Subgroups — C₃×#2735 #84599

Order 384.

552 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#397	C₂	1	2	yes	maximal, complemented
4	C₃×#396	C₂	1	2	yes	maximal, complemented
5	C₃×#388	C₂	1	2	yes	maximal, complemented
6	C₃×#388	C₂	1	2	yes	maximal, complemented
7	C₃×#388	C₂	1	2	yes	maximal, complemented
8	C₃×#379	C₂	1	1	yes	maximal, characteristic
9	C₃×#379	C₂	1	2	yes	maximal
10	#2735	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂⁵×C₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	complemented
15	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂²	1	2	yes	complemented
27	C₃×#95	C₂²	1	2	yes	complemented
28	C₃×#95	C₂²	1	1	yes	characteristic, complemented
29	C₃×#95	C₂²	1	2	yes	complemented
30	C₃×#95	C₂²	1	2	yes	complemented
31	C₃×#95	C₂²	1	1	yes	characteristic, complemented
32	#399	C₂×C₃	1	1	yes	characteristic, complemented
33	#397	C₂×C₃	1	1	yes	characteristic, complemented
34	#397	C₂×C₃	1	2	yes	complemented
35	#396	C₂×C₃	1	2	yes	complemented
36	#388	C₂×C₃	1	2	yes	complemented
37	#388	C₂×C₃	1	2	yes	complemented
38	#388	C₂×C₃	1	2	yes	complemented
39	#379	C₂×C₃	1	1	yes	characteristic
40	#379	C₂×C₃	1	2	yes	—
41	C₂⁴×C₃	—	2	2	no	—
42	C₂⁴×C₃	—	2	4	no	—
43	C₂⁴×C₃	—	4	2	no	—
44	C₂⁴×C₃	—	4	1	no	—
45	C₂⁴×C₃	—	4	1	no	—
46	C₂⁴×C₃	C₂³	1	2	yes	—
47	C₂⁴×C₃	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	C₂³	1	2	yes	—
52	C₂²×C₃×C₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	C₂³	1	2	yes	—
54	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
55	C₃×C₄⋊C₄	—	4	1	no	—
56	C₃×C₄⋊C₄	—	4	2	no	—
57	C₃×C₄⋊C₄	—	4	2	no	—
58	C₃×C₄⋊C₄	—	4	1	no	—
59	C₃×C₂²⋊C₄	—	2	4	no	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	—	2	4	no	—
62	C₃×C₂²⋊C₄	—	4	2	no	—
63	C₃×C₂²⋊C₄	—	4	2	no	—
64	C₃×C₂²⋊C₄	—	4	2	no	—
65	C₃×C₂²⋊C₄	—	4	1	no	—
66	C₃×C₂²⋊C₄	—	4	2	no	—
67	C₃×C₂²⋊C₄	—	4	2	no	—
68	C₃×C₂²⋊C₄	—	4	1	no	—
69	C₃×C₄²	—	4	2	no	—
70	C₂⁵	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
72	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
73	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
74	C₂×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
75	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
84	C₂×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄²	C₂²×C₃	1	2	yes	complemented
86	#95	C₂²×C₃	1	2	yes	complemented
87	#95	C₂²×C₃	1	1	yes	characteristic, complemented
88	#95	C₂²×C₃	1	2	yes	complemented
89	#95	C₂²×C₃	1	2	yes	complemented
90	#95	C₂²×C₃	1	1	yes	characteristic, complemented
91	C₂³×C₃	—	2	2	no	—
92	C₂³×C₃	—	2	2	no	—
93	C₂³×C₃	—	2	2	no	—
94	C₂³×C₃	—	2	1	no	—
95	C₂³×C₃	—	2	2	no	—
96	C₂³×C₃	—	2	2	no	—
97	C₂³×C₃	—	2	1	no	—
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	4	8	no	—
102	C₂³×C₃	—	4	4	no	—
103	C₂³×C₃	—	4	8	no	—
104	C₂³×C₃	—	4	4	no	—
105	C₂³×C₃	—	4	4	no	—
106	C₂³×C₃	#41	1	4	yes	—
107	C₂³×C₃	#41	1	2	yes	—
108	C₂³×C₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂⁴	—	2	2	no	—
129	C₂⁴	—	2	4	no	—
130	C₂⁴	—	4	2	no	—
131	C₂⁴	—	4	1	no	—
132	C₂⁴	—	4	1	no	—
133	C₂⁴	C₂³×C₃	1	2	yes	—
134	C₂⁴	C₂³×C₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₄⋊C₄	—	4	1	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₂²⋊C₄	—	4	2	no	—
