Subgroups — C₃×#90342 #13060997

Order 1344.

392 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#123	C₂	1	1	yes	maximal, characteristic
2	C₃×#6499	C₂	1	1	yes	maximal, characteristic
3	C₃×#6498	C₂	1	1	yes	maximal, characteristic
4	C₃×#6487	C₂	1	2	yes	maximal, complemented
5	C₃×#6452	C₂	1	2	yes	maximal, complemented
6	C₃×#6451	C₂	1	2	yes	maximal, complemented
7	C₃×#6442	C₂	1	2	yes	maximal, complemented
8	C₃×#6442	C₂	1	2	yes	maximal, complemented
9	C₃×#6441	C₂	1	2	yes	maximal, complemented
10	#90342	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
15	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
16	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
20	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic
21	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
23	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₄⋊Dic₇	C₂²	1	2	yes	—
25	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
26	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄×Dic₇	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
32	C₇×#123	C₂×C₃	1	1	yes	characteristic
33	#6499	C₂×C₃	1	1	yes	characteristic
34	#6498	C₂×C₃	1	1	yes	characteristic
35	#6487	C₂×C₃	1	2	yes	complemented
36	#6452	C₂×C₃	1	2	yes	complemented
37	#6451	C₂×C₃	1	2	yes	complemented
38	#6442	C₂×C₃	1	2	yes	complemented
39	#6442	C₂×C₃	1	2	yes	complemented
40	#6441	C₂×C₃	1	2	yes	complemented
41	C₃×#563	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
44	C₃×C₇×D₄	—	4	1	no	—
45	C₂×C₃×C₄×C₇	—	4	1	no	—
46	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
49	C₂×C₃×Dic₇	—	4	1	no	—
50	C₂×C₃×Dic₇	C₂³	1	2	yes	—
51	C₂×C₃×Dic₇	C₂³	1	2	yes	—
52	C₂×C₃×Dic₇	C₂³	1	2	yes	—
53	C₂×C₃×Dic₇	C₂³	1	2	yes	—
54	C₃×Dic₁₄	—	4	1	no	—
55	C₃×Dic₁₄	—	4	2	no	—
56	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
58	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
59	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
60	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
61	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
63	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
65	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
66	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
67	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
68	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
69	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
70	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
71	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
72	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
73	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	C₄×Dic₇	C₂²×C₃	1	2	yes	—
76	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₃×C₄⋊Q₈	—	7	2	no	—
78	C₃×#126	—	7	2	no	—
79	C₃×#126	—	7	2	no	—
80	C₃×#125	—	7	2	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#123	—	7	1	no	—
83	C₃×#123	D₇	1	1	yes	characteristic
84	C₃×C₂²⋊Q₈	—	7	2	no	—
85	C₃×C₂²⋊Q₈	—	7	2	no	—
86	C₂²×C₃×C₇	—	2	1	no	—
87	C₂²×C₃×C₇	—	2	1	no	—
88	C₂²×C₃×C₇	—	2	2	no	—
89	C₂²×C₃×C₇	—	2	2	no	—
90	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
91	C₃×C₄×C₇	—	2	2	no	—
92	C₃×C₄×C₇	—	2	1	no	—
93	C₃×C₄×C₇	—	2	2	no	—
94	C₃×Dic₇	—	2	2	no	—
95	C₃×Dic₇	—	2	2	no	—
96	C₃×Dic₇	—	2	2	no	—
97	C₃×Dic₇	—	2	2	no	—
98	#563	—	7	1	no	Sylow
99	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
100	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
101	C₇×D₄	—	4	1	no	—
102	C₂×C₄×C₇	—	4	1	no	—
103	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
104	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
106	C₂×Dic₇	—	4	1	no	—
107	C₂×Dic₇	C₂³×C₃	1	2	yes	—
108	C₂×Dic₇	C₂³×C₃	1	2	yes	—
109	C₂×Dic₇	C₂³×C₃	1	2	yes	—
110	C₂×Dic₇	C₂³×C₃	1	2	yes	—
111	Dic₁₄	—	4	1	no	—
112	Dic₁₄	—	4	2	no	—
113	C₂×C₃×Q₈	—	7	2	no	—
114	C₂×C₃×Q₈	—	7	1	no	—
115	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
116	C₂²×C₃×C₄	—	7	1	no	—
117	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
118	C₃×C₄⋊C₄	—	7	2	no	—
119	C₃×C₄⋊C₄	—	7	2	no	—
120	C₃×C₄⋊C₄	—	7	2	no	—
121	C₃×C₄⋊C₄	—	7	2	no	—
122	C₃×C₄⋊C₄	—	7	2	no	—
123	C₃×C₄⋊C₄	—	7	2	no	—
124	C₃×C₄⋊C₄	—	7	2	no	—
125	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
