Subgroups — #24995751

Order 11008.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145181	C₂	1	1	yes	maximal, characteristic
2	#19135538	C₂	1	2	yes	maximal
3	#19133093	C₂	1	2	yes	maximal
4	C₈×#10621877	C₂	1	1	yes	maximal, characteristic
5	C₄×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	2	yes	—
7	#14583304	C₄	1	1	yes	characteristic, complemented
8	#14583279	C₄	1	1	yes	characteristic, complemented
9	#14583250	C₄	1	2	yes	complemented
10	#14583140	—	2	1	no	—
11	#14583140	—	2	1	no	—
12	C₄×Dic₁₇₂	—	2	1	no	—
13	C₄×Dic₁₇₂	—	2	1	no	—
14	#14582952	C₂²	1	1	yes	characteristic
15	C₄×#10621877	C₂²	1	1	yes	characteristic
16	#14582659	—	2	4	no	—
17	#14582635	—	2	4	no	—
18	#14582621	—	2	1	no	—
19	C₈×C₄₃⋊C₈	—	2	1	no	—
20	C₂×#107×C₄₃	—	2	1	no	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
23	C₄₃×#340	—	2	1	no	—
24	C₂×Dic₁₇₂	—	2	1	no	—
25	C₂×Dic₁₇₂	—	2	1	no	—
26	C₂×#10621877	D₄	1	2	yes	—
27	#13063325	C₂×C₄	1	2	yes	—
28	#13063325	C₂×C₄	1	2	yes	—
29	#13063317	—	2	2	no	—
30	#13063304	—	2	2	no	—
31	#13063304	—	2	2	no	—
32	C₄×Dic₈₆	—	2	2	no	—
33	C₄×Dic₈₆	—	2	2	no	—
34	#13063284	—	2	2	no	—
35	#13063259	—	2	1	no	—
36	#13063259	—	2	1	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
38	#13063250	—	2	2	no	—
39	#13063250	—	2	2	no	—
40	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
41	#13063236	—	2	1	no	—
42	C₄×C₄₃⋊C₈	—	2	1	no	—
43	C₄×C₄₃⋊C₈	—	2	1	no	—
44	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
47	C₂×C₄²×C₄₃	—	2	2	no	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	#107×C₄₃	—	2	4	no	—
50	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
54	#95×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	4	2	no	—
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₈₆	—	2	2	no	—
60	C₂×Dic₈₆	—	2	2	no	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	—	4	1	no	—
63	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₄₃	—	2	4	no	—
68	C₄⋊C₄×C₄₃	—	4	2	no	—
69	C₄²×C₄₃	—	4	4	no	—
70	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₃	#41	1	1	yes	characteristic
73	C₄²×C₄₃	#41	1	1	yes	characteristic
74	C₂²⋊Dic₄₃	—	4	2	no	—
75	C₄⋊Dic₄₃	—	2	2	no	—
76	C₄⋊Dic₄₃	—	2	2	no	—
77	C₄⋊Dic₄₃	—	2	2	no	—
78	C₄⋊Dic₄₃	—	2	2	no	—
79	Dic₄₃⋊C₄	—	4	2	no	—
80	Dic₄₃⋊C₄	—	4	2	no	—
81	C₄×Dic₄₃	—	4	2	no	—
82	#10621877	—	2	2	no	—
83	#10621877	Q₁₆	1	4	yes	complemented
84	C₂×C₄₃⋊C₈	—	2	2	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	Dic₁₇₂	—	4	2	no	—
87	Dic₁₇₂	—	4	2	no	—
88	C₂³×C₄₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₄₃	—	2	1	no	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	4	1	no	—
92	C₂×C₄×C₄₃	—	4	4	no	—
93	C₂×C₄×C₄₃	—	8	2	no	—
94	C₂×C₄×C₄₃	—	8	2	no	—
95	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₄₃	#135	1	2	yes	—
103	C₂×Dic₄₃	—	4	2	no	—
104	C₂×Dic₄₃	—	4	2	no	—
105	Dic₈₆	—	4	2	no	—
106	Dic₈₆	—	4	2	no	—
107	Dic₈₆	—	8	2	no	—
108	C₈×C₄₃	—	2	1	no	—
109	C₈×C₄₃	—	2	1	no	—
110	C₈×C₄₃	—	2	1	no	—
111	C₈×C₄₃	#104	1	2	yes	—
112	C₄₃⋊C₈	—	2	4	no	—
113	C₄₃⋊C₈	—	4	2	no	—
114	#9718	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	—	2	1	no	—
117	C₂²×C₄₃	#390	1	1	yes	characteristic
118	C₂²×C₄₃	#494	1	1	yes	characteristic
119	C₂²×C₄₃	#495	1	1	yes	characteristic
120	C₄×C₄₃	—	2	1	no	—
121	C₄×C₄₃	—	2	1	no	—
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	8	4	no	—
124	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
125	C₄×C₄₃	#421	1	1	yes	characteristic
126	C₄×C₄₃	C₄×Q₁₆	1	2	yes	—
127	C₄×C₄₃	#467	1	2	yes	—
128	Dic₄₃	—	8	2	no	—
129	Dic₄₃	—	8	2	no	—
130	#3170	—	43	1	no	—
131	#1986	—	43	2	no	—
132	C₄×#107	D₄₃	1	1	yes	characteristic
133	#1503	—	43	2	no	—
134	C₈×M₁₆	—	43	1	no	—
135	C₂×C₄₃	—	2	1	no	—
136	C₂×C₄₃	—	2	1	no	—
137	C₂×C₄₃	#1969	1	1	yes	characteristic
138	C₂×C₄₃	#1976	1	1	yes	characteristic
139	C₂×C₄₃	#1980	1	1	yes	characteristic
