Subgroups — C₃×#81794 #11877023

Order 1152.

777 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5728	C₂	1	2	yes	maximal
2	C₃²×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5189	C₂	1	4	yes	maximal
4	C₃×#5184	C₂	1	4	yes	maximal, complemented
5	C₃×#5079	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
7	C₃×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3185	—	3	1	no	maximal
9	#81794	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×Q₁₆	C₂²	1	4	yes	—
11	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×#946	C₂²	1	4	yes	—
13	C₃×Q₈×Dic₃	C₂²	1	4	yes	—
14	C₂×C₃×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
15	C₃×#897	—	2	2	no	—
16	C₃×#892	—	2	2	no	—
17	C₃×#890	—	2	2	no	—
18	C₃×#887	—	2	2	no	—
19	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
20	C₃×#869	C₂²	1	2	yes	—
21	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₃×#835	C₂²	1	8	yes	—
23	C₃×#817	C₂²	1	4	yes	—
24	C₃×#815	C₂²	1	2	yes	—
25	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₃×#533	—	3	2	no	—
27	#5728	C₂×C₃	1	2	yes	—
28	C₃×#498	—	3	1	no	—
29	C₃×#494	—	2	1	no	—
30	C₃×#494	D₃	1	1	yes	characteristic, complemented
31	C₃×#494	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#487	—	3	2	no	—
33	C₃×#470	—	3	4	no	—
34	C₃×#441	—	3	4	no	—
35	C₃×C₄×M₁₆	—	3	1	no	—
36	#5189	C₂×C₃	1	4	yes	—
37	#5184	C₂×C₃	1	4	yes	complemented
38	#5079	C₂×C₃	1	2	yes	complemented
39	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
40	#4680	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃²×Q₈	C₂³	1	4	yes	—
42	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
43	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
44	C₂×C₃×Dic₆	—	2	2	no	—
45	C₃²×Q₁₆	—	2	8	no	—
46	C₂×C₃²×C₈	C₂³	1	2	yes	—
47	C₃²×C₄⋊C₄	—	2	2	no	—
48	C₃²×C₄⋊C₄	—	2	2	no	—
49	C₃²×C₄²	C₂³	1	1	yes	characteristic
50	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
51	C₃×C₄⋊Dic₃	—	2	2	no	—
52	C₃×C₄⋊Dic₃	—	2	4	no	—
53	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
54	C₃×Dic₃⋊C₄	—	2	4	no	—
55	C₃×C₄×Dic₃	—	2	4	no	—
56	C₃×C₄×Dic₃	D₄	1	2	yes	—
57	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
59	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
60	#3185	—	3	1	no	Sylow
61	C₂×C₃×Q₁₆	—	2	4	no	—
62	C₂×C₃×Q₁₆	C₂×D₃	1	4	yes	—
63	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
64	C₂×C₃×M₁₆	—	3	2	no	—
65	C₃×C₄⋊Q₈	—	2	2	no	—
66	C₃×C₄⋊Q₈	—	3	2	no	—
67	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
68	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
69	C₃×#125	—	6	2	no	—
70	C₃×C₂²⋊Q₈	—	3	4	no	—
71	C₃×C₂²⋊Q₈	—	6	2	no	—
72	C₃×C₄×Q₈	—	3	4	no	—
73	C₃×C₄×Q₈	—	6	2	no	—
74	C₃×#117	—	6	2	no	—
75	C₂×C₃×C₄²	—	3	1	no	—
76	#946	C₂²×C₃	1	4	yes	—
77	Q₈×Dic₃	C₂²×C₃	1	4	yes	—
78	C₂×ΓL(1,𝔽₂₅)	C₂²×C₃	1	2	yes	—
79	#897	—	2	2	no	—
80	#892	—	2	2	no	—
81	#890	—	2	2	no	—
82	#887	—	2	2	no	—
83	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
84	#869	C₂²×C₃	1	2	yes	—
85	C₃×#106	—	3	4	no	—
86	C₃×#103	—	3	8	no	—
87	C₃×#97	—	3	2	no	—
88	C₃×C₄×C₈	—	2	1	no	—
89	C₃×C₄×C₈	—	3	1	no	—
90	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
91	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
92	#835	C₂²×C₃	1	8	yes	—
93	#817	C₂²×C₃	1	4	yes	—
94	#815	C₂²×C₃	1	2	yes	—