126	C₃×C₂²⋊C₄	—	7	2	no	—
127	C₃×C₂²⋊C₄	—	7	2	no	—
128	C₃×C₂²⋊C₄	—	7	2	no	—
129	C₃×C₂²⋊C₄	—	7	1	no	—
130	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
131	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
132	C₃×C₄²	—	7	1	no	—
133	C₃×C₄²	—	7	2	no	—
134	C₂×C₃×C₇	—	4	1	no	—
135	C₂×C₃×C₇	—	4	1	no	—
136	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
138	C₄⋊Q₈	—	7	2	no	—
139	#126	—	7	2	no	—
140	#126	—	7	2	no	—
141	#125	—	7	2	no	—
142	#124	—	7	1	no	—
143	#123	—	7	1	no	—
144	#123	C₃×D₇	1	1	yes	characteristic
145	C₂²⋊Q₈	—	7	2	no	—
146	C₂²⋊Q₈	—	7	2	no	—
147	C₂²×C₇	—	2	1	no	—
148	C₂²×C₇	—	2	1	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	2	2	no	—
151	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	2	2	no	—
155	Dic₇	—	2	2	no	—
156	Dic₇	—	2	2	no	—
157	Dic₇	—	2	2	no	—
158	Dic₇	—	2	2	no	—
159	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
160	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
161	C₃×Q₈	—	28	2	no	—
162	C₃×Q₈	—	28	1	no	—
163	C₃×D₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	1	no	—
165	C₂×C₃×C₄	—	7	2	no	—
166	C₂×C₃×C₄	—	7	2	no	—
167	C₂×C₃×C₄	—	7	2	no	—
168	C₂×C₃×C₄	—	7	2	no	—
169	C₂×C₃×C₄	—	28	1	no	—
170	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
171	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
173	C₃×C₇	#563	1	1	yes	characteristic, complemented, Hall
174	C₂×Q₈	—	7	2	no	—
175	C₂×Q₈	—	7	1	no	—
176	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
177	C₂²×C₄	—	7	1	no	—
178	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
179	C₄⋊C₄	—	7	2	no	—
180	C₄⋊C₄	—	7	2	no	—
181	C₄⋊C₄	—	7	2	no	—
182	C₄⋊C₄	—	7	2	no	—
183	C₄⋊C₄	—	7	2	no	—
184	C₄⋊C₄	—	7	2	no	—
185	C₄⋊C₄	—	7	2	no	—
186	C₄⋊C₄	C₂×C₃×D₇	1	2	yes	—
187	C₂²⋊C₄	—	7	2	no	—
188	C₂²⋊C₄	—	7	2	no	—
189	C₂²⋊C₄	—	7	2	no	—
190	C₂²⋊C₄	—	7	1	no	—
191	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
192	C₂²⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
193	C₄²	—	7	1	no	—
194	C₄²	—	7	2	no	—
195	C₂×C₇	—	4	1	no	—
196	C₂×C₇	—	4	1	no	—
197	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
199	C₂²×C₃	—	2	1	no	—
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	2	no	—
203	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	14	2	no	—
208	C₃×C₄	—	14	2	no	—
209	C₃×C₄	—	14	2	no	—
210	C₃×C₄	—	14	2	no	—
211	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
212	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
213	Q₈	—	28	2	no	—
214	Q₈	—	28	1	no	—
215	D₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	7	2	no	—
218	C₂×C₄	—	7	2	no	—
219	C₂×C₄	—	7	2	no	—
220	C₂×C₄	—	7	2	no	—
221	C₂×C₄	—	28	1	no	—
222	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
223	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
225	C₇	C₃×#563	1	1	yes	characteristic, complemented, Sylow
226	C₂×C₃	—	4	1	no	—
227	C₂×C₃	—	4	1	no	—
228	C₂×C₃	#6548	1	1	yes	characteristic
229	C₂×C₃	#6554	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	14	2	no	—
239	C₄	—	14	2	no	—
240	C₄	—	14	2	no	—
241	C₄	—	14	2	no	—
242	C₃	#90342	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	4	1	no	—
244	C₂	—	4	1	no	—
245	C₂	C₃×#6548	1	1	yes	characteristic
246	C₂	C₃×#6554	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×#123	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×C₇×D₄	C₂²
hypercenter	C₃×#123	D₇
hypocenter	C₇	C₃×#563
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#123

Complements

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

N	H ≅ G/N	type
4. C₃×#6487	C₂	semidirect
5. C₃×#6452	C₂	semidirect
6. C₃×#6451	C₂	semidirect
7. C₃×#6442	C₂	semidirect
8. C₃×#6442	C₂	semidirect
9. C₃×#6441	C₂	semidirect
10. #90342	C₃	direct (N × H)
35. #6487	C₂×C₃	semidirect
36. #6452	C₂×C₃	semidirect
37. #6451	C₂×C₃	semidirect
38. #6442	C₂×C₃	semidirect
39. #6442	C₂×C₃	semidirect
40. #6441	C₂×C₃	semidirect
173. C₃×C₇	#563	semidirect
225. C₇	C₃×#563	semidirect
242. C₃	#90342	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×#123	2
C₃×#6499	2
C₃×#6498	2
C₃×#6487	2
C₃×#6452	2
C₃×#6451	2
C₃×#6442	2
C₃×#6441	2
#90342	3
C₃×#563	7

Subgroups — C3×#90342 #13060997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90342 #13060997