Subgroups — #25106925

Order 11008.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143040	C₂	1	2	yes	maximal
2	#19134802	C₂	1	8	yes	maximal, complemented
3	#19134637	C₂	1	2	yes	maximal, complemented
4	#19134633	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₃×#408	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	4	yes	—
7	C₂×#13063261	C₂²	1	4	yes	—
8	C₂×#13063261	C₂²	1	4	yes	—
9	#14582866	—	2	4	no	—
10	#14582866	—	2	4	no	—
11	#14582863	C₂²	1	4	yes	—
12	#14582863	C₂²	1	4	yes	—
13	#14582664	C₂²	1	4	yes	—
14	#14582664	C₂²	1	4	yes	—
15	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#408	C₄	1	4	yes	—
19	C₄₃×#408	C₄	1	4	yes	—
20	C₂×C₂²⋊C₈×C₄₃	C₂²	1	4	yes	—
21	C₂×#10621877	—	2	4	no	—
22	C₂×#10621877	—	4	4	no	—
23	C₂×#10621877	—	4	4	no	—
24	C₂×#10621877	C₂³	1	8	yes	—
25	C₂²×C₄₃⋊C₈	—	2	4	no	—
26	#13063261	—	4	4	no	—
27	#13063261	—	4	4	no	—
28	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
31	C₂×#41×C₄₃	—	4	2	no	—
32	C₂×#41×C₄₃	—	4	2	no	—
33	C₂×#41×C₄₃	C₂×C₄	1	2	yes	—
34	C₂×#41×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×#41×C₄₃	D₄	1	2	yes	—
36	C₂×#41×C₄₃	D₄	1	2	yes	—
37	C₂²×Q₈×C₄₃	Q₈	1	1	yes	characteristic
38	C₂²×D₄×C₄₃	Q₈	1	1	yes	characteristic
39	C₂²×D₄×C₄₃	Q₈	1	2	yes	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
42	C₂×M₁₆×C₄₃	—	4	2	no	—
43	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₄₃	—	4	2	no	—
46	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂²⋊C₈×C₄₃	C₂×C₄	1	16	yes	—
51	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	C₂⁴×C₄₃	C₂×Q₈	1	2	yes	—
53	#41×C₄₃	—	4	2	no	—
54	#41×C₄₃	—	4	2	no	—
55	#41×C₄₃	—	4	2	no	—
56	#41×C₄₃	—	4	2	no	—
57	#41×C₄₃	—	4	2	no	—
58	#41×C₄₃	—	4	4	no	—
59	#41×C₄₃	—	4	2	no	—
60	C₂×Q₈×C₄₃	—	4	2	no	—
61	C₂×Q₈×C₄₃	C₄⋊C₄	1	2	yes	—
62	C₂×Q₈×C₄₃	C₄⋊C₄	1	2	yes	—
63	C₂×D₄×C₄₃	—	4	4	no	—
64	C₂×D₄×C₄₃	—	4	2	no	—
65	C₂×D₄×C₄₃	C₄⋊C₄	1	2	yes	—
66	C₂×D₄×C₄₃	C₄⋊C₄	1	2	yes	—
67	C₂×D₄×C₄₃	C₄⋊C₄	1	4	yes	—
68	C₂×D₄×C₄₃	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	—	2	2	no	—
71	C₂²×C₄×C₄₃	—	2	4	no	—
72	C₂²×C₄×C₄₃	—	2	4	no	—
73	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₄₃	—	4	2	no	—
83	M₁₆×C₄₃	—	4	2	no	—
84	C₂×C₈×C₄₃	—	2	4	no	—
85	C₂×C₈×C₄₃	—	2	4	no	—
86	C₂×C₈×C₄₃	—	4	2	no	—
87	C₂×C₈×C₄₃	—	4	2	no	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
90	#10621877	—	2	8	no	—
91	#10621877	—	2	8	no	—
92	#10621877	—	4	8	no	—
93	C₂×C₄₃⋊C₈	—	2	8	no	—
94	C₂×C₄₃⋊C₈	—	4	8	no	—
95	C₂³×C₄₃	—	2	2	no	—
96	C₂³×C₄₃	—	2	4	no	—
97	C₂³×C₄₃	—	4	2	no	—
98	C₂³×C₄₃	—	4	4	no	—
99	C₂³×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
103	Q₈×C₄₃	—	4	2	no	—
104	Q₈×C₄₃	—	4	2	no	—
105	D₄×C₄₃	—	4	2	no	—
106	D₄×C₄₃	—	4	2	no	—
107	D₄×C₄₃	—	4	4	no	—
108	D₄×C₄₃	—	4	4	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	4	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	2	4	no	—
118	C₂×C₄×C₄₃	—	2	4	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
128	C₈×C₄₃	—	2	4	no	—
129	C₈×C₄₃	—	2	4	no	—
130	C₄₃⋊C₈	—	4	8	no	—
131	#21950	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	2	1	no	—
134	C₂²×C₄₃	—	2	1	no	—
135	C₂²×C₄₃	—	2	4	no	—
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	2	2	no	—
138	C₂²×C₄₃	—	2	8	no	—
139	C₂²×C₄₃	—	4	4	no	—
140	C₂²×C₄₃	—	8	4	no	—
141	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₃	#543	1	2	yes	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	2	4	no	—
146	C₄×C₄₃	#518	1	1	yes	characteristic
147	C₄×C₄₃	#518	1	1	yes	characteristic
148	C₄×C₄₃	#534	1	2	yes	—
149	#2879	—	43	2	no	—
150	C₂×#408	D₄₃	1	1	yes	characteristic
151	#1855	—	43	8	no	—
152	#1811	—	43	2	no	—
153	#1810	—	43	2	no	—
154	C₂×C₄₃	—	2	2	no	—
155	C₂×C₄₃	—	4	1	no	—
156	C₂×C₄₃	—	4	1	no	—
157	C₂×C₄₃	—	4	4	no	—
158	C₂×C₄₃	#2315	1	1	yes	characteristic
159	C₂×C₄₃	#2916	1	2	yes	—
160	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
161	C₂²×M₁₆	—	43	4	no	—
162	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
