Subgroups — C₃×#90263 #13060914

Order 1344.

532 conjugacy classes of subgroups.

The subgroup lattice has 184 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
2	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6455	C₂	1	3	yes	maximal
5	C₃×#6450	C₂	1	3	yes	maximal, complemented
6	C₃×#6435	C₂	1	3	yes	maximal
7	C₃×C₄×Dic₁₄	C₂	1	3	yes	maximal, complemented
8	#90263	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₃×Q₈×D₇	C₄	1	8	yes	complemented
11	C₂×C₃×C₄×D₇	C₂²	1	3	yes	—
12	C₂×C₃×Dic₁₄	C₂²	1	3	yes	—
13	C₃×C₇×C₄⋊C₄	C₂²	1	3	yes	—
14	C₃×C₄²×C₇	C₂²	1	3	yes	—
15	C₃×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
16	C₃×(C₂×D₇)⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Dic₇	C₂²	1	3	yes	—
18	C₃×Dic₇⋊C₄	C₂²	1	3	yes	—
19	C₃×Dic₇⋊C₄	C₂²	1	3	yes	—
20	C₃×C₄×Dic₇	C₂²	1	6	yes	—
21	C₃×C₄×Dic₇	C₂²	1	3	yes	—
22	C₂×Q₈×D₇	C₂×C₃	1	1	yes	characteristic
23	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
24	Q₈×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
25	#6455	C₂×C₃	1	3	yes	—
26	#6450	C₂×C₃	1	3	yes	complemented
27	#6435	C₂×C₃	1	3	yes	—
28	C₄×Dic₁₄	C₂×C₃	1	3	yes	complemented
29	C₃×#519	—	7	1	no	maximal, Hall
30	C₂²×C₃×D₇	C₂³	1	1	yes	characteristic
31	C₃×C₇×Q₈	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
35	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
36	C₂×C₃×Dic₇	C₂³	1	3	yes	—
37	C₂×C₃×Dic₇	C₂³	1	3	yes	—
38	C₃×C₄×D₇	C₂×C₄	1	12	yes	—
39	C₃×Dic₁₄	C₂×C₄	1	12	yes	—
40	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
41	Q₈×D₇	C₃×C₄	1	8	yes	complemented
42	C₂×C₄×D₇	C₂²×C₃	1	3	yes	—
43	C₂×Dic₁₄	C₂²×C₃	1	3	yes	—
44	C₇×C₄⋊C₄	C₂²×C₃	1	3	yes	—
45	C₄²×C₇	C₂²×C₃	1	3	yes	—
46	(C₂×D₇)⋊C₄	C₂²×C₃	1	1	yes	characteristic
47	(C₂×D₇)⋊C₄	C₂²×C₃	1	3	yes	—
48	C₄⋊Dic₇	C₂²×C₃	1	3	yes	—
49	Dic₇⋊C₄	C₂²×C₃	1	3	yes	—
50	Dic₇⋊C₄	C₂²×C₃	1	3	yes	—
51	C₄×Dic₇	C₂²×C₃	1	6	yes	—
52	C₄×Dic₇	C₂²×C₃	1	3	yes	—
53	C₂²×C₃×Q₈	—	7	1	no	—
54	C₃×C₄×Q₈	—	7	1	no	—
55	C₃×C₄×Q₈	—	7	3	no	—
56	C₃×C₄×Q₈	—	7	3	no	—
57	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
58	C₃×#117	—	7	3	no	—
59	C₃×#117	—	7	3	no	—
60	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
61	C₂×C₃×D₇	—	2	1	no	—
62	C₂×C₃×D₇	—	2	1	no	—
63	C₂×C₃×D₇	C₂²×C₄	1	2	yes	—
64	C₃×C₄×C₇	—	2	1	no	—
65	C₃×C₄×C₇	—	2	3	no	—
66	C₃×C₄×C₇	C₂²×C₄	1	6	yes	—
67	C₃×Dic₇	—	2	3	no	—
68	C₃×Dic₇	—	2	1	no	—
69	C₃×Dic₇	C₂²×C₄	1	6	yes	—
70	#519	—	7	1	no	Sylow
71	C₂²×D₇	C₂³×C₃	1	1	yes	characteristic
72	C₇×Q₈	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
74	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
76	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
77	C₂×Dic₇	C₂³×C₃	1	3	yes	—
78	C₂×Dic₇	C₂³×C₃	1	3	yes	—
79	C₄×D₇	C₂×C₃×C₄	1	12	yes	—
80	Dic₁₄	C₂×C₃×C₄	1	12	yes	—
81	C₂×C₃×Q₈	—	7	8	no	—
82	C₂×C₃×Q₈	—	7	3	no	—
83	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
84	C₂²×C₃×C₄	—	7	3	no	—
85	C₃×C₄⋊C₄	—	7	3	no	—
86	C₃×C₄⋊C₄	—	7	3	no	—
87	C₃×C₄⋊C₄	—	7	3	no	—
88	C₃×C₄⋊C₄	C₂×D₇	1	3	yes	—
89	C₃×C₂²⋊C₄	—	7	3	no	—
90	C₃×C₂²⋊C₄	—	7	1	no	—
91	C₃×C₄²	—	7	6	no	—
92	C₃×C₄²	—	7	3	no	—
93	C₃×C₄²	C₂×D₇	1	3	yes	—
94	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
97	C₃×D₇	—	2	2	no	—
98	C₂²×Q₈	—	7	1	no	—
99	C₄×Q₈	—	7	1	no	—
100	C₄×Q₈	—	7	3	no	—
101	C₄×Q₈	—	7	3	no	—