150	C₂²⋊C₄	—	4	2	no	—
151	C₂²⋊C₄	—	4	2	no	—
152	C₂²⋊C₄	—	4	1	no	—
153	C₂²⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₂²⋊C₄	—	4	1	no	—
156	C₄²	—	4	2	no	—
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	—	2	2	no	—
159	C₂²×C₃	—	2	4	no	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	—	4	16	no	—
163	C₂²×C₃	—	4	2	no	—
164	C₂²×C₃	—	4	2	no	—
165	C₂²×C₃	—	4	2	no	—
166	C₂²×C₃	—	4	2	no	—
167	C₂²×C₃	—	4	1	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	—	4	1	no	—
170	C₂²×C₃	C₂×#41	1	2	yes	—
171	C₂²×C₃	C₂×#41	1	1	yes	characteristic
172	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
174	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
175	C₃×C₄	—	4	2	no	—
176	C₃×C₄	—	4	2	no	—
177	C₃×C₄	—	4	2	no	—
178	C₃×C₄	—	4	2	no	—
179	C₃×C₄	—	4	1	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	1	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	1	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	8	no	—
193	C₂³	—	4	4	no	—
194	C₂³	—	4	8	no	—
195	C₂³	—	4	4	no	—
196	C₂³	—	4	4	no	—
197	C₂³	C₃×#41	1	4	yes	—
198	C₂³	C₃×#41	1	2	yes	—
199	C₂³	C₂⁴×C₃	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₃	—	4	4	no	—
220	C₂×C₃	—	4	2	no	—
221	C₂×C₃	#538	1	2	yes	—
222	C₂×C₃	#538	1	1	yes	characteristic
223	C₂×C₃	#551	1	2	yes	—
224	C₂×C₃	#551	1	1	yes	characteristic
225	C₂×C₃	#560	1	1	yes	characteristic
226	C₂²	—	2	4	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	4	16	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₂×C₃×#41	1	2	yes	—
240	C₂²	C₂×C₃×#41	1	1	yes	characteristic
241	C₂²	C₃×H(5,𝔽₂)	1	2	yes	—
242	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
244	C₄	—	4	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	1	no	—
251	C₃	#2735	1	1	yes	characteristic, direct product factor, Sylow
252	C₂	—	4	4	no	—
253	C₂	—	4	2	no	—
254	C₂	C₃×#538	1	2	yes	—
255	C₂	C₃×#538	1	1	yes	characteristic
256	C₂	C₃×#551	1	2	yes	—
257	C₂	C₃×#551	1	1	yes	characteristic
258	C₂	C₃×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#399	C₂	semidirect
2. C₃×#397	C₂	semidirect
3. C₃×#397	C₂	semidirect
4. C₃×#396	C₂	semidirect
5. C₃×#388	C₂	semidirect
6. C₃×#388	C₂	semidirect
7. C₃×#388	C₂	semidirect
10. #2735	C₃	direct (N × H)
12. C₂×C₃×C₄⋊C₄	C₂²	semidirect
13. C₂×C₃×C₄⋊C₄	C₂²	semidirect
14. C₂×C₃×C₄⋊C₄	C₂²	semidirect
15. C₂×C₃×C₄⋊C₄	C₂²	semidirect
26. C₂×C₃×C₄²	C₂²	semidirect
27. C₃×#95	C₂²	semidirect
28. C₃×#95	C₂²	semidirect
29. C₃×#95	C₂²	semidirect
30. C₃×#95	C₂²	semidirect
31. C₃×#95	C₂²	semidirect
32. #399	C₂×C₃	semidirect
33. #397	C₂×C₃	semidirect
34. #397	C₂×C₃	semidirect
35. #396	C₂×C₃	semidirect
36. #388	C₂×C₃	semidirect
37. #388	C₂×C₃	semidirect
38. #388	C₂×C₃	semidirect
71. C₂×C₄⋊C₄	C₂²×C₃	semidirect
72. C₂×C₄⋊C₄	C₂²×C₃	semidirect
73. C₂×C₄⋊C₄	C₂²×C₃	semidirect
74. C₂×C₄⋊C₄	C₂²×C₃	semidirect
85. C₂×C₄²	C₂²×C₃	semidirect
86. #95	C₂²×C₃	semidirect
87. #95	C₂²×C₃	semidirect
88. #95	C₂²×C₃	semidirect
89. #95	C₂²×C₃	semidirect
90. #95	C₂²×C₃	semidirect
251. C₃	#2735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#399	2
C₃×#397	2
C₃×#396	2
C₃×#388	2
C₃×#379	2
#2735	3

Subgroups — C3×#2735 #84599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2735 #84599