃⋊C₄	—	4	2	no	—
111	C₄×Dic₄₃	—	2	2	no	—
112	C₄×Dic₄₃	—	2	2	no	—
113	C₄×Dic₄₃	—	4	2	no	—
114	C₂×C₄₃⋊C₈	—	2	2	no	—
115	C₂×C₄₃⋊C₈	—	4	2	no	—
116	Dic₁₇₂	—	4	8	no	—
117	C₂³×C₄₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	4	1	no	—
120	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₄₃	#137	1	2	yes	—
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
128	C₂×Dic₄₃	—	2	4	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	—	2	2	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	—	4	2	no	—
133	C₂×Dic₄₃	—	4	2	no	—
134	Dic₈₆	—	2	4	no	—
135	Dic₈₆	—	2	2	no	—
136	Dic₈₆	—	4	4	no	—
137	Dic₈₆	—	8	2	no	—
138	Dic₈₆	#135	1	4	yes	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	4	4	no	—
142	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
143	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
144	C₄₃⋊C₈	—	4	2	no	—
145	#19118	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	#462	1	2	yes	—
149	C₂²×C₄₃	C₂×#121	1	1	yes	characteristic
150	C₄×C₄₃	—	2	1	no	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	#462	1	2	yes	—
154	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
155	C₄×C₄₃	C₂×#135	1	2	yes	—
156	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
157	Dic₄₃	—	2	4	no	—
158	Dic₄₃	—	4	2	no	—
159	Dic₄₃	—	8	2	no	—
160	Dic₄₃	—	8	2	no	—
161	#3160	—	43	1	no	—
162	#3104	—	43	2	no	—
163	C₂×C₈×Q₈	—	43	1	no	—
164	#1676	—	43	2	no	—
165	#1674	—	43	2	no	—
166	#1672	—	43	4	no	—
167	#1564	D₄₃	1	1	yes	characteristic
168	#1542	—	43	2	no	—
169	C₂×C₄₃	—	2	2	no	—
170	C₂×C₄₃	C₂×#462	1	1	yes	characteristic
171	C₂×C₄₃	#3050	1	1	yes	characteristic
172	C₂×C₄₃	#3067	1	1	yes	characteristic
173	#533	—	43	2	no	—
174	C₂×C₄×Q₈	—	43	1	no	—
175	#500	—	43	1	no	—
176	#500	—	43	1	no	—
177	#499	—	86	1	no	—
178	#499	—	86	1	no	—
179	#498	—	43	2	no	—
180	#484	—	86	2	no	—
181	#477	—	43	4	no	—
182	#462	—	43	4	no	—
183	#451	—	86	2	no	—
184	C₈×Q₈	—	43	2	no	—
185	C₈×Q₈	—	43	2	no	—
186	C₈×Q₈	—	86	1	no	—
187	C₈×Q₈	—	86	1	no	—
188	C₄×Q₁₆	—	86	4	no	—
189	#431	C₂×D₄₃	1	2	yes	—
190	#427	—	86	1	no	—
191	#427	—	86	1	no	—
192	C₂×C₄⋊C₈	—	43	1	no	—
193	C₂×C₄⋊C₈	—	86	1	no	—
194	C₂×C₄×C₈	—	86	1	no	—
195	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
196	#335	C₂×D₄₃	1	2	yes	—
197	#334	C₂×D₄₃	1	2	yes	—
198	#333	—	43	4	no	—
199	#332	—	43	4	no	—
200	C₄₃	#19118	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	43	1	no	—
202	C₂×Q₁₆	—	86	4	no	—
203	C₂²×C₈	—	86	1	no	—
204	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
205	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
206	C₄⋊Q₈	—	43	2	no	—
207	C₄⋊Q₈	—	43	2	no	—
208	#125	—	86	2	no	—
209	C₂²⋊Q₈	—	86	2	no	—
210	C₂²⋊Q₈	—	86	2	no	—
211	C₄×Q₈	—	43	2	no	—
212	C₄×Q₈	—	43	2	no	—
213	C₄×Q₈	—	86	1	no	—
214	C₄×Q₈	—	86	1	no	—
215	C₄×Q₈	—	86	2	no	—
216	C₄×Q₈	—	86	2	no	—
217	#117	—	86	2	no	—
218	C₂×C₄⋊C₄	—	43	1	no	—
219	C₂×C₄⋊C₄	—	86	1	no	—
220	C₂×C₄²	—	86	1	no	—
221	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	#107	—	86	4	no	—
223	#107	—	86	1	no	—
224	#107	—	86	1	no	—
225	#107	—	86	1	no	—
226	#107	—	86	1	no	—
227	#106	—	86	2	no	—
228	#106	—	86	2	no	—
229	C₄⋊C₈	—	43	2	no	—