102	C₄×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
103	#117	—	7	3	no	—
104	#117	—	7	3	no	—
105	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
106	C₂×D₇	—	2	1	no	—
107	C₂×D₇	—	2	1	no	—
108	C₂×D₇	C₂²×C₃×C₄	1	2	yes	—
109	C₄×C₇	—	2	1	no	—
110	C₄×C₇	—	2	3	no	—
111	C₄×C₇	C₂²×C₃×C₄	1	6	yes	—
112	Dic₇	—	2	3	no	—
113	Dic₇	—	2	1	no	—
114	Dic₇	C₂²×C₃×C₄	1	6	yes	—
115	C₂³×C₃	—	7	1	no	—
116	C₃×Q₈	—	7	12	no	—
117	C₃×Q₈	C₄×D₇	1	4	yes	—
118	C₂×C₃×C₄	—	7	3	no	—
119	C₂×C₃×C₄	—	7	12	no	—
120	C₂×C₃×C₄	—	7	3	no	—
121	C₂×C₃×C₄	—	7	1	no	—
122	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
123	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
125	C₃×C₇	#519	1	1	yes	characteristic, complemented, Hall
126	C₂×Q₈	—	7	8	no	—
127	C₂×Q₈	—	7	3	no	—
128	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
129	C₂²×C₄	—	7	3	no	—
130	C₄⋊C₄	—	7	3	no	—
131	C₄⋊C₄	—	7	3	no	—
132	C₄⋊C₄	—	7	3	no	—
133	C₄⋊C₄	C₂×C₃×D₇	1	3	yes	—
134	C₂²⋊C₄	—	7	3	no	—
135	C₂²⋊C₄	—	7	1	no	—
136	C₄²	—	7	6	no	—
137	C₄²	—	7	3	no	—
138	C₄²	C₂×C₃×D₇	1	3	yes	—
139	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
140	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
142	D₇	—	2	2	no	—
143	C₂²×C₃	—	7	2	no	—
144	C₂²×C₃	—	14	1	no	—
145	C₂²×C₃	—	14	1	no	—
146	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
147	C₃×C₄	—	2	1	no	—
148	C₃×C₄	—	2	3	no	—
149	C₃×C₄	—	7	6	no	—
150	C₃×C₄	—	14	3	no	—
151	C₃×C₄	—	14	1	no	—
152	C₃×C₄	C₂×C₄×D₇	1	6	yes	—
153	C₂³	—	7	1	no	—
154	Q₈	—	7	12	no	—
155	Q₈	C₃×C₄×D₇	1	4	yes	—
156	C₂×C₄	—	7	3	no	—
157	C₂×C₄	—	7	12	no	—
158	C₂×C₄	—	7	3	no	—
159	C₂×C₄	—	7	1	no	—
160	C₂×C₄	C₂²×C₃×D₇	1	3	yes	—
161	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
162	C₂×C₄	C₂²×C₃×D₇	1	3	yes	—
163	C₇	C₃×#519	1	1	yes	characteristic, complemented, Sylow
164	C₂×C₃	—	14	2	no	—
165	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
166	C₂×C₃	#6551	1	1	yes	characteristic
167	C₂×C₃	#6554	1	1	yes	characteristic
168	C₂²	—	7	2	no	—
169	C₂²	—	14	1	no	—
170	C₂²	—	14	1	no	—
171	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	3	no	—
174	C₄	—	7	6	no	—
175	C₄	—	14	3	no	—
176	C₄	—	14	1	no	—
177	C₄	C₂×C₃×C₄×D₇	1	6	yes	—
178	C₃	#90263	1	1	yes	characteristic, direct product factor, Sylow
179	C₂	—	14	2	no	—
180	C₂	C₂²×C₃×C₄×D₇	1	1	yes	characteristic
181	C₂	C₃×#6551	1	1	yes	characteristic
182	C₂	C₃×#6554	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₄×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂²×C₃×D₇	C₂³
cosocle	C₂²×C₇	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₃×#519
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₇×Q₈	C₂	semidirect
3. C₃×Q₈×Dic₇	C₂	semidirect
5. C₃×#6450	C₂	semidirect
7. C₃×C₄×Dic₁₄	C₂	semidirect
8. #90263	C₃	direct (N × H)
10. C₃×Q₈×D₇	C₄	semidirect
23. C₄×C₇×Q₈	C₂×C₃	semidirect
24. Q₈×Dic₇	C₂×C₃	semidirect
26. #6450	C₂×C₃	semidirect
28. C₄×Dic₁₄	C₂×C₃	semidirect
41. Q₈×D₇	C₃×C₄	semidirect
57. C₃×C₄×Q₈	D₇	semidirect
102. C₄×Q₈	C₃×D₇	semidirect
125. C₃×C₇	#519	semidirect
163. C₇	C₃×#519	semidirect
178. C₃	#90263	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
C₃×C₄×C₇×Q₈	2
C₃×Q₈×Dic₇	2
C₃×#6455	2
C₃×#6450	2
C₃×#6435	2
C₃×C₄×Dic₁₄	2
#90263	3
C₃×#519	7

Subgroups — C3×#90263 #13060914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90263 #13060914