Subgroups — #19103251

Order 5103.

758 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#10588065	C₃	1	1	yes	maximal, characteristic
2	#13133595	C₃	1	1	yes	maximal, characteristic, complemented
3	#13133594	C₃	1	1	yes	maximal, characteristic, complemented
4	#13133592	C₃	1	2	yes	maximal, complemented
5	#13133591	C₃	1	2	yes	maximal, complemented
6	#13133586	C₃	1	2	yes	maximal, complemented
7	#13133576	C₃	1	3	yes	maximal, complemented
8	#13133575	C₃	1	3	yes	maximal, complemented
9	#13133572	C₃	1	3	yes	maximal, complemented
10	#13133559	C₃	1	3	yes	maximal, complemented
11	#13133558	C₃	1	3	yes	maximal, complemented
12	#13133547	C₃	1	3	yes	maximal
13	3₋¹⁺²×C₇⋊C₉	C₃	1	3	yes	maximal, complemented
14	#4427⋊C₉	C₃	1	3	yes	maximal, complemented
15	#4428⋊C₉	C₃	1	3	yes	maximal, complemented
16	#13133529	C₃	1	2	yes	maximal, complemented
17	C₉²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
18	C₇×#6900	C₃	1	1	yes	maximal, characteristic, complemented
19	#10624879	—	7	1	no	maximal, Sylow
20	C₃×C₇×3₋¹⁺²	C₃²	1	3	yes	—
21	C₃×C₇×3₋¹⁺²	C₃²	1	1	yes	characteristic
22	C₃×C₇⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
23	C₃×#4430	C₃²	1	1	yes	characteristic
24	C₃²×C₇⋊C₉	C₃²	1	1	yes	characteristic
25	#10588065	—	3	9	no	—
26	C₃×#4428	C₃²	1	3	yes	—
27	C₃×#4428	C₃²	1	1	yes	characteristic
28	C₃×#4428	C₃²	1	1	yes	characteristic
29	C₃×#4428	C₃²	1	1	yes	characteristic
30	C₃×#4427	C₃²	1	1	yes	characteristic
31	C₃×#4427	C₃²	1	3	yes	—
32	C₃×#4427	C₃²	1	1	yes	characteristic
33	C₃×#4427	C₃²	1	1	yes	characteristic
34	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
35	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
36	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	3	yes	—
37	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
38	C₇×C₉⋊C₉	C₃²	1	3	yes	—
39	C₇×C₉⋊C₉	C₃²	1	3	yes	—
40	C₇×C₉⋊C₉	C₃²	1	2	yes	—
41	C₇×C₉²	C₃²	1	1	yes	characteristic
42	C₃²⋊(C₇⋊C₉)	C₃²	1	6	yes	—
43	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	6	yes	—
44	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	6	yes	—
45	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	6	yes	—
46	#10588036	C₃²	1	3	yes	—
47	#10588036	C₃²	1	6	yes	—
48	#10588036	C₃²	1	3	yes	—
49	#10588036	C₃²	1	2	yes	—
50	#10588035	C₃²	1	3	yes	—
51	#10588035	C₃²	1	6	yes	—
52	#10588035	C₃²	1	3	yes	—
53	#10588035	C₃²	1	2	yes	—
54	(C₇⋊C₉)⋊C₉	C₃²	1	6	yes	—
55	(C₇⋊C₉)⋊C₉	C₃²	1	6	yes	—
56	(C₇⋊C₉)⋊C₉	C₃²	1	6	yes	—
57	(C₇⋊C₉)⋊C₉	C₃²	1	6	yes	—
58	(C₇⋊C₉)⋊C₉	C₃²	1	6	yes	—
59	C₉×C₇⋊C₉	C₃²	1	3	yes	—
60	C₉×C₇⋊C₉	C₃²	1	2	yes	—
61	C₉×C₇⋊C₉	C₃²	1	3	yes	—
62	C₉×C₇⋊C₉	C₃²	1	6	yes	—
63	C₃×#737	—	7	1	no	—
64	#6900	—	7	1	no	—
65	#6900	—	7	1	no	—
66	#6900	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
67	#6899	—	7	2	no	—
68	#6899	—	7	2	no	—
69	#6899	—	7	2	no	—
70	#6896	—	7	3	no	—
71	#6896	—	7	3	no	—
72	#6896	—	7	3	no	—
73	#6894	—	7	3	no	—
74	#6894	—	7	3	no	—
75	C₉⋊H(3,𝔽₃)	—	7	3	no	—
76	C₉×3₋¹⁺²	—	7	3	no	—
77	C₉×3₋¹⁺²	—	7	3	no	—
78	C₉×3₋¹⁺²	—	7	3	no	—
