Subgroups — C₃×#90278 #13060930

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#120	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊Dic₇	C₂	1	2	yes	maximal, complemented
3	C₃×#6499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6498	C₂	1	2	yes	maximal, complemented
6	C₃×D₄×Dic₇	C₂	1	2	yes	maximal, complemented
7	C₃×#6442	C₂	1	2	yes	maximal, complemented
8	C₃×#6441	C₂	1	2	yes	maximal, complemented
9	C₃×#6440	C₂	1	2	yes	maximal, complemented
10	#90278	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
13	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
21	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	complemented
25	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
26	C₃×C₄⋊Dic₇	C₂²	1	2	yes	complemented
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
30	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₄×Dic₇	C₂²	1	2	yes	complemented
32	C₇×#120	C₂×C₃	1	1	yes	characteristic
33	C₂×C₂²⋊Dic₇	C₂×C₃	1	2	yes	complemented
34	#6499	C₂×C₃	1	1	yes	characteristic, complemented
35	#6498	C₂×C₃	1	1	yes	characteristic, complemented
36	#6498	C₂×C₃	1	2	yes	complemented
37	D₄×Dic₇	C₂×C₃	1	2	yes	complemented
38	#6442	C₂×C₃	1	2	yes	complemented
39	#6441	C₂×C₃	1	2	yes	complemented
40	#6440	C₂×C₃	1	2	yes	complemented
41	C₃×#551	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	—	2	4	no	—
43	C₂³×C₃×C₇	—	4	1	no	—
44	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
45	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
46	C₂³×C₃×C₇	C₂³	1	2	yes	—
47	C₃×C₇×D₄	—	2	4	no	—
48	C₃×C₇×D₄	—	4	1	no	—
49	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×Dic₇	—	2	4	no	—
52	C₂×C₃×Dic₇	—	4	2	no	—
53	C₂×C₃×Dic₇	—	4	1	no	—
54	C₂×C₃×Dic₇	C₂³	1	2	yes	—
55	C₂×C₃×Dic₇	C₂³	1	2	yes	—
56	C₂×C₃×Dic₇	C₂³	1	2	yes	—
57	C₂×C₃×Dic₇	C₂³	1	2	yes	—
58	C₃×Dic₁₄	—	4	1	no	—
59	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
61	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
63	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
64	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic, complemented
66	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
67	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
69	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
70	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
71	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
72	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	complemented
73	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
75	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
76	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
77	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
78	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
79	C₄×Dic₇	C₂²×C₃	1	2	yes	complemented
80	C₃×#126	—	7	2	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#123	—	7	2	no	—
83	C₃×#123	—	7	1	no	—
84	C₃×C₂²⋊Q₈	—	7	2	no	—
85	C₃×#120	D₇	1	1	yes	characteristic
86	C₃×C₄×D₄	—	7	2	no	—
87	C₃×#117	—	7	2	no	—
88	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
89	C₂²×C₃×C₇	—	2	2	no	—
90	C₂²×C₃×C₇	—	2	2	no	—
91	C₂²×C₃×C₇	—	2	2	no	—
92	C₂²×C₃×C₇	—	2	2	no	—
93	C₂²×C₃×C₇	—	2	1	no	—
94	C₂²×C₃×C₇	—	2	1	no	—
95	C₂²×C₃×C₇	—	4	4	no	—
96	C₂²×C₃×C₇	#41	1	4	yes	—
