Subgroups — C₃×#90359 #13061019

Order 1344.

620 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#1161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1159	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6507	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6503	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6447	C₂	1	4	yes	maximal, complemented
7	C₃×(C₂²⋊D₇)⋊C₄	C₂	1	4	yes	maximal, complemented
8	C₃×#6438	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄²×D₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#90359	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊D₇	C₂²	1	4	yes	—
14	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
16	C₃×#1161	—	2	4	no	—
17	C₃×#1159	—	2	4	no	—
18	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic
19	C₂×C₃×D₂₈	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
21	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
23	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
24	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
26	C₃×(C₂×D₇)⋊C₄	C₂²	1	4	yes	—
27	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
28	C₃×C₄×Dic₇	C₂²	1	2	yes	—
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₂×#1161	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#1159	C₂×C₃	1	1	yes	characteristic, complemented
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6507	C₂×C₃	1	1	yes	characteristic, complemented
34	#6503	C₂×C₃	1	1	yes	characteristic, complemented
35	#6447	C₂×C₃	1	4	yes	complemented
36	(C₂²⋊D₇)⋊C₄	C₂×C₃	1	4	yes	complemented
37	#6438	C₂×C₃	1	1	yes	characteristic, complemented
38	C₄²×D₇	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#532	—	7	1	no	maximal, Hall
40	C₂³×C₃×C₇	C₂³	1	2	yes	—
41	C₂²×C₃×D₇	C₂³	1	2	yes	—
42	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
43	C₃×C₇×Q₈	—	2	2	no	—
44	C₃×C₇×D₄	—	2	2	no	—
45	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₃×C₂²⋊D₇	—	2	4	no	—
49	C₃×C₂²⋊D₇	—	2	8	no	—
50	C₂×C₃×Dic₇	—	2	4	no	—
51	C₂×C₃×Dic₇	C₂³	1	2	yes	—
52	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
53	C₂×C₃×Dic₇	C₂³	1	2	yes	—
54	C₃×D₂₈	—	2	4	no	—
55	C₃×D₂₈	—	2	2	no	—
56	C₃×C₄×D₇	—	2	4	no	—
57	C₃×C₄×D₇	—	2	4	no	—
58	C₃×C₄×D₇	D₄	1	4	yes	complemented
59	C₃×Dic₁₄	—	2	2	no	—
60	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₂²⋊D₇	C₂²×C₃	1	4	yes	—
63	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
64	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
65	#1161	—	2	4	no	—
66	#1159	—	2	4	no	—
67	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×D₂₈	C₂²×C₃	1	2	yes	—
69	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
70	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
71	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
72	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
73	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
74	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
75	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
76	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
77	C₄×Dic₇	C₂²×C₃	1	2	yes	—
78	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×#41	—	7	1	no	—
80	C₂×C₃×#41	—	7	1	no	—
81	C₃×C₄⋊Q₈	—	7	1	no	—
82	C₃×#127	—	7	1	no	—
83	C₃×#124	—	7	1	no	—
