Subgroups — #24995405

Order 11008.

558 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134712	C₂	1	1	yes	maximal, characteristic, complemented
2	#19134341	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×D₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
5	C₄×#124×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
8	#14582654	—	2	2	no	—
9	#14582654	—	2	2	no	—
10	Q₈⋊(C₄₃⋊C₈)	—	2	4	no	—
11	D₄⋊(C₄₃⋊C₈)	—	2	4	no	—
12	C₂×#124×C₄₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#391	C₄	1	1	yes	characteristic
16	C₄₃×#390	C₄	1	1	yes	characteristic
17	C₄₃×#388	—	2	2	no	—
18	C₄₃×#382	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊C₄×C₄₃	—	2	2	no	—
20	C₄³×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₄×C₄₃⋊C₈	—	2	2	no	—
26	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
27	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₃	—	2	1	no	—
30	#124×C₄₃	—	2	2	no	—
31	#124×C₄₃	—	2	2	no	—
32	C₄×Q₈×C₄₃	—	2	2	no	—
33	C₄×Q₈×C₄₃	—	2	2	no	—
34	C₄×D₄×C₄₃	—	2	2	no	—
35	C₄×D₄×C₄₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
42	C₂×C₄²×C₄₃	—	2	1	no	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	D₄	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	#95×C₄₃	—	2	2	no	—
49	#95×C₄₃	—	2	2	no	—
50	C₂⁴×C₄₃	—	2	1	no	—
51	C₂×Q₈×C₄₃	—	2	1	no	—
52	C₂×Q₈×C₄₃	—	2	2	no	—
53	C₂×Q₈×C₄₃	—	2	2	no	—
54	C₂×Q₈×C₄₃	M₁₆	1	2	yes	—
55	C₂×D₄×C₄₃	—	2	1	no	—
56	C₂×D₄×C₄₃	—	2	2	no	—
57	C₂×D₄×C₄₃	—	2	2	no	—
58	C₂×D₄×C₄₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	—	4	1	no	—
63	C₂²×C₄×C₄₃	—	4	1	no	—
64	C₂²×C₄×C₄₃	—	4	4	no	—
65	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	—	2	2	no	—
71	C₄⋊C₄×C₄₃	—	2	2	no	—
72	C₄⋊C₄×C₄₃	—	4	2	no	—
73	C₂²⋊C₄×C₄₃	—	2	8	no	—
74	C₂²⋊C₄×C₄₃	—	4	4	no	—
75	C₂²⋊C₄×C₄₃	—	4	2	no	—
76	C₄²×C₄₃	—	2	1	no	—
77	C₄²×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	—	2	4	no	—
80	C₄²×C₄₃	—	4	2	no	—
81	C₄²×C₄₃	—	4	1	no	—
82	C₄²×C₄₃	—	4	1	no	—
83	C₄²×C₄₃	C₂×D₄	1	2	yes	—
84	C₄²×C₄₃	#41	1	2	yes	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	4	4	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	4	4	no	—
89	C₂³×C₄₃	—	2	2	no	—
90	C₂³×C₄₃	—	4	1	no	—
91	C₂³×C₄₃	—	4	1	no	—
92	C₂³×C₄₃	—	4	2	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₄₃	—	2	4	no	—
96	Q₈×C₄₃	—	4	2	no	—
97	D₄×C₄₃	—	2	4	no	—
98	D₄×C₄₃	—	4	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	2	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	4	1	no	—
108	C₂×C₄×C₄₃	—	4	1	no	—
109	C₂×C₄×C₄₃	—	4	1	no	—
110	C₂×C₄×C₄₃	—	4	1	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	8	2	no	—
114	C₂×C₄×C₄₃	—	8	2	no	—
115	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₃	#121	1	2	yes	—
117	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#131	1	2	yes	—
122	C₂×C₄×C₄₃	#135	1	4	yes	—
123	C₄₃⋊C₈	—	4	4	no	—
124	C₄₃⋊C₈	—	8	2	no	—
125	#9645	—	43	1	no	maximal, Sylow
126	C₂²×C₄₃	—	4	2	no	—
127	C₂²×C₄₃	—	4	4	no	—
128	C₂²×C₄₃	—	8	1	no	—
129	C₂²×C₄₃	—	8	1	no	—
130	C₂²×C₄₃	—	8	2	no	—
131	C₂²×C₄₃	#390	1	1	yes	characteristic
132	C₂²×C₄₃	#407	1	1	yes	characteristic
133	C₂²×C₄₃	#408	1	1	yes	characteristic
134	C₂²×C₄₃	#416	1	2	yes	—
135	C₂²×C₄₃	#421	1	2	yes	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	—	4	2	no	—
138	C₄×C₄₃	—	4	2	no	—
139	C₄×C₄₃	—	4	2	no	—
140	C₄×C₄₃	—	8	1	no	—
141	C₄×C₄₃	—	8	1	no	—
142	C₄×C₄₃	#435	1	2	yes	—
143	C₄×C₄₃	#435	1	2	yes	—
144	C₄×C₄₃	#488	1	2	yes	—
145	C₄×C₄₃	#495	1	2	yes	—
146	C₄×#124	D₄₃	1	1	yes	characteristic