97	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
98	C₃×C₄×C₇	—	2	2	no	—
99	C₃×C₄×C₇	—	2	1	no	—
100	C₃×Dic₇	—	2	2	no	—
101	C₃×Dic₇	—	2	2	no	—
102	C₃×Dic₇	—	2	2	no	—
103	C₃×Dic₇	—	2	2	no	—
104	#551	—	7	1	no	Sylow
105	C₂³×C₇	—	2	4	no	—
106	C₂³×C₇	—	4	1	no	—
107	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
108	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
109	C₂³×C₇	C₂³×C₃	1	2	yes	—
110	C₇×D₄	—	2	4	no	—
111	C₇×D₄	—	4	1	no	—
112	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
113	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
114	C₂×Dic₇	—	2	4	no	—
115	C₂×Dic₇	—	4	2	no	—
116	C₂×Dic₇	—	4	1	no	—
117	C₂×Dic₇	C₂³×C₃	1	2	yes	—
118	C₂×Dic₇	C₂³×C₃	1	2	yes	—
119	C₂×Dic₇	C₂³×C₃	1	2	yes	—
120	C₂×Dic₇	C₂³×C₃	1	2	yes	—
121	Dic₁₄	—	4	1	no	—
122	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
123	C₂×C₃×Q₈	—	7	1	no	—
124	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
125	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₂²×C₃×C₄	—	7	1	no	—
129	C₃×C₄⋊C₄	—	7	2	no	—
130	C₃×C₄⋊C₄	—	7	2	no	—
131	C₃×C₄⋊C₄	—	7	2	no	—
132	C₃×C₄⋊C₄	—	7	2	no	—
133	C₃×C₂²⋊C₄	—	7	2	no	—
134	C₃×C₂²⋊C₄	—	7	2	no	—
135	C₃×C₂²⋊C₄	—	7	2	no	—
136	C₃×C₂²⋊C₄	—	7	2	no	—
137	C₃×C₂²⋊C₄	—	7	2	no	—
138	C₃×C₂²⋊C₄	—	7	1	no	—
139	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
140	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
141	C₃×C₄²	—	7	1	no	—
142	C₃×C₄²	—	7	2	no	—
143	C₂×C₃×C₇	—	2	4	no	—
144	C₂×C₃×C₇	—	4	1	no	—
145	C₂×C₃×C₇	—	4	1	no	—
146	C₂×C₃×C₇	C₂×#41	1	2	yes	—
147	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
148	#126	—	7	2	no	—
149	#124	—	7	1	no	—
150	#123	—	7	2	no	—
151	#123	—	7	1	no	—
152	C₂²⋊Q₈	—	7	2	no	—
153	#120	C₃×D₇	1	1	yes	characteristic
154	C₄×D₄	—	7	2	no	—
155	#117	—	7	2	no	—
156	C₂×C₂²⋊C₄	—	7	2	no	—
157	C₂²×C₇	—	2	2	no	—
158	C₂²×C₇	—	2	2	no	—
159	C₂²×C₇	—	2	2	no	—
160	C₂²×C₇	—	2	2	no	—
161	C₂²×C₇	—	2	1	no	—
162	C₂²×C₇	—	2	1	no	—
163	C₂²×C₇	—	4	4	no	—
164	C₂²×C₇	C₃×#41	1	4	yes	—
165	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
166	C₄×C₇	—	2	2	no	—
167	C₄×C₇	—	2	1	no	—
168	Dic₇	—	2	2	no	—
169	Dic₇	—	2	2	no	—
170	Dic₇	—	2	2	no	—
171	Dic₇	—	2	2	no	—
172	C₂³×C₃	—	2	4	no	—
173	C₂³×C₃	—	4	1	no	—
174	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
175	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
176	C₂³×C₃	C₂²×D₇	1	2	yes	—
177	C₃×Q₈	—	28	1	no	—
178	C₃×D₄	—	2	4	no	—
179	C₃×D₄	—	4	1	no	—
180	C₂×C₃×C₄	—	7	2	no	—
181	C₂×C₃×C₄	—	7	2	no	—
182	C₂×C₃×C₄	—	7	2	no	—
183	C₂×C₃×C₄	—	7	2	no	—
184	C₂×C₃×C₄	—	14	4	no	—
185	C₂×C₃×C₄	—	28	1	no	—
186	C₂×C₃×C₄	—	28	2	no	—
187	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
188	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
189	C₃×C₇	#551	1	1	yes	characteristic, complemented, Hall
190	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic
191	C₂×Q₈	—	7	1	no	—
192	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
193	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
194	C₂²×C₄	—	7	2	no	—
195	C₂²×C₄	—	7	2	no	—
196	C₂²×C₄	—	7	1	no	—
197	C₄⋊C₄	—	7	2	no	—
198	C₄⋊C₄	—	7	2	no	—
199	C₄⋊C₄	—	7	2	no	—
200	C₄⋊C₄	—	7	2	no	—
201	C₂²⋊C₄	—	7	2	no	—
202	C₂²⋊C₄	—	7	2	no	—
203	C₂²⋊C₄	—	7	2	no	—
204	C₂²⋊C₄	—	7	2	no	—
205	C₂²⋊C₄	—	7	2	no	—
206	C₂²⋊C₄	—	7	1	no	—
207	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
