Subgroups — #25196560

Order 11008.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145119	C₂	1	1	yes	maximal, characteristic, complemented
2	#19135767	C₂	1	4	yes	maximal, complemented
3	#19135494	C₂	1	2	yes	maximal
4	#19134720	C₂	1	2	yes	maximal, complemented
5	#19134649	C₂	1	2	yes	maximal, complemented
6	#19134641	C₂	1	2	yes	maximal, complemented
7	#19134479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#444	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	#14583308	—	2	4	no	—
11	C₂×#13063261	C₂²	1	1	yes	characteristic
12	#14582937	C₂²	1	2	yes	—
13	#14582937	C₂²	1	1	yes	characteristic
14	#14582937	C₂²	1	2	yes	—
15	#14582916	C₂²	1	4	yes	—
16	#14582873	C₂²	1	4	yes	—
17	#14582854	C₂²	1	2	yes	—
18	#14582671	C₂²	1	4	yes	—
19	#14582659	C₂²	1	2	yes	—
20	#14582656	C₂²	1	4	yes	—
21	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#131×C₄₃	C₂²	1	2	yes	—
23	C₄₃×#444	—	2	4	no	—
24	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#104×C₄₃	C₂²	1	2	yes	—
26	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621877	C₂³	1	2	yes	—
28	#13063325	—	2	2	no	—
29	#13063317	—	2	2	no	—
30	#13063304	—	4	2	no	—
31	C₄×Dic₈₆	—	4	2	no	—
32	#13063284	C₂³	1	2	yes	—
33	#13063284	C₂³	1	4	yes	—
34	#13063261	—	2	2	no	—
35	#13063259	—	2	2	no	—
36	#13063259	—	2	2	no	—
37	#13063259	—	2	1	no	—
38	#13063257	—	2	2	no	—
39	#13063257	—	2	2	no	—
40	#13063257	—	2	1	no	—
41	#13063252	—	4	4	no	—
42	#13063240	—	4	4	no	—
43	#13063235	—	2	2	no	—
44	C₈×Dic₄₃	—	2	2	no	—
45	C₂×#41×C₄₃	C₂³	1	2	yes	—
46	#135×C₄₃	—	2	2	no	—
47	#135×C₄₃	—	2	2	no	—
48	C₂×Q₁₆×C₄₃	Q₈	1	1	yes	characteristic, complemented
49	C₂×SD₁₆×C₄₃	Q₈	1	2	yes	complemented
50	C₂×D₈×C₄₃	Q₈	1	1	yes	characteristic, complemented
51	#131×C₄₃	—	2	2	no	—
52	#131×C₄₃	—	2	2	no	—
53	#131×C₄₃	—	2	2	no	—
54	#131×C₄₃	—	2	2	no	—
55	C₂×M₁₆×C₄₃	—	2	2	no	—
56	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₃	—	2	2	no	—
58	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
61	#108×C₄₃	—	2	2	no	—
62	#104×C₄₃	—	2	4	no	—
63	#97×C₄₃	D₄	1	4	yes	—
64	#41×C₄₃	—	2	2	no	—
65	#41×C₄₃	—	2	2	no	—
66	#41×C₄₃	—	2	2	no	—
67	#41×C₄₃	—	2	2	no	—
68	C₂×Q₈×C₄₃	C₂×Q₈	1	2	yes	—
69	C₂×D₄×C₄₃	—	2	2	no	—
70	C₂×D₄×C₄₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₃	—	2	2	no	—
72	C₂²×C₄×C₄₃	—	2	1	no	—
73	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₈₆	—	4	2	no	—
75	Q₁₆×C₄₃	—	2	2	no	—
76	SD₁₆×C₄₃	—	2	4	no	—
77	D₈×C₄₃	—	2	2	no	—
78	M₁₆×C₄₃	—	2	2	no	—
79	M₁₆×C₄₃	—	2	2	no	—
80	M₁₆×C₄₃	—	4	2	no	—
81	M₁₆×C₄₃	—	4	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	4	2	no	—
84	C₂×C₈×C₄₃	—	4	2	no	—
85	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₄₃	#41	1	2	yes	—
90	C₂×C₈×C₄₃	#41	1	2	yes	—
91	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
93	C₄²×C₄₃	—	2	1	no	—
94	C₄²×C₄₃	C₂×D₄	1	2	yes	—
95	C₂²⋊Dic₄₃	—	2	2	no	—
96	C₂²⋊Dic₄₃	—	2	4	no	—
97	C₄⋊Dic₄₃	—	2	4	no	—
98	C₄⋊Dic₄₃	—	2	2	no	—
