Subgroups — #19103229

Order 5103.

662 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133590	C₃	1	2	yes	maximal, complemented
2	#13133589	C₃	1	2	yes	maximal, complemented
3	#13133585	C₃	1	2	yes	maximal, complemented
4	#13133583	C₃	1	2	yes	maximal, complemented
5	#13133581	C₃	1	2	yes	maximal, complemented
6	#13133580	C₃	1	2	yes	maximal, complemented
7	#13133577	C₃	1	2	yes	maximal, complemented
8	#13133574	C₃	1	2	yes	maximal, complemented
9	#13133571	C₃	1	1	yes	maximal, characteristic, complemented
10	#13133569	C₃	1	2	yes	maximal, complemented
11	#13133562	C₃	1	2	yes	maximal, complemented
12	#13133561	C₃	1	2	yes	maximal, complemented
13	#13133555	C₃	1	1	yes	maximal, characteristic
14	#13133549	C₃	1	2	yes	maximal
15	#13133540	C₃	1	2	yes	maximal, complemented
16	#4427⋊C₉	C₃	1	2	yes	maximal, complemented
17	#4427⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
18	#4428⋊C₉	C₃	1	2	yes	maximal, complemented
19	#4428⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
20	#13133531	C₃	1	1	yes	maximal, characteristic
21	#13133529	C₃	1	2	yes	maximal, complemented
22	#13133527	C₃	1	1	yes	maximal, characteristic, complemented
23	C₉²×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
24	C₇×#6898	C₃	1	1	yes	maximal, characteristic, complemented
25	#10624877	—	7	1	no	maximal, Sylow
26	C₃×C₇⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
27	C₃×#4430	C₃²	1	2	yes	—
28	C₃²×C₇⋊C₉	C₃²	1	1	yes	characteristic
29	C₃×#4428	C₃²	1	2	yes	—
30	C₃×#4428	C₃²	1	1	yes	characteristic
31	C₃×#4427	C₃²	1	2	yes	—
32	C₃×#4427	C₃²	1	1	yes	characteristic
33	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
34	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
35	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	2	yes	—
36	C₃×C₉×ΓL(1,𝔽₈)	C₃²	1	2	yes	—
37	C₇×C₉⋊C₉	C₃²	1	2	yes	—
38	C₇×C₉⋊C₉	C₃²	1	2	yes	—
39	C₇×C₉⋊C₉	C₃²	1	2	yes	—
40	C₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
41	C₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
42	C₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
43	C₇×C₃²⋊C₉	C₃²	1	2	yes	—
44	C₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
45	C₇×C₉²	C₃²	1	1	yes	characteristic
46	C₃²⋊(C₇⋊C₉)	C₃²	1	1	yes	characteristic
47	C₃²⋊(C₇⋊C₉)	C₃²	1	2	yes	—
48	C₃²⋊(C₇⋊C₉)	C₃²	1	2	yes	—
49	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
50	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
51	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
52	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
53	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
54	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
55	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
56	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
57	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
58	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
59	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
60	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃²	1	2	yes	—
61	#10588036	C₃²	1	2	yes	—
62	#10588036	C₃²	1	1	yes	characteristic
63	#10588036	C₃²	1	2	yes	—
64	#10588035	C₃²	1	2	yes	—