84	C₃×#124	D₇	1	1	yes	characteristic, complemented
85	C₃×#121	—	7	4	no	—
86	C₃×C₄×D₄	—	7	4	no	—
87	C₂×C₃×C₄²	—	7	1	no	—
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	4	no	—
90	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
91	C₂×C₃×D₇	—	2	2	no	—
92	C₂×C₃×D₇	—	2	2	no	—
93	C₂×C₃×D₇	—	2	4	no	—
94	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
95	C₃×C₄×C₇	—	2	2	no	—
96	C₃×C₄×C₇	—	2	2	no	—
97	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
98	C₃×Dic₇	—	2	2	no	—
99	C₃×Dic₇	C₂×D₄	1	2	yes	—
100	C₃×Dic₇	#41	1	4	yes	complemented
101	#532	—	7	1	no	Sylow
102	C₂³×C₇	C₂³×C₃	1	2	yes	—
103	C₂²×D₇	C₂³×C₃	1	2	yes	—
104	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
105	C₇×Q₈	—	2	2	no	—
106	C₇×D₄	—	2	2	no	—
107	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
108	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
109	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
110	C₂²⋊D₇	—	2	4	no	—
111	C₂²⋊D₇	—	2	8	no	—
112	C₂×Dic₇	—	2	4	no	—
113	C₂×Dic₇	C₂³×C₃	1	2	yes	—
114	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
115	C₂×Dic₇	C₂³×C₃	1	2	yes	—
116	D₂₈	—	2	4	no	—
117	D₂₈	—	2	2	no	—
118	C₄×D₇	—	2	4	no	—
119	C₄×D₇	—	2	4	no	—
120	C₄×D₇	C₃×D₄	1	4	yes	complemented
121	Dic₁₄	—	2	2	no	—
122	C₃×#41	—	14	4	no	—
123	C₃×#41	—	14	4	no	—
124	C₂×C₃×Q₈	—	7	1	no	—
125	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
126	C₂×C₃×D₄	—	7	4	no	—
127	C₂×C₃×D₄	—	7	2	no	—
128	C₂×C₃×D₄	—	7	2	no	—
129	C₂×C₃×D₄	—	7	1	no	—
130	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
131	C₂²×C₃×C₄	—	7	1	no	—
132	C₂²×C₃×C₄	—	7	2	no	—
133	C₂²×C₃×C₄	—	7	2	no	—
134	C₂²×C₃×C₄	—	7	2	no	—
135	C₃×C₄⋊C₄	—	7	4	no	—
136	C₃×C₂²⋊C₄	—	7	4	no	—
137	C₃×C₂²⋊C₄	C₂×D₇	1	4	yes	—
138	C₃×C₄²	—	7	2	no	—
139	C₃×C₄²	—	7	1	no	—
140	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
141	C₂×C₃×C₇	—	4	2	no	—
142	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₃×C₇	C₂×#41	1	2	yes	—
144	C₃×D₇	—	2	2	no	—
145	C₃×D₇	—	4	2	no	—
146	C₂×#41	—	7	1	no	—
147	C₂×#41	—	7	1	no	—
148	C₄⋊Q₈	—	7	1	no	—
149	#127	—	7	1	no	—
150	#124	—	7	1	no	—
151	#124	C₃×D₇	1	1	yes	characteristic, complemented
152	#121	—	7	4	no	—
153	C₄×D₄	—	7	4	no	—
154	C₂×C₄²	—	7	1	no	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	4	no	—
157	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
158	C₂×D₇	—	2	2	no	—
159	C₂×D₇	—	2	2	no	—
160	C₂×D₇	—	2	4	no	—
161	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
162	C₄×C₇	—	2	2	no	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
165	Dic₇	—	2	2	no	—
166	Dic₇	C₂×C₃×D₄	1	2	yes	—
167	Dic₇	C₃×#41	1	4	yes	complemented
168	C₂³×C₃	—	7	1	no	—
169	C₂³×C₃	—	7	2	no	—
170	C₂³×C₃	C₂²×D₇	1	2	yes	—
171	C₃×Q₈	—	2	2	no	—
172	C₃×Q₈	—	14	2	no	—
173	C₃×D₄	—	2	2	no	—
174	C₃×D₄	—	14	4	no	—
175	C₃×D₄	—	14	8	no	—
176	C₃×D₄	—	14	4	no	—
177	C₃×D₄	—	14	2	no	—
178	C₂×C₃×C₄	—	7	2	no	—
179	C₂×C₃×C₄	—	7	1	no	—
180	C₂×C₃×C₄	—	7	4	no	—
181	C₂×C₃×C₄	—	7	2	no	—
182	C₂×C₃×C₄	—	14	4	no	—
183	C₂×C₃×C₄	—	14	4	no	—
184	C₂×C₃×C₄	—	14	4	no	—
185	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
187	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
188	C₃×C₇	#532	1	1	yes	characteristic, complemented, Hall
189	#41	—	14	4	no	—
190	#41	—	14	4	no	—
191	C₂×Q₈	—	7	1	no	—