99	C₄⋊Dic₄₃	—	4	2	no	—
100	Dic₄₃⋊C₄	—	4	4	no	—
101	C₄×Dic₄₃	—	2	2	no	—
102	C₄×Dic₄₃	—	2	4	no	—
103	#10621877	—	2	2	no	—
104	#10621877	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
108	Q₈×C₄₃	—	2	2	no	—
109	Q₈×C₄₃	—	2	2	no	—
110	D₄×C₄₃	—	2	2	no	—
111	D₄×C₄₃	—	2	2	no	—
112	D₄×C₄₃	—	4	2	no	—
113	D₄×C₄₃	—	4	2	no	—
114	C₂×C₄×C₄₃	—	2	2	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	8	1	no	—
119	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×Dic₄₃	—	4	4	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	Dic₈₆	—	8	4	no	—
128	C₈×C₄₃	—	4	2	no	—
129	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
131	C₈×C₄₃	#124	1	2	yes	—
132	C₈×C₄₃	#124	1	2	yes	—
133	C₄₃⋊C₈	—	4	2	no	—
134	#29155	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	—	4	1	no	—
136	C₂²×C₄₃	—	4	2	no	—
137	C₂²×C₄₃	—	8	2	no	—
138	C₂²×C₄₃	#535	1	1	yes	characteristic
139	C₂²×C₄₃	#554	1	1	yes	characteristic
140	C₂²×C₄₃	#555	1	1	yes	characteristic
141	C₄×C₄₃	—	4	2	no	—
142	C₄×C₄₃	—	4	2	no	—
143	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
145	C₄×C₄₃	#554	1	1	yes	characteristic
146	C₄×C₄₃	#555	1	1	yes	characteristic
147	Dic₄₃	—	8	4	no	—
148	Dic₄₃	—	8	2	no	—
149	#3162	—	43	1	no	—
150	C₂×#444	D₄₃	1	1	yes	characteristic
151	#2031	—	43	4	no	—
152	#1976	—	43	2	no	—
153	#1833	—	43	2	no	—
154	#1814	—	43	2	no	—
155	#1812	—	43	2	no	—
156	#1773	—	43	1	no	—
157	C₂×C₄₃	—	2	1	no	—
158	C₂×C₄₃	—	2	1	no	—
159	C₂×C₄₃	—	2	1	no	—
160	C₂×C₄₃	—	8	2	no	—
161	C₂×C₄₃	#2563	1	1	yes	characteristic
162	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
163	C₂×#131	C₂×D₄₃	1	2	yes	—
164	#533	—	43	2	no	—
165	#501	—	86	4	no	—
166	#444	—	2	4	no	—
167	C₂×#108	—	43	1	no	—
168	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
169	#427	—	43	1	no	—
170	#427	—	43	2	no	—
171	#427	—	43	2	no	—
172	#421	—	43	4	no	—
173	C₂×#104	C₂×D₄₃	1	2	yes	—
174	#410	—	43	4	no	—
175	#405	—	43	2	no	—
176	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
177	#344	—	43	4	no	—
178	#339	—	43	2	no	—
179	H(3,ℤ/4)	—	43	4	no	—
180	C₄₃	#29155	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	C₂²×D₄₃	1	2	yes	—
182	#135	—	2	2	no	—
183	#135	—	2	2	no	—
184	C₂×Q₁₆	Dic₈₆	1	1	yes	characteristic, complemented
185	C₂×SD₁₆	Dic₈₆	1	2	yes	complemented
186	C₂×D₈	Dic₈₆	1	1	yes	characteristic, complemented
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	43	2	no	—
193	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	86	2	no	—
198	#125	—	86	2	no	—
199	C₂²⋊Q₈	—	172	2	no	—
200	C₄×Q₈	—	172	2	no	—
201	#117	—	43	4	no	—
202	#117	—	43	2	no	—
203	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	#108	—	2	2	no	—
205	#108	—	86	2	no	—
206	#107	—	86	1	no	—
207	#107	—	86	2	no	—
208	#107	—	86	2	no	—
209	#106	—	86	1	no	—
210	#106	—	86	2	no	—
211	#106	—	86	2	no	—
212	#104	—	2	4	no	—
213	#104	—	172	4	no	—
214	#99	—	172	4	no	—
215	#97	—	86	2	no	—
216	#97	C₂²⋊D₄₃	1	4	yes	—