65	#10588035	C₃²	1	1	yes	characteristic
66	#10588035	C₃²	1	2	yes	—
67	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
68	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
69	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
70	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
71	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
72	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
73	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
74	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
75	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
76	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
77	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
78	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
79	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
80	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
81	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
82	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
83	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
84	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
85	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
86	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
87	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
88	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
89	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
90	(C₇⋊C₉)⋊C₉	C₃²	1	2	yes	—
91	C₉×C₇⋊C₉	C₃²	1	1	yes	characteristic
92	C₉×C₇⋊C₉	C₃²	1	2	yes	—
93	C₉×C₇⋊C₉	C₃²	1	2	yes	—
94	C₉×C₇⋊C₉	C₃²	1	1	yes	characteristic
95	C₉×C₇⋊C₉	C₃²	1	2	yes	—
96	C₉×C₇⋊C₉	C₃²	1	1	yes	characteristic
97	C₉×C₇⋊C₉	C₃²	1	1	yes	characteristic
98	C₉×C₇⋊C₉	C₃²	1	2	yes	—
99	C₉×C₇⋊C₉	C₃²	1	2	yes	—
100	#6899	—	7	2	no	—
101	#6899	—	7	2	no	—
102	#6899	—	7	2	no	—
103	#6898	—	7	2	no	—
104	#6898	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
105	#6897	—	7	2	no	—
106	#6897	—	7	2	no	—
107	#6897	—	7	2	no	—
108	#6896	—	7	2	no	—
109	#6896	—	7	1	no	—
110	#6895	—	7	2	no	—
111	#6894	—	7	2	no	—
112	#6894	—	7	2	no	—
113	#6893	—	7	1	no	—
114	C₉⋊H(3,𝔽₃)	—	7	2	no	—
115	C₉×3₋¹⁺²	—	7	2	no	—
116	C₉×3₋¹⁺²	—	7	1	no	—
117	C₉×3₋¹⁺²	—	7	2	no	—
118	C₉×3₋¹⁺²	—	7	1	no	—
119	C₉×3₋¹⁺²	—	7	2	no	—
120	C₉×H(3,𝔽₃)	—	7	1	no	—
121	#6887	—	7	2	no	—
122	#6887	—	7	1	no	—
123	C₃×C₉²	—	7	1	no	—
124	C₃³×C₇	C₃³	1	1	yes	characteristic
125	C₃²×ΓL(1,𝔽₈)	C₃³	1	2	yes	—
126	C₃²×ΓL(1,𝔽₈)	C₃³	1	1	yes	characteristic
127	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
128	C₃×C₇×C₉	C₃³	1	2	yes	—
129	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
130	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
131	C₃×C₇×C₉	C₃³	1	2	yes	—
132	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
133	C₃×C₇×C₉	C₃³	1	2	yes	—
134	C₃×C₇×C₉	C₃³	1	2	yes	—
135	C₇⋊H(3,𝔽₃)	—	3	3	no	—
136	#4430	—	3	6	no	—
137	C₃×C₇⋊C₉	—	3	3	no	—
138	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
139	C₃×C₇⋊C₉	C₃³	1	2	yes	—
140	C₃×C₇⋊C₉	C₃³	1	2	yes	—
141	C₃×C₇⋊C₉	C₃³	1	2	yes	—
142	C₃×C₇⋊C₉	C₃³	1	2	yes	—
143	C₃×C₇⋊C₉	C₃³	1	2	yes	—
144	C₃×C₇⋊C₉	C₃³	1	2	yes	—
145	C₃×C₇⋊C₉	C₃³	1	2	yes	—
146	C₃×C₇⋊C₉	C₃³	1	2	yes	—
147	C₃×C₇⋊C₉	C₃³	1	2	yes	—