147	#1831	—	43	1	no	—
148	#1744	—	43	1	no	—
149	C₂×Q₈⋊C₈	—	43	2	no	—
150	C₂×D₄⋊C₈	—	43	2	no	—
151	C₂×C₄₃	—	8	2	no	—
152	C₂×C₄₃	#1482	1	2	yes	—
153	C₂×C₄₃	#1488	1	2	yes	—
154	C₂×C₄₃	#1955	1	1	yes	characteristic
155	C₂×C₄₃	#1958	1	1	yes	characteristic
156	C₂×C₄₃	#1963	1	1	yes	characteristic
157	C₂×#124	Dic₄₃	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
159	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	43	2	no	—
161	C₂×C₄×C₈	—	43	2	no	—
162	#391	Dic₄₃	1	1	yes	characteristic
163	#390	Dic₄₃	1	1	yes	characteristic
164	#388	—	2	2	no	—
165	#382	Dic₄₃	1	1	yes	characteristic
166	C₄×C₂²⋊C₄	—	2	2	no	—
167	C₄³	C₂×D₄₃	1	1	yes	characteristic
168	#336	—	86	2	no	—
169	#336	—	86	2	no	—
170	Q₈⋊C₈	—	86	4	no	—
171	D₄⋊C₈	—	86	4	no	—
172	C₄₃	#9645	1	1	yes	characteristic, complemented, Sylow
173	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
174	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
175	C₂³×C₄	—	2	1	no	—
176	C₂²×C₈	—	86	2	no	—
177	C₂²×C₈	—	86	2	no	—
178	#124	—	2	2	no	—
179	#124	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×Q₈	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
194	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
195	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
196	C₄⋊C₈	—	86	2	no	—
197	C₄⋊C₈	—	86	2	no	—
198	C₄×C₈	—	43	4	no	—
199	C₄×C₈	—	86	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	C₂⁴	—	2	1	no	—
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	#10621877	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	#10621877	1	2	yes	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
218	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂×C₈	—	86	4	no	—
223	C₂×C₈	—	172	2	no	—
224	C₂×C₈	—	172	4	no	—
225	C₂×C₈	—	172	4	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	D₄×D₄₃	1	2	yes	—
240	C₄²	#10621900	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	4	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	D₄	—	2	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
268	C₂×C₄	#13063296	1	2	yes	—
269	C₂×C₄	#13063312	1	1	yes	characteristic
270	C₂×C₄	#13063321	1	1	yes	characteristic
271	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
272	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
273	C₂×C₄	#13063332	1	2	yes	—
274	C₂×C₄	#13063345	1	4	yes	—
275	C₈	—	344	2	no	—
276	C₈	—	172	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	1	no	—
281	C₂²	—	8	1	no	—
282	C₂²	#14582802	1	1	yes	characteristic
283	C₂²	#14582863	1	1	yes	characteristic
284	C₂²	#14582866	1	1	yes	characteristic
285	C₂²	#14582895	1	2	yes	—
286	C₂²	#14582916	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	4	2	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	#14582971	1	2	yes	—
294	C₄	#14582971	1	2	yes	—
295	C₄	#14583253	1	2	yes	—
296	C₄	#14583281	1	2	yes	—
297	C₂	—	8	2	no	—
298	C₂	#19133003	1	2	yes	—
299	C₂	#19133029	1	2	yes	—
300	C₂	#19135394	1	1	yes	characteristic
301	C₂	#19135407	1	1	yes	characteristic
302	C₂	#19135428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#124×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#124	D₄₃
hypocenter	C₄₃	#9645
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #19134712	C₂	semidirect
2. #19134341	C₂	semidirect
3. C₂×Q₈⋊(C₄₃⋊C₈)	C₂	semidirect
172. C₄₃	#9645	semidirect

Maximal subgroups

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

subgroup	index
#19134712	2
#19134341	2
C₂×Q₈⋊(C₄₃⋊C₈)	2
C₂×D₄⋊(C₄₃⋊C₈)	2
C₄×#124×C₄₃	2
#9645	43