230	C₄⋊C₈	—	86	2	no	—
231	C₄⋊C₈	—	86	2	no	—
232	C₄⋊C₈	—	86	1	no	—
233	C₄⋊C₈	C₂²×D₄₃	1	4	yes	—
234	#103	—	86	4	no	—
235	#103	—	86	4	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	86	2	no	—
239	C₄×C₈	—	86	2	no	—
240	C₄×C₈	D₁₇₂	1	1	yes	characteristic
241	C₄×C₈	D₁₇₂	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×Q₈	—	43	2	no	—
245	C₂×Q₈	—	86	4	no	—
246	C₂×Q₈	—	86	1	no	—
247	C₂×Q₈	—	86	2	no	—
248	C₂×Q₈	—	86	2	no	—
249	C₂²×C₄	—	86	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	#10621902	1	1	yes	characteristic
254	Q₁₆	—	172	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	—	172	2	no	—
259	C₂×C₈	Dic₁₇₂	1	4	yes	complemented
260	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
261	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
262	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
263	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
264	C₄⋊C₄	—	43	2	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	86	2	no	—
267	C₄⋊C₄	—	86	2	no	—
268	C₄⋊C₄	—	86	2	no	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	86	2	no	—
271	C₄⋊C₄	—	86	2	no	—
272	C₄⋊C₄	—	172	2	no	—
273	C₄⋊C₄	—	172	2	no	—
274	C₂²⋊C₄	—	172	2	no	—
275	C₄²	—	86	2	no	—
276	C₄²	—	86	2	no	—
277	C₄²	—	172	2	no	—
278	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
279	C₄²	D₄×D₄₃	1	1	yes	characteristic
280	C₄²	#10621902	1	1	yes	characteristic
281	C₄²	C₂³×D₄₃	1	1	yes	characteristic
282	C₂³	#13063300	1	1	yes	characteristic
283	Q₈	—	43	4	no	—
284	Q₈	—	86	4	no	—
285	Q₈	—	86	2	no	—
286	Q₈	—	344	2	no	—
287	Q₈	—	172	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	86	4	no	—
291	C₂×C₄	—	86	2	no	—
292	C₂×C₄	—	86	2	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	#13063300	1	1	yes	characteristic
297	C₂×C₄	#13063300	1	1	yes	characteristic
298	C₂×C₄	#13063300	1	1	yes	characteristic
299	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
300	C₂×C₄	#13063358	1	2	yes	—
301	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	4	no	—
307	C₈	—	172	2	no	—
308	C₈	C₂×Dic₁₇₂	1	2	yes	—
309	C₈	C₂×Dic₁₇₂	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#14583110	1	2	yes	—
313	C₂²	C₂×#13063300	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	86	4	no	—
318	C₄	—	344	2	no	—
319	C₄	—	344	2	no	—
320	C₄	—	172	2	no	—
321	C₄	#14583110	1	2	yes	—
322	C₄	C₂²×Dic₁₇₂	1	1	yes	characteristic
323	C₄	#135×D₄₃	1	2	yes	—
324	C₄	C₂×#13063358	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	C₂×#14583110	1	1	yes	characteristic
327	C₂	#19144177	1	1	yes	characteristic
328	C₂	#19144316	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063300
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×#121
prime radical	C₂³×C₄₃	#121
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#19118
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #19134000	C₂	semidirect
5. #19133990	C₂	semidirect
6. #19133978	C₂	semidirect
91. C₂×C₈×C₄₃	Q₁₆	semidirect
200. C₄₃	#19118	semidirect
259. C₂×C₈	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19144635	2
C₂×C₈×Dic₈₆	2
#19134000	2
#19133990	2
#19133978	2
#19133250	2
C₄₃×#1564	2
#19118	43