Subgroups — C₃×#90366 #13061018

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1159	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6507	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6499	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6446	C₂	1	4	yes	maximal, complemented
7	C₃×#6441	C₂	1	4	yes	maximal, complemented
8	C₃×#6435	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6432	C₂	1	1	yes	maximal, characteristic, complemented
10	#90366	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	2	yes	—
14	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
15	C₃×Q₈×D₇	—	2	4	no	—
16	C₃×#1159	—	2	4	no	—
17	C₂×C₃×C₄×D₇	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
19	C₂×C₃×Dic₁₄	C₂²	1	2	yes	—
20	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
22	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
25	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₇	C₂²	1	4	yes	—
27	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
28	C₃×Dic₇⋊C₄	C₂²	1	4	yes	—
29	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#1159	C₂×C₃	1	1	yes	characteristic
32	C₇×#124	C₂×C₃	1	1	yes	characteristic, complemented
33	#6507	C₂×C₃	1	1	yes	characteristic, complemented
34	#6499	C₂×C₃	1	1	yes	characteristic, complemented
35	#6446	C₂×C₃	1	4	yes	complemented
36	#6441	C₂×C₃	1	4	yes	complemented
37	#6435	C₂×C₃	1	1	yes	characteristic, complemented
38	#6432	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#536	—	7	1	no	maximal, Hall
40	C₂³×C₃×C₇	C₂³	1	2	yes	—
41	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
42	C₃×C₇×Q₈	—	2	2	no	—
43	C₃×C₇×D₄	—	2	2	no	—
44	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
45	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₃×C₂²⋊D₇	—	2	4	no	—
48	C₂×C₃×Dic₇	—	2	4	no	—
49	C₂×C₃×Dic₇	C₂³	1	2	yes	—
50	C₂×C₃×Dic₇	C₂³	1	2	yes	—
51	C₂×C₃×Dic₇	C₂³	1	2	yes	—
52	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
53	C₃×C₄×D₇	—	2	4	no	—
54	C₃×C₄×D₇	D₄	1	4	yes	—
55	C₃×Dic₁₄	—	2	2	no	—
56	C₃×Dic₁₄	—	2	2	no	—
57	C₃×Dic₁₄	—	2	4	no	—
58	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
61	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
62	Q₈×D₇	—	2	4	no	—
63	#1159	—	2	4	no	—
64	C₂×C₄×D₇	C₂²×C₃	1	2	yes	—
65	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
66	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
69	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
70	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
71	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
72	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	—
73	C₄⋊Dic₇	C₂²×C₃	1	4	yes	—
74	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
75	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
76	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×#41	—	7	1	no	—
78	C₂²×C₃×Q₈	—	7	1	no	—
79	C₃×C₄⋊Q₈	—	7	1	no	—
80	C₃×C₄⋊Q₈	—	7	1	no	—
81	C₃×#124	—	7	1	no	—
82	C₃×#124	D₇	1	1	yes	characteristic, complemented
83	C₃×#123	—	7	4	no	—
84	C₃×C₂²⋊Q₈	—	7	4	no	—
85	C₃×#117	—	7	1	no	—
86	C₂²×C₃×C₇	—	2	2	no	—
87	C₂²×C₃×C₇	—	2	4	no	—
88	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₃×D₇	—	2	2	no	—
90	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
91	C₃×C₄×C₇	—	2	2	no	—
92	C₃×C₄×C₇	—	2	2	no	—
93	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
94	C₃×Dic₇	—	2	2	no	—
95	C₃×Dic₇	—	2	2	no	—
96	C₃×Dic₇	—	2	2	no	—
97	C₃×Dic₇	C₂×D₄	1	2	yes	—
98	#536	—	7	1	no	Sylow
99	C₂³×C₇	C₂³×C₃	1	2	yes	—
100	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
101	C₇×Q₈	—	2	2	no	—
102	C₇×D₄	—	2	2	no	—
103	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
104	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
105	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
106	C₂²⋊D₇	—	2	4	no	—
107	C₂×Dic₇	—	2	4	no	—
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	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
112	C₄×D₇	—	2	4	no	—
113	C₄×D₇	C₃×D₄	1	4	yes	—
114	Dic₁₄	—	2	2	no	—
115	Dic₁₄	—	2	2	no	—
116	Dic₁₄	—	2	4	no	—
117	C₃×#41	—	14	4	no	—
118	C₂×C₃×Q₈	—	7	2	no	—
119	C₂×C₃×Q₈	—	7	1	no	—
120	C₂×C₃×Q₈	—	7	1	no	—
121	C₂×C₃×Q₈	—	14	4	no	—
122	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
123	C₂×C₃×D₄	—	7	2	no	—
124	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	7	1	no	—
126	C₂²×C₃×C₄	—	7	2	no	—
127	C₂²×C₃×C₄	—	7	2	no	—
128	C₃×C₄⋊C₄	—	7	2	no	—
129	C₃×C₄⋊C₄	—	7	4	no	—
130	C₃×C₄⋊C₄	—	7	4	no	—
131	C₃×C₂²⋊C₄	—	7	4	no	—
132	C₃×C₂²⋊C₄	—	7	2	no	—
133	C₃×C₂²⋊C₄	C₂×D₇	1	4	yes	—
134	C₃×C₄²	—	7	1	no	—