217	C₄×C₈	—	86	2	no	—
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	—	172	2	no	—
223	C₂×Q₈	C₂×Dic₈₆	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×Dic₈₆	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	2	4	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	86	2	no	—
237	M₁₆	—	86	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	86	2	no	—
242	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
243	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
244	C₂×C₈	#10621898	1	2	yes	—
245	C₂×C₈	#10621900	1	1	yes	characteristic
246	C₂×C₈	#10621900	1	1	yes	characteristic
247	C₂×C₈	#10621900	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
250	C₄⋊C₄	—	86	4	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	172	4	no	—
253	C₄⋊C₄	—	172	2	no	—
254	C₂²⋊C₄	—	86	2	no	—
255	C₂²⋊C₄	—	86	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	86	2	no	—
258	C₄²	—	86	4	no	—
259	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	#13063371	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	344	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	172	2	no	—
275	C₂×C₄	—	172	4	no	—
276	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
278	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
279	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
280	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
281	C₂×C₄	#13063371	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	172	2	no	—
284	C₈	#13063304	1	2	yes	—
285	C₈	#13063304	1	2	yes	—
286	C₈	#13063312	1	2	yes	—
287	C₈	#13063312	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	#14583463	1	1	yes	characteristic
292	C₂²	#14583606	1	1	yes	characteristic
293	C₂²	#14583614	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	344	4	no	—
297	C₄	—	344	2	no	—
298	C₄	C₂×#13063304	1	1	yes	characteristic
299	C₄	C₂×#13063312	1	1	yes	characteristic
300	C₄	#14583606	1	1	yes	characteristic
301	C₄	#14583614	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	#19139768	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583614
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#444	C₂
socle	C₂×C₄₃	#2563
prime radical	C₂×#135×C₄₃	C₂²
hypercenter	C₂×#444	D₄₃
hypocenter	C₄₃	#29155
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#444

Complements

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

N	H ≅ G/N	type
1. #19145119	C₂	semidirect
2. #19135767	C₂	semidirect
4. #19134720	C₂	semidirect
5. #19134649	C₂	semidirect
6. #19134641	C₂	semidirect
7. #19134479	C₂	semidirect
48. C₂×Q₁₆×C₄₃	Q₈	semidirect
49. C₂×SD₁₆×C₄₃	Q₈	semidirect
50. C₂×D₈×C₄₃	Q₈	semidirect
180. C₄₃	#29155	semidirect
184. C₂×Q₁₆	Dic₈₆	semidirect
185. C₂×SD₁₆	Dic₈₆	semidirect
186. C₂×D₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145119	2
#19135767	2
#19135494	2
#19134720	2
#19134649	2
#19134641	2
#19134479	2
C₂×C₄₃×#444	2
#29155	43