79	#6887	—	7	2	no	—
80	C₃×C₉²	—	7	1	no	—
81	C₃³×C₇	C₃³	1	1	yes	characteristic
82	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
83	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
84	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
85	C₇×3₋¹⁺²	—	3	3	no	—
86	C₇×3₋¹⁺²	—	3	9	no	—
87	C₃×C₇×C₉	C₃³	1	6	yes	—
88	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
89	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
90	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
91	C₃×C₇×C₉	C₃³	1	3	yes	—
92	C₇⋊H(3,𝔽₃)	—	3	3	no	—
93	#4430	—	3	3	no	—
94	C₃×C₇⋊C₉	—	3	3	no	—
95	C₃×C₇⋊C₉	C₃³	1	6	yes	—
96	C₃×C₇⋊C₉	C₃³	1	6	yes	—
97	C₃×C₇⋊C₉	C₃³	1	6	yes	—
98	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
99	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
100	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
101	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
102	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
103	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
104	#4428	—	3	3	no	—
105	#4428	—	3	3	no	—
106	#4428	—	3	9	no	—
107	#4428	—	3	3	no	—
108	#4427	—	3	3	no	—
109	#4427	—	3	3	no	—
110	#4427	—	3	9	no	—
111	#4427	—	3	3	no	—
112	C₉×ΓL(1,𝔽₈)	—	3	9	no	—
113	C₉×ΓL(1,𝔽₈)	—	3	3	no	—
114	C₉×ΓL(1,𝔽₈)	—	3	3	no	—
115	C₉×ΓL(1,𝔽₈)	3₋¹⁺²	1	9	yes	complemented
116	#737	—	21	9	no	—
117	C₃×3₋¹⁺²	—	7	1	no	—
118	C₃×3₋¹⁺²	—	7	1	no	—
119	C₃×3₋¹⁺²	—	7	1	no	—
120	C₃×3₋¹⁺²	—	7	1	no	—
121	C₃×3₋¹⁺²	—	7	1	no	—
122	C₃×3₋¹⁺²	—	7	1	no	—
123	C₃×3₋¹⁺²	—	7	1	no	—
124	C₃×3₋¹⁺²	—	7	3	no	—
125	C₃×3₋¹⁺²	—	7	3	no	—
126	C₃×3₋¹⁺²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
127	C₃×3₋¹⁺²	C₃×ΓL(1,𝔽₈)	1	3	yes	—
128	C₃×H(3,𝔽₃)	—	7	1	no	—
129	C₃²×C₉	—	7	1	no	—
130	C₃²×C₉	—	7	1	no	—
131	C₃²×C₉	—	7	1	no	—
132	C₃²×C₉	—	7	1	no	—
133	C₃²×C₉	—	7	3	no	—
134	C₉⋊C₉	—	7	6	no	—
135	C₉⋊C₉	—	7	6	no	—
136	C₉⋊C₉	—	7	6	no	—
137	C₉⋊C₉	—	7	6	no	—
138	C₉⋊C₉	—	7	6	no	—
139	C₉⋊C₉	—	7	2	no	—
140	C₉⋊C₉	—	7	6	no	—
141	C₉⋊C₉	—	7	3	no	—
142	C₉⋊C₉	—	7	3	no	—
143	C₉⋊C₉	—	7	2	no	—
144	C₉⋊C₉	—	7	6	no	—
145	C₉⋊C₉	—	7	3	no	—
146	C₉⋊C₉	—	7	3	no	—
147	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
148	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
149	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	3	yes	—
150	C₃²⋊C₉	—	7	6	no	—
151	C₃²⋊C₉	—	7	6	no	—
152	C₃²⋊C₉	—	7	6	no	—
153	C₃²⋊C₉	—	7	6	no	—
154	C₉²	—	7	3	no	—
155	C₉²	—	7	3	no	—
156	C₉²	—	7	2	no	—
157	C₉²	—	7	6	no	—
158	C₉²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
159	C₃²×C₇	—	3	1	no	—
160	C₃²×C₇	—	3	3	no	—
161	C₃²×C₇	C₃⁴	1	1	yes	characteristic
162	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
163	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
164	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
165	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
166	C₃×ΓL(1,𝔽₈)	—	3	3	no	—
167	C₃×ΓL(1,𝔽₈)	C₃×3₋¹⁺²	1	3	yes	—
168	C₇×C₉	—	3	1	no	—
169	C₇×C₉	—	3	1	no	—
170	C₇×C₉	—	3	6	no	—
171	C₇×C₉	C₃×3₋¹⁺²	1	3	yes	—