163	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
164	C₂×#108	—	43	4	no	—
165	C₂×#108	—	43	4	no	—
166	#408	—	86	4	no	—
167	#408	—	86	4	no	—
168	#408	Dic₄₃	1	4	yes	—
169	#408	Dic₄₃	1	4	yes	—
170	#407	—	43	4	no	—
171	#407	—	43	4	no	—
172	C₂×C₂²⋊C₈	C₂×D₄₃	1	4	yes	—
173	#341	—	43	4	no	—
174	#341	—	43	4	no	—
175	C₄₃	#21950	1	1	yes	characteristic, complemented, Sylow
176	C₂×#41	—	4	2	no	—
177	C₂×#41	—	4	2	no	—
178	C₂×#41	D₁₇₂	1	2	yes	—
179	C₂×#41	D₁₇₂	1	2	yes	—
180	C₂×#41	C₂×Dic₄₃	1	2	yes	—
181	C₂×#41	C₂×Dic₄₃	1	2	yes	—
182	C₂²×Q₈	Dic₈₆	1	1	yes	characteristic
183	C₂²×D₄	Dic₈₆	1	1	yes	characteristic
184	C₂²×D₄	Dic₈₆	1	2	yes	—
185	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
187	C₂×M₁₆	—	4	2	no	—
188	C₂×M₁₆	—	43	8	no	—
189	C₂×M₁₆	—	86	4	no	—
190	C₂×M₁₆	—	172	4	no	—
191	C₂×M₁₆	—	172	4	no	—
192	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
193	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	—	86	4	no	—
196	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
197	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
198	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
199	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
200	#108	—	172	4	no	—
201	#108	—	172	4	no	—
202	C₂²⋊C₈	—	86	4	no	—
203	C₂²⋊C₈	—	86	4	no	—
204	C₂²⋊C₈	—	86	8	no	—
205	C₂²⋊C₈	C₂×Dic₄₃	1	16	yes	—
206	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
207	C₂⁴	C₂×Dic₈₆	1	2	yes	—
208	#41	—	4	2	no	—
209	#41	—	4	4	no	—
210	#41	—	4	2	no	—
211	#41	—	4	2	no	—
212	#41	—	4	2	no	—
213	#41	—	4	2	no	—
214	#41	—	4	2	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	C₄⋊Dic₄₃	1	2	yes	—
217	C₂×Q₈	C₄⋊Dic₄₃	1	2	yes	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	C₄⋊Dic₄₃	1	2	yes	—
221	C₂×D₄	C₄⋊Dic₄₃	1	2	yes	—
222	C₂×D₄	C₄⋊Dic₄₃	1	4	yes	—
223	C₂×D₄	C₄⋊Dic₄₃	1	4	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
230	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₁₇₂	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
234	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
236	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
237	M₁₆	—	4	2	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	86	8	no	—
240	M₁₆	—	86	8	no	—
241	M₁₆	—	172	8	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	86	8	no	—
247	C₂×C₈	—	172	8	no	—
248	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
249	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	4	no	—
254	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
255	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
256	C₂³	C₂×C₄⋊Dic₄₃	1	4	yes	—
257	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
258	Q₈	—	4	2	no	—
259	Q₈	—	4	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
279	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
281	C₂×C₄	#13063370	1	2	yes	—
282	C₂×C₄	#13063370	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	4	no	—
285	C₈	—	172	8	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
296	C₂²	#14583525	1	2	yes	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	4	no	—
300	C₄	#14583374	1	1	yes	characteristic
301	C₄	#14583374	1	1	yes	characteristic
302	C₄	#14583461	1	2	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	4	1	no	—
305	C₂	—	4	1	no	—
306	C₂	—	4	4	no	—
307	C₂	#19137209	1	1	yes	characteristic
308	C₂	#19143206	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#408	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#408	D₄₃
hypocenter	C₄₃	#21950
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂×#408

Complements

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

N	H ≅ G/N	type
2. #19134802	C₂	semidirect
3. #19134637	C₂	semidirect
4. #19134633	C₂	semidirect
175. C₄₃	#21950	semidirect

Maximal subgroups

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

subgroup	index
#19143040	2
#19134802	2
#19134637	2
#19134633	2
C₂×C₄₃×#408	2
#21950	43