135	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
136	C₂×C₃×C₇	—	4	2	no	—
137	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
139	C₃×D₇	—	2	2	no	—
140	C₂×#41	—	7	1	no	—
141	C₂²×Q₈	—	7	1	no	—
142	C₄⋊Q₈	—	7	1	no	—
143	C₄⋊Q₈	—	7	1	no	—
144	#124	—	7	1	no	—
145	#124	C₃×D₇	1	1	yes	characteristic, complemented
146	#123	—	7	4	no	—
147	C₂²⋊Q₈	—	7	4	no	—
148	#117	—	7	1	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	2	4	no	—
151	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
152	C₂×D₇	—	2	2	no	—
153	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
157	Dic₇	—	2	2	no	—
158	Dic₇	—	2	2	no	—
159	Dic₇	—	2	2	no	—
160	Dic₇	C₂×C₃×D₄	1	2	yes	—
161	C₂³×C₃	—	7	1	no	—
162	C₂³×C₃	C₂²×D₇	1	2	yes	—
163	C₃×Q₈	—	2	2	no	—
164	C₃×Q₈	—	14	4	no	—
165	C₃×Q₈	—	14	2	no	—
166	C₃×Q₈	—	14	2	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	14	4	no	—
169	C₂×C₃×C₄	—	7	1	no	—
170	C₂×C₃×C₄	—	7	4	no	—
171	C₂×C₃×C₄	—	7	2	no	—
172	C₂×C₃×C₄	—	7	2	no	—
173	C₂×C₃×C₄	—	7	2	no	—
174	C₂×C₃×C₄	—	14	4	no	—
175	C₂×C₃×C₄	—	14	4	no	—
176	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
177	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
178	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
179	C₃×C₇	#536	1	1	yes	characteristic, complemented, Hall
180	#41	—	14	4	no	—
181	C₂×Q₈	—	7	2	no	—
182	C₂×Q₈	—	7	1	no	—
183	C₂×Q₈	—	7	1	no	—
184	C₂×Q₈	—	14	4	no	—
185	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
186	C₂×D₄	—	7	2	no	—
187	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
188	C₂²×C₄	—	7	1	no	—
189	C₂²×C₄	—	7	2	no	—
190	C₂²×C₄	—	7	2	no	—
191	C₄⋊C₄	—	7	2	no	—
192	C₄⋊C₄	—	7	4	no	—
193	C₄⋊C₄	—	7	4	no	—
194	C₂²⋊C₄	—	7	4	no	—
195	C₂²⋊C₄	—	7	2	no	—
196	C₂²⋊C₄	C₂×C₃×D₇	1	4	yes	—
197	C₄²	—	7	1	no	—
198	C₄²	C₂×C₃×D₇	1	1	yes	characteristic
199	C₂×C₇	—	4	2	no	—
200	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
201	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
202	D₇	—	2	2	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	4	no	—
205	C₂²×C₃	—	7	2	no	—
206	C₂²×C₃	—	14	2	no	—
207	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
208	C₃×C₄	—	2	2	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	7	2	no	—
211	C₃×C₄	—	14	2	no	—
212	C₃×C₄	—	14	2	no	—
213	C₃×C₄	—	14	2	no	—
214	C₃×C₄	D₄×D₇	1	2	yes	—
215	C₂³	—	7	1	no	—
216	C₂³	C₂²×C₃×D₇	1	2	yes	—
217	Q₈	—	2	2	no	—
218	Q₈	—	14	4	no	—
219	Q₈	—	14	2	no	—
220	Q₈	—	14	2	no	—
221	D₄	—	2	2	no	—
222	D₄	—	14	4	no	—
223	C₂×C₄	—	7	1	no	—
224	C₂×C₄	—	7	4	no	—
225	C₂×C₄	—	7	2	no	—
226	C₂×C₄	—	7	2	no	—
227	C₂×C₄	—	7	2	no	—
228	C₂×C₄	—	14	4	no	—
229	C₂×C₄	—	14	4	no	—
230	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
231	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
232	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
233	C₇	C₃×#536	1	1	yes	characteristic, complemented, Sylow
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	—	14	2	no	—
236	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
237	C₂×C₃	#6554	1	2	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	7	2	no	—
241	C₂²	—	14	2	no	—
242	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	7	2	no	—
246	C₄	—	14	2	no	—
247	C₄	—	14	2	no	—
248	C₄	—	14	2	no	—
249	C₄	C₃×D₄×D₇	1	2	yes	—
250	C₃	#90366	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	2	no	—
252	C₂	—	14	2	no	—
253	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic
254	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×#124	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂×C₃×#1159	C₂
hypercenter	C₃×#124	D₇
hypocenter	C₇	C₃×#536
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×Q₈×D₇	C₂	semidirect
3. C₃×C₇×#124	C₂	semidirect
4. C₃×#6507	C₂	semidirect
5. C₃×#6499	C₂	semidirect
6. C₃×#6446	C₂	semidirect
7. C₃×#6441	C₂	semidirect
8. C₃×#6435	C₂	semidirect
9. C₃×#6432	C₂	semidirect
10. #90366	C₃	direct (N × H)
30. C₂×Q₈×D₇	C₂×C₃	semidirect
32. C₇×#124	C₂×C₃	semidirect
33. #6507	C₂×C₃	semidirect
34. #6499	C₂×C₃	semidirect
35. #6446	C₂×C₃	semidirect
36. #6441	C₂×C₃	semidirect
37. #6435	C₂×C₃	semidirect
38. #6432	C₂×C₃	semidirect
82. C₃×#124	D₇	semidirect
145. #124	C₃×D₇	semidirect
179. C₃×C₇	#536	semidirect
233. C₇	C₃×#536	semidirect
250. C₃	#90366	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×Q₈×D₇	2
C₂×C₃×#1159	2
C₃×C₇×#124	2
C₃×#6507	2
C₃×#6499	2
C₃×#6446	2
C₃×#6441	2
C₃×#6435	2
C₃×#6432	2
#90366	3
C₃×#536	7

Subgroups — C3×#90366 #13061018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90366 #13061018