148	C₃×C₇⋊C₉	C₃³	1	2	yes	—
149	C₃×C₇⋊C₉	C₃³	1	1	yes	characteristic
150	C₃×C₇⋊C₉	C₃³	1	2	yes	—
151	#4428	—	3	3	no	—
152	#4428	—	3	6	no	—
153	#4427	—	3	3	no	—
154	#4427	—	3	6	no	—
155	C₉×ΓL(1,𝔽₈)	—	3	3	no	—
156	C₉×ΓL(1,𝔽₈)	—	3	3	no	—
157	C₉×ΓL(1,𝔽₈)	—	3	6	no	—
158	C₉×ΓL(1,𝔽₈)	—	3	6	no	—
159	C₃×3₋¹⁺²	—	7	2	no	—
160	C₃×3₋¹⁺²	—	7	2	no	—
161	C₃×3₋¹⁺²	—	7	1	no	—
162	C₃×3₋¹⁺²	—	7	2	no	—
163	C₃×3₋¹⁺²	—	7	1	no	—
164	C₃×H(3,𝔽₃)	—	7	1	no	—
165	C₃²×C₉	—	7	1	no	—
166	C₃²×C₉	—	7	1	no	—
167	C₃²×C₉	—	7	2	no	—
168	C₃²×C₉	—	7	1	no	—
169	C₃²×C₉	—	7	2	no	—
170	C₉⋊C₉	—	7	2	no	—
171	C₉⋊C₉	—	7	2	no	—
172	C₉⋊C₉	—	7	2	no	—
173	C₉⋊C₉	—	7	2	no	—
174	C₉⋊C₉	—	7	2	no	—
175	C₉⋊C₉	—	7	2	no	—
176	C₉⋊C₉	—	7	2	no	—
177	C₉⋊C₉	—	7	2	no	—
178	C₉⋊C₉	—	7	2	no	—
179	C₉⋊C₉	—	7	2	no	—
180	C₉⋊C₉	—	7	2	no	—
181	C₉⋊C₉	—	7	2	no	—
182	C₉⋊C₉	—	7	2	no	—
183	C₉⋊C₉	—	7	2	no	—
184	C₉⋊C₉	—	7	2	no	—
185	C₉⋊C₉	—	7	2	no	—
186	C₉⋊C₉	—	7	2	no	—
187	C₉⋊C₉	—	7	2	no	—
188	C₉⋊C₉	—	7	2	no	—
189	C₉⋊C₉	—	7	2	no	—
190	C₉⋊C₉	—	7	2	no	—
191	C₉⋊C₉	—	7	2	no	—
192	C₉⋊C₉	—	7	2	no	—
193	C₉⋊C₉	—	7	2	no	—
194	C₉⋊C₉	—	7	2	no	—
195	C₉⋊C₉	—	7	2	no	—
196	C₉⋊C₉	—	7	1	no	—
197	C₉⋊C₉	—	7	2	no	—
198	C₉⋊C₉	—	7	2	no	—
199	C₉⋊C₉	—	7	1	no	—
200	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
201	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
202	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
203	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
204	C₉⋊C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
205	C₃²⋊C₉	—	7	2	no	—
206	C₃²⋊C₉	—	7	1	no	—
207	C₃²⋊C₉	—	7	2	no	—
208	C₃²⋊C₉	—	7	2	no	—
209	C₃²⋊C₉	—	7	2	no	—
210	C₃²⋊C₉	—	7	2	no	—
211	C₃²⋊C₉	—	7	2	no	—
212	C₃²⋊C₉	—	7	2	no	—
213	C₃²⋊C₉	—	7	2	no	—
214	C₃²⋊C₉	—	7	2	no	—
215	C₃²⋊C₉	—	7	2	no	—
216	C₃²⋊C₉	—	7	2	no	—
217	C₃²⋊C₉	—	7	2	no	—
218	C₃²⋊C₉	—	7	2	no	—
219	C₃²⋊C₉	—	7	2	no	—
220	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
221	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
222	C₃²⋊C₉	C₃×ΓL(1,𝔽₈)	1	2	yes	—
223	C₉²	—	7	1	no	—
224	C₉²	—	7	2	no	—
225	C₉²	—	7	1	no	—
226	C₉²	—	7	2	no	—
227	C₉²	—	7	2	no	—
228	C₉²	—	7	1	no	—
229	C₉²	—	7	1	no	—
230	C₉²	—	7	2	no	—
231	C₉²	—	7	2	no	—
232	C₉²	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
233	C₃²×C₇	—	3	1	no	—
234	C₃²×C₇	—	3	2	no	—
235	C₃²×C₇	—	3	1	no	—
236	C₃²×C₇	C₃⁴	1	1	yes	characteristic
237	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
238	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
239	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
240	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
241	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
242	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
243	C₃×ΓL(1,𝔽₈)	#737	1	3	yes	—
244	C₇×C₉	—	3	1	no	—
245	C₇×C₉	—	3	1	no	—
246	C₇×C₉	—	3	2	no	—
247	C₇×C₉	—	3	2	no	—
248	C₇×C₉	—	3	1	no	—
249	C₇×C₉	—	3	2	no	—
250	C₇×C₉	—	3	1	no	—
251	C₇×C₉	—	3	2	no	—
252	C₇⋊C₉	—	3	2	no	—
253	C₇⋊C₉	—	3	1	no	—
254	C₇⋊C₉	—	3	2	no	—
255	C₇⋊C₉	—	3	2	no	—
256	C₇⋊C₉	—	3	2	no	—
257	C₇⋊C₉	—	3	2	no	—
258	C₇⋊C₉	—	3	2	no	—
259	C₇⋊C₉	—	3	2	no	—
260	C₇⋊C₉	—	3	2	no	—
261	C₇⋊C₉	—	3	2	no	—
262	C₇⋊C₉	—	3	2	no	—
263	C₇⋊C₉	#737	1	3	yes	—