95	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
96	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
97	C₃²×Q₈	—	2	8	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
100	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×Dic₃	—	2	4	no	—
102	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
103	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
104	C₃×C₄×D₃	—	2	4	no	—
105	C₃×C₄×D₃	—	2	2	no	—
106	C₃×Dic₆	—	4	4	no	—
107	C₃²×C₈	#41	1	4	yes	—
108	C₃×C₃⋊C₈	—	2	2	no	—
109	#533	—	3	2	no	—
110	#498	—	3	1	no	—
111	#494	C₃×D₃	1	1	yes	characteristic, complemented
112	#487	—	3	2	no	—
113	#470	—	3	4	no	—
114	#441	—	3	4	no	—
115	C₄×M₁₆	—	3	1	no	—
116	C₂×C₃×Q₈	—	2	4	no	—
117	C₂×C₃×Q₈	—	6	2	no	—
118	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
119	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
120	C₂²×C₃×C₄	—	3	2	no	—
121	C₂²×C₃×C₄	—	3	1	no	—
122	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
123	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
124	C₂×Dic₆	—	2	2	no	—
125	C₃×Q₁₆	—	2	8	no	—
126	C₃×Q₁₆	—	2	8	no	—
127	C₃×Q₁₆	—	4	8	no	—
128	C₃×M₁₆	—	6	4	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	3	2	no	—
131	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
132	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	3	4	no	—
138	C₃×C₄⋊C₄	—	4	2	no	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₄⋊C₄	—	6	4	no	—
141	C₃×C₄⋊C₄	—	6	4	no	—
142	C₃×C₄⋊C₄	—	6	2	no	—
143	C₃×C₂²⋊C₄	—	6	4	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	3	2	no	—
146	C₃×C₄²	—	3	1	no	—
147	C₃×C₄²	—	6	4	no	—
148	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
150	(C₂×D₃)⋊C₄	—	2	4	no	—
151	C₄⋊Dic₃	—	2	2	no	—
152	C₄⋊Dic₃	—	2	4	no	—
153	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
154	Dic₃⋊C₄	—	2	4	no	—
155	C₄×Dic₃	—	2	4	no	—
156	C₄×Dic₃	C₃×D₄	1	2	yes	—
157	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
159	ΓL(1,𝔽₂₅)	—	2	4	no	—
160	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₃×D₃	—	2	1	no	—
162	C₂×C₃×D₃	—	2	2	no	—
163	C₃²×C₄	—	4	4	no	—
164	C₃²×C₄	#137	1	4	yes	—
165	C₃²×C₄	C₂×#41	1	2	yes	—
166	C₃×Dic₃	—	2	1	no	—
167	C₃×Dic₃	—	2	2	no	—
168	C₃×Dic₃	—	4	4	no	—
169	C₂×Q₁₆	C₂×C₃×D₃	1	4	yes	—
170	C₂×M₁₆	—	3	2	no	—
171	C₄⋊Q₈	—	3	2	no	—
172	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
173	#125	—	6	2	no	—
174	C₂²⋊Q₈	—	3	4	no	—
175	C₂²⋊Q₈	—	6	2	no	—
176	C₄×Q₈	—	3	4	no	—
177	C₄×Q₈	—	6	2	no	—
178	#117	—	6	2	no	—
179	C₂×C₄²	—	3	1	no	—
180	#106	—	3	4	no	—
181	#103	—	3	8	no	—
182	#97	—	3	2	no	—
183	C₄×C₈	—	3	1	no	—
184	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
185	C₂³×C₃	—	3	1	no	—
186	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
187	C₃×Q₈	—	2	8	no	—
188	C₃×Q₈	—	2	8	no	—
189	C₃×Q₈	—	4	8	no	—
190	C₃×Q₈	—	12	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	3	2	no	—
196	C₂×C₃×C₄	—	3	1	no	—
197	C₂×C₃×C₄	—	4	4	no	—
198	C₂×C₃×C₄	—	6	4	no	—
199	C₂×C₃×C₄	—	6	2	no	—
200	C₂×C₃×C₄	—	6	4	no	—
201	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
202	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
203	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
205	C₂×Dic₃	—	2	4	no	—
206	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
207	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