192	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
193	C₂×D₄	—	7	4	no	—
194	C₂×D₄	—	7	2	no	—
195	C₂×D₄	—	7	2	no	—
196	C₂×D₄	—	7	1	no	—
197	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
198	C₂²×C₄	—	7	1	no	—
199	C₂²×C₄	—	7	2	no	—
200	C₂²×C₄	—	7	2	no	—
201	C₂²×C₄	—	7	2	no	—
202	C₄⋊C₄	—	7	4	no	—
203	C₂²⋊C₄	—	7	4	no	—
204	C₂²⋊C₄	C₂×C₃×D₇	1	4	yes	—
205	C₄²	—	7	2	no	—
206	C₄²	—	7	1	no	—
207	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
208	C₂×C₇	—	4	2	no	—
209	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
210	C₂×C₇	C₂×C₃×#41	1	2	yes	—
211	D₇	—	2	2	no	—
212	D₇	—	4	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	7	2	no	—
216	C₂²×C₃	—	14	2	no	—
217	C₂²×C₃	—	14	2	no	—
218	C₂²×C₃	—	14	4	no	—
219	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	7	2	no	—
223	C₃×C₄	—	7	4	no	—
224	C₃×C₄	—	14	2	no	—
225	C₃×C₄	D₄×D₇	1	2	yes	—
226	C₂³	—	7	1	no	—
227	C₂³	—	7	2	no	—
228	C₂³	C₂²×C₃×D₇	1	2	yes	—
229	Q₈	—	2	2	no	—
230	Q₈	—	14	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	14	4	no	—
233	D₄	—	14	8	no	—
234	D₄	—	14	4	no	—
235	D₄	—	14	2	no	—
236	C₂×C₄	—	7	2	no	—
237	C₂×C₄	—	7	1	no	—
238	C₂×C₄	—	7	4	no	—
239	C₂×C₄	—	7	2	no	—
240	C₂×C₄	—	14	4	no	—
241	C₂×C₄	—	14	4	no	—
242	C₂×C₄	—	14	4	no	—
243	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
245	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
246	C₇	C₃×#532	1	1	yes	characteristic, complemented, Sylow
247	C₂×C₃	—	4	2	no	—
248	C₂×C₃	—	14	2	no	—
249	C₂×C₃	—	28	2	no	—
250	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
251	C₂×C₃	D₇×#41	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	7	2	no	—
255	C₂²	—	14	2	no	—
256	C₂²	—	14	2	no	—
257	C₂²	—	14	4	no	—
258	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	7	2	no	—
262	C₄	—	7	4	no	—
263	C₄	—	14	2	no	—
264	C₄	C₃×D₄×D₇	1	2	yes	—
265	C₃	#90359	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	4	2	no	—
267	C₂	—	14	2	no	—
268	C₂	—	28	2	no	—
269	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
270	C₂	C₃×D₇×#41	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#124	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₃×#532
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#1161	C₂	semidirect
2. C₂×C₃×#1159	C₂	semidirect
3. C₃×C₇×#124	C₂	semidirect
4. C₃×#6507	C₂	semidirect
5. C₃×#6503	C₂	semidirect
6. C₃×#6447	C₂	semidirect
7. C₃×(C₂²⋊D₇)⋊C₄	C₂	semidirect
8. C₃×#6438	C₂	semidirect
9. C₃×C₄²×D₇	C₂	semidirect
10. #90359	C₃	direct (N × H)
30. C₂×#1161	C₂×C₃	semidirect
31. C₂×#1159	C₂×C₃	semidirect
32. C₇×#124	C₂×C₃	semidirect
33. #6507	C₂×C₃	semidirect
34. #6503	C₂×C₃	semidirect
35. #6447	C₂×C₃	semidirect
36. (C₂²⋊D₇)⋊C₄	C₂×C₃	semidirect
37. #6438	C₂×C₃	semidirect
38. C₄²×D₇	C₂×C₃	semidirect
58. C₃×C₄×D₇	D₄	semidirect
84. C₃×#124	D₇	semidirect
100. C₃×Dic₇	#41	semidirect
120. C₄×D₇	C₃×D₄	semidirect
151. #124	C₃×D₇	semidirect
167. Dic₇	C₃×#41	semidirect
188. C₃×C₇	#532	semidirect
246. C₇	C₃×#532	semidirect
265. C₃	#90359	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#1161	2
C₂×C₃×#1159	2
C₃×C₇×#124	2
C₃×#6507	2
C₃×#6503	2
C₃×#6447	2
C₃×(C₂²⋊D₇)⋊C₄	2
C₃×#6438	2
C₃×C₄²×D₇	2
#90359	3
C₃×#532	7

Subgroups — C3×#90359 #13061019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90359 #13061019