208	C₂²⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
209	C₄²	—	7	1	no	—
210	C₄²	—	7	2	no	—
211	C₂×C₇	—	2	4	no	—
212	C₂×C₇	—	4	1	no	—
213	C₂×C₇	—	4	1	no	—
214	C₂×C₇	C₂×C₃×#41	1	2	yes	—
215	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	4	4	no	—
223	C₂²×C₃	#1159	1	4	yes	—
224	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	14	2	no	—
228	C₃×C₄	—	14	2	no	—
229	C₃×C₄	—	14	2	no	—
230	C₃×C₄	—	14	2	no	—
231	C₂³	—	2	4	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
234	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
235	C₂³	C₂²×C₃×D₇	1	2	yes	—
236	Q₈	—	28	1	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	1	no	—
239	C₂×C₄	—	7	2	no	—
240	C₂×C₄	—	7	2	no	—
241	C₂×C₄	—	7	2	no	—
242	C₂×C₄	—	7	2	no	—
243	C₂×C₄	—	14	4	no	—
244	C₂×C₄	—	28	1	no	—
245	C₂×C₄	—	28	2	no	—
246	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
247	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
248	C₇	C₃×#551	1	1	yes	characteristic, complemented, Sylow
249	C₂×C₃	—	2	4	no	—
250	C₂×C₃	—	4	1	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	C₂×#1159	1	2	yes	—
253	C₂×C₃	#6548	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	4	4	no	—
261	C₂²	C₃×#1159	1	4	yes	—
262	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	1	no	—
265	C₄	—	14	2	no	—
266	C₄	—	14	2	no	—
267	C₄	—	14	2	no	—
268	C₄	—	14	2	no	—
269	C₃	#90278	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	—	2	4	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	1	no	—
273	C₂	C₂×C₃×#1159	1	2	yes	—
274	C₂	C₃×#6548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting, prime radical	C₃×C₇×#120	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#120	D₇
hypocenter	C₇	C₃×#551
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#120

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₂²⋊Dic₇	C₂	semidirect
3. C₃×#6499	C₂	semidirect
4. C₃×#6498	C₂	semidirect
5. C₃×#6498	C₂	semidirect
6. C₃×D₄×Dic₇	C₂	semidirect
7. C₃×#6442	C₂	semidirect
8. C₃×#6441	C₂	semidirect
9. C₃×#6440	C₂	semidirect
10. #90278	C₃	direct (N × H)
16. C₂²×C₃×Dic₇	C₂²	semidirect
17. C₂×C₃×Dic₁₄	C₂²	semidirect
24. C₃×C₂²⋊Dic₇	C₂²	semidirect
25. C₃×C₂²⋊Dic₇	C₂²	semidirect
26. C₃×C₄⋊Dic₇	C₂²	semidirect
27. C₃×Dic₇⋊C₄	C₂²	semidirect
28. C₃×Dic₇⋊C₄	C₂²	semidirect
29. C₃×Dic₇⋊C₄	C₂²	semidirect
30. C₃×C₄×Dic₇	C₂²	semidirect
31. C₃×C₄×Dic₇	C₂²	semidirect
33. C₂×C₂²⋊Dic₇	C₂×C₃	semidirect
34. #6499	C₂×C₃	semidirect
35. #6498	C₂×C₃	semidirect
36. #6498	C₂×C₃	semidirect
37. D₄×Dic₇	C₂×C₃	semidirect
38. #6442	C₂×C₃	semidirect
39. #6441	C₂×C₃	semidirect
40. #6440	C₂×C₃	semidirect
64. C₂²×Dic₇	C₂²×C₃	semidirect
65. C₂×Dic₁₄	C₂²×C₃	semidirect
72. C₂²⋊Dic₇	C₂²×C₃	semidirect
73. C₂²⋊Dic₇	C₂²×C₃	semidirect
74. C₄⋊Dic₇	C₂²×C₃	semidirect
75. Dic₇⋊C₄	C₂²×C₃	semidirect
76. Dic₇⋊C₄	C₂²×C₃	semidirect
77. Dic₇⋊C₄	C₂²×C₃	semidirect
78. C₄×Dic₇	C₂²×C₃	semidirect
79. C₄×Dic₇	C₂²×C₃	semidirect
189. C₃×C₇	#551	semidirect
248. C₇	C₃×#551	semidirect
269. C₃	#90278	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×C₇×#120	2
C₂×C₃×C₂²⋊Dic₇	2
C₃×#6499	2
C₃×#6498	2
C₃×D₄×Dic₇	2
C₃×#6442	2
C₃×#6441	2
C₃×#6440	2
#90278	3
C₃×#551	7

Subgroups — C3×#90278 #13060930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90278 #13060930