264	C₇⋊C₉	#737	1	6	yes	—
265	C₃³	—	7	1	no	—
266	C₃³	—	7	2	no	—
267	C₃³	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
268	3₋¹⁺²	—	21	6	no	—
269	3₋¹⁺²	—	21	6	no	—
270	3₋¹⁺²	—	21	3	no	—
271	3₋¹⁺²	—	21	3	no	—
272	3₋¹⁺²	—	21	6	no	—
273	H(3,𝔽₃)	—	21	3	no	—
274	C₃×C₉	—	7	1	no	—
275	C₃×C₉	—	7	2	no	—
276	C₃×C₉	—	7	2	no	—
277	C₃×C₉	—	7	2	no	—
278	C₃×C₉	—	7	1	no	—
279	C₃×C₉	—	7	2	no	—
280	C₃×C₉	—	7	2	no	—
281	C₃×C₉	—	7	2	no	—
282	C₃×C₉	—	7	2	no	—
283	C₃×C₉	—	7	2	no	—
284	C₃×C₉	—	7	2	no	—
285	C₃×C₉	—	7	2	no	—
286	C₃×C₉	—	7	2	no	—
287	C₃×C₉	—	21	3	no	—
288	C₃×C₉	—	21	6	no	—
289	C₃×C₉	—	21	6	no	—
290	C₃×C₉	—	21	3	no	—
291	C₃×C₉	—	21	3	no	—
292	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
293	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	2	yes	—
294	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
295	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	2	yes	—
296	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
297	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	2	yes	—
298	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
299	C₃×C₉	C₃²×ΓL(1,𝔽₈)	1	2	yes	—
300	C₃×C₇	—	9	1	no	—
301	C₃×C₇	C₃×#737	1	1	yes	characteristic
302	C₃×C₇	C₃×#737	1	1	yes	characteristic
303	C₃×C₇	C₃×#737	1	2	yes	—
304	ΓL(1,𝔽₈)	—	3	3	no	—
305	ΓL(1,𝔽₈)	—	9	2	no	—
306	C₃²	—	3	1	no	—
307	C₃²	—	3	1	no	—
308	C₃²	—	3	2	no	—
309	C₃²	—	7	3	no	—
310	C₃²	—	21	1	no	—
311	C₃²	—	21	2	no	—
312	C₃²	—	21	2	no	—
313	C₃²	—	21	2	no	—
314	C₃²	—	21	2	no	—
315	C₃²	—	21	2	no	—
316	C₃²	C₃³×ΓL(1,𝔽₈)	1	1	yes	characteristic
317	C₉	—	3	1	no	—
318	C₉	—	3	2	no	—
319	C₉	—	3	2	no	—
320	C₉	—	3	1	no	—
321	C₉	—	3	2	no	—
322	C₉	—	3	2	no	—
323	C₉	—	3	1	no	—
324	C₉	—	3	1	no	—
325	C₉	—	7	3	no	—
326	C₉	—	7	6	no	—
327	C₉	—	21	1	no	—
328	C₉	—	21	2	no	—
329	C₉	—	21	2	no	—
330	C₉	—	21	2	no	—
331	C₉	—	21	2	no	—
332	C₉	—	21	2	no	—
333	C₉	—	21	2	no	—
334	C₉	—	21	2	no	—
335	C₉	—	21	2	no	—
336	C₉	—	21	2	no	—
337	C₉	—	21	2	no	—
338	C₇	#10624877	1	1	yes	characteristic, complemented, Sylow
339	C₃	—	9	1	no	—
340	C₃	—	21	3	no	—
341	C₃	—	63	2	no	—
342	C₃	ΓL(1,𝔽₈)×#737	1	1	yes	characteristic
343	C₃	ΓL(1,𝔽₈)×#737	1	2	yes	—
344	C₃	ΓL(1,𝔽₈)×#737	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×ΓL(1,𝔽₈)
commutator, socle	C₃²×C₇	C₃⁴
Fitting	C₇×#6898	C₃
hypercenter	#6898	ΓL(1,𝔽₈)
hypocenter	C₇	#10624877
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ #6898

Complements

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

N	H ≅ G/N	type
1. #13133590	C₃	semidirect
2. #13133589	C₃	semidirect
3. #13133585	C₃	semidirect
4. #13133583	C₃	semidirect
5. #13133581	C₃	semidirect
6. #13133580	C₃	semidirect
7. #13133577	C₃	semidirect
8. #13133574	C₃	semidirect
9. #13133571	C₃	semidirect
10. #13133569	C₃	semidirect
11. #13133562	C₃	semidirect
12. #13133561	C₃	semidirect
15. #13133540	C₃	semidirect
16. #4427⋊C₉	C₃	semidirect
17. #4427⋊C₉	C₃	semidirect
18. #4428⋊C₉	C₃	semidirect
19. #4428⋊C₉	C₃	semidirect
21. #13133529	C₃	semidirect
22. #13133527	C₃	semidirect
23. C₉²×ΓL(1,𝔽₈)	C₃	semidirect
24. C₇×#6898	C₃	semidirect
104. #6898	ΓL(1,𝔽₈)	semidirect
338. C₇	#10624877	semidirect