140	#533	—	43	2	no	—
141	#500	—	43	1	no	—
142	#494	—	43	1	no	—
143	#487	—	43	2	no	—
144	#467	—	86	1	no	—
145	#467	—	86	1	no	—
146	C₄×Q₁₆	—	86	1	no	—
147	C₄×Q₁₆	—	86	1	no	—
148	#431	—	43	1	no	—
149	C₂×#107	—	2	1	no	—
150	C₄×M₁₆	—	43	1	no	—
151	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
153	#340	—	2	1	no	—
154	#339	—	86	4	no	—
155	#328	—	86	4	no	—
156	#322	—	86	1	no	—
157	C₈²	—	86	1	no	—
158	C₄₃	#9718	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₁₆	—	86	1	no	—
160	C₂×Q₁₆	—	86	1	no	—
161	C₂×M₁₆	—	43	2	no	—
162	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	43	2	no	—
165	C₄⋊Q₈	—	43	2	no	—
166	#125	—	86	2	no	—
167	C₂²⋊Q₈	—	86	2	no	—
168	C₂²⋊Q₈	—	86	2	no	—
169	C₄×Q₈	—	86	2	no	—
170	C₄×Q₈	—	86	2	no	—
171	#117	—	86	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	—	86	1	no	—
177	#107	—	86	1	no	—
178	C₄⋊C₈	—	86	1	no	—
179	#103	—	86	2	no	—
180	#103	—	86	2	no	—
181	C₂²⋊C₈	—	43	2	no	—
182	#97	—	86	1	no	—
183	C₄×C₈	—	86	1	no	—
184	C₄×C₈	—	86	1	no	—
185	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
186	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
187	C₄×C₈	D₁₇₂	1	1	yes	characteristic
188	C₄×C₈	D₁₇₂	1	1	yes	characteristic
189	#95	—	2	2	no	—
190	C₂×Q₈	—	86	2	no	—
191	C₂×Q₈	—	86	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
194	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
196	Q₁₆	—	172	2	no	—
197	Q₁₆	—	172	2	no	—
198	M₁₆	—	43	4	no	—
199	M₁₆	—	86	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	86	2	no	—
203	C₂×C₈	—	86	2	no	—
204	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	86	2	no	—
211	C₄⋊C₄	—	86	2	no	—
212	C₄⋊C₄	—	86	2	no	—
213	C₄⋊C₄	—	86	2	no	—
214	C₄⋊C₄	—	172	2	no	—
215	C₄⋊C₄	—	172	2	no	—
216	C₂²⋊C₄	—	172	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	—	172	2	no	—
219	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
220	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
221	C₄²	#10621902	1	1	yes	characteristic
222	C₄²	#10621902	1	1	yes	characteristic
223	C₂³	#13063321	1	1	yes	characteristic
224	Q₈	—	344	2	no	—
225	Q₈	—	172	2	no	—
226	Q₈	—	172	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	—	172	2	no	—
235	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
236	C₂×C₄	#13063287	1	1	yes	characteristic
237	C₂×C₄	#13063287	1	1	yes	characteristic
238	C₂×C₄	#13063300	1	1	yes	characteristic
239	C₂×C₄	#13063300	1	1	yes	characteristic
240	C₂×C₄	#13063314	1	1	yes	characteristic
241	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
242	C₂×C₄	#13063344	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	86	4	no	—
247	C₈	—	172	2	no	—
248	C₈	#13063253	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#14582800	1	1	yes	characteristic
252	C₂²	#14583277	1	1	yes	characteristic
253	C₂²	#14583282	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	4	no	—
258	C₄	—	344	2	no	—
259	C₄	—	344	2	no	—
260	C₄	C₂×#13063253	1	1	yes	characteristic
261	C₄	#14582917	1	1	yes	characteristic
262	C₄	#14582993	1	2	yes	—
263	C₄	#14583141	1	2	yes	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#19135454	1	1	yes	characteristic
267	C₂	#19135491	1	1	yes	characteristic
268	C₂	#19135513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063300
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#107×C₄₃	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	#127
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#107	D₄₃
hypocenter	C₄₃	#9718
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. #14583304	C₄	semidirect
8. #14583279	C₄	semidirect
9. #14583250	C₄	semidirect
83. #10621877	Q₁₆	semidirect
158. C₄₃	#9718	semidirect

Maximal subgroups

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

subgroup	index
#19145181	2
#19135538	2
#19133093	2
C₈×#10621877	2
C₄×#107×C₄₃	2
#9718	43