208	C₄×D₃	—	2	4	no	—
209	C₄×D₃	—	2	2	no	—
210	Dic₆	—	4	4	no	—
211	C₃×C₈	—	2	4	no	—
212	C₃×C₈	—	6	2	no	—
213	C₃×C₈	#237	1	4	yes	—
214	C₃×C₈	C₃×#41	1	4	yes	—
215	C₃⋊C₈	—	2	2	no	—
216	C₂×C₃²	#532	1	1	yes	characteristic
217	C₂×C₃²	C₂×#137	1	2	yes	—
218	C₃×D₃	—	4	1	no	—
219	C₂×Q₈	—	6	2	no	—
220	C₂×Q₈	C₂²×C₃×D₃	1	4	yes	—
221	C₂²×C₄	—	3	1	no	—
222	C₂²×C₄	—	3	2	no	—
223	Q₁₆	—	2	8	no	—
224	M₁₆	—	6	4	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	3	4	no	—
230	C₄⋊C₄	—	6	4	no	—
231	C₄⋊C₄	—	6	4	no	—
232	C₄⋊C₄	—	6	2	no	—
233	C₂²⋊C₄	—	6	4	no	—
234	C₄²	—	3	1	no	—
235	C₄²	—	3	2	no	—
236	C₄²	—	6	4	no	—
237	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	6	1	no	—
240	C₂²×C₃	—	6	2	no	—
241	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
242	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
243	C₂×D₃	—	2	2	no	—
244	C₂×D₃	—	2	1	no	—
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	—	4	4	no	—
249	C₃×C₄	—	6	1	no	—
250	C₃×C₄	—	6	2	no	—
251	C₃×C₄	—	8	4	no	—
252	C₃×C₄	—	12	4	no	—
253	C₃×C₄	#918	1	4	yes	—
254	C₃×C₄	C₃×#137	1	4	yes	—
255	C₃×C₄	C₂×#237	1	2	yes	—
256	C₃×C₄	C₂×C₃×#41	1	2	yes	—
257	Dic₃	—	2	2	no	—
258	Dic₃	—	2	1	no	—
259	Dic₃	—	4	4	no	—
260	C₃²	#3185	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	3	1	no	—
262	Q₈	—	2	8	no	—
263	Q₈	—	12	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	3	2	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	6	2	no	—
269	C₂×C₄	—	6	4	no	—
270	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
271	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
272	C₈	—	6	2	no	—
273	C₈	C₃×#237	1	4	yes	—
274	C₂×C₃	—	2	2	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	12	1	no	—
277	C₂×C₃	#5716	1	1	yes	characteristic
278	C₂×C₃	C₂×#918	1	2	yes	—
279	C₂×C₃	C₃×#532	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃×#137	1	2	yes	—
281	D₃	—	4	1	no	—
282	C₂²	—	6	2	no	—
283	C₂²	—	6	1	no	—
284	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
285	C₄	—	4	4	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	1	no	—
288	C₄	—	12	4	no	—
289	C₄	C₃×#918	1	4	yes	—
290	C₄	C₂×C₃×#237	1	2	yes	—
291	C₃	—	2	1	no	—
292	C₃	#81794	1	1	yes	characteristic, direct product factor
293	C₃	C₃×#3185	1	1	yes	characteristic, complemented
294	C₂	—	12	1	no	—
295	C₂	C₃×#5716	1	1	yes	characteristic
296	C₂	C₂×C₃×#918	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#494	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×C₃×D₃	C₂×D₄
hypercenter	C₃×#494	D₃
hypocenter	C₃	C₃×#3185
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#494

Complements

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

N	H ≅ G/N	type
2. C₃²×#494	C₂	semidirect
4. C₃×#5184	C₂	semidirect
5. C₃×#5079	C₂	semidirect
7. C₃×#4680	C₂	semidirect
9. #81794	C₃	direct (N × H)
30. C₃×#494	D₃	semidirect
31. C₃×#494	C₂×C₃	semidirect
37. #5184	C₂×C₃	semidirect
38. #5079	C₂×C₃	semidirect
40. #4680	C₂×C₃	semidirect
111. #494	C₃×D₃	semidirect
260. C₃²	#3185	semidirect
292. C₃	#81794	direct (N × H)
293. C₃	C₃×#3185	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5728	2
C₃²×#494	2
C₃×#5189	2
C₃×#5184	2
C₃×#5079	2
C₃×C₄×ΓL(1,𝔽₂₅)	2
C₃×#4680	2
C₃×#3185	3
#81794	3

Subgroups — C3×#81794 #11877023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#81794 #11877023