172	C₇×C₉	#737	1	9	yes	complemented
173	C₇⋊C₉	—	3	1	no	—
174	C₇⋊C₉	—	3	1	no	—
175	C₇⋊C₉	—	3	1	no	—
176	C₇⋊C₉	—	3	1	no	—
177	C₇⋊C₉	—	3	6	no	—
178	C₇⋊C₉	—	3	6	no	—
179	C₇⋊C₉	—	3	6	no	—
180	C₇⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
181	C₇⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
182	C₃³	—	7	1	no	—
183	C₃³	—	7	1	no	—
184	C₃³	—	7	1	no	—
185	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
186	3₋¹⁺²	—	3	3	no	—
187	3₋¹⁺²	—	3	9	no	—
188	3₋¹⁺²	—	21	3	no	—
189	3₋¹⁺²	—	21	3	no	—
190	3₋¹⁺²	—	21	3	no	—
191	3₋¹⁺²	—	21	3	no	—
192	3₋¹⁺²	—	21	3	no	—
193	3₋¹⁺²	—	21	3	no	—
194	3₋¹⁺²	—	21	3	no	—
195	3₋¹⁺²	—	21	9	no	—
196	3₋¹⁺²	—	21	9	no	—
197	H(3,𝔽₃)	—	21	3	no	—
198	C₃×C₉	—	7	1	no	—
199	C₃×C₉	—	7	1	no	—
200	C₃×C₉	—	7	9	no	—
201	C₃×C₉	—	7	6	no	—
202	C₃×C₉	—	7	6	no	—
203	C₃×C₉	—	7	6	no	—
204	C₃×C₉	—	7	1	no	—
205	C₃×C₉	—	7	1	no	—
206	C₃×C₉	—	7	1	no	—
207	C₃×C₉	—	7	1	no	—
208	C₃×C₉	—	21	3	no	—
209	C₃×C₉	—	21	3	no	—
210	C₃×C₉	—	21	3	no	—
211	C₃×C₉	—	21	9	no	—
212	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	6	yes	—
213	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
214	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
215	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
216	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	3	yes	—
217	C₃×C₇	—	9	1	no	—
218	C₃×C₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
219	C₃×C₇	C₃×#737	1	3	yes	—
220	ΓL(1,𝔽₈)	—	3	3	no	—
221	ΓL(1,𝔽₈)	—	9	1	no	—
222	ΓL(1,𝔽₈)	—	9	1	no	—
223	C₃²	—	3	1	no	—
224	C₃²	—	3	3	no	—
225	C₃²	—	7	3	no	—
226	C₃²	—	21	3	no	—
227	C₃²	—	21	1	no	—
228	C₃²	—	21	1	no	—
229	C₃²	—	21	3	no	—
230	C₃²	—	21	3	no	—
231	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
232	C₉	—	3	1	no	—
233	C₉	—	3	1	no	—
234	C₉	—	3	6	no	—
235	C₉	—	7	3	no	—
236	C₉	—	7	3	no	—
237	C₉	—	21	1	no	—
238	C₉	—	21	1	no	—
239	C₉	—	21	1	no	—
240	C₉	—	21	1	no	—
241	C₉	—	21	6	no	—
242	C₉	—	21	6	no	—
243	C₉	—	21	6	no	—
244	C₉	ΓL(1,𝔽₈)×3₋¹⁺²	1	3	yes	—
245	C₉	#10588065	1	9	yes	complemented
246	C₇	#10624879	1	1	yes	characteristic, complemented, Sylow
247	C₃	—	9	1	no	—
248	C₃	—	21	3	no	—
249	C₃	—	63	1	no	—
250	C₃	—	63	1	no	—
251	C₃	C₃×ΓL(1,𝔽₈)×3₋¹⁺²	1	1	yes	characteristic
252	C₃	C₃×#10588065	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×#6900	C₃
hypercenter	#6900	ΓL(1,𝔽₈)
hypocenter	C₇	#10624879
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
2. #13133595	C₃	semidirect
3. #13133594	C₃	semidirect
4. #13133592	C₃	semidirect
5. #13133591	C₃	semidirect
6. #13133586	C₃	semidirect
7. #13133576	C₃	semidirect
8. #13133575	C₃	semidirect
9. #13133572	C₃	semidirect
10. #13133559	C₃	semidirect
11. #13133558	C₃	semidirect
13. 3₋¹⁺²×C₇⋊C₉	C₃	semidirect
14. #4427⋊C₉	C₃	semidirect
15. #4428⋊C₉	C₃	semidirect
16. #13133529	C₃	semidirect
17. C₉²×ΓL(1,𝔽₈)	C₃	semidirect
18. C₇×#6900	C₃	semidirect
66. #6900	ΓL(1,𝔽₈)	semidirect
115. C₉×ΓL(1,𝔽₈)	3₋¹⁺²	semidirect
172. C₇×C₉	#737	semidirect
245. C₉	#10588065	semidirect
246. C₇	#10624879	semidirect