Subgroups — C₃×#90265 #13060916

Order 1344.

588 conjugacy classes of subgroups.

The subgroup lattice has 188 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
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×#6456	C₂	1	3	yes	maximal, complemented
5	C₃×D₇×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₃×C₄×D₂₈	C₂	1	3	yes	maximal, complemented
7	C₃×#6435	C₂	1	3	yes	maximal, complemented
8	#90265	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₃×#1161	C₄	1	8	yes	complemented
11	C₂×C₃×D₂₈	C₂²	1	3	yes	—
12	C₂×C₃×C₄×D₇	C₂²	1	6	yes	—
13	C₂×C₃×C₄×D₇	C₂²	1	3	yes	—
14	C₃×C₇×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×C₄²×C₇	C₂²	1	3	yes	—
16	C₃×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
17	C₃×(C₂×D₇)⋊C₄	C₂²	1	3	yes	—
18	C₃×C₄⋊Dic₇	C₂²	1	3	yes	—
19	C₃×Dic₇⋊C₄	C₂²	1	3	yes	—
20	C₃×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₄×Dic₇	C₂²	1	3	yes	—
22	C₂×#1161	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	#6456	C₂×C₃	1	3	yes	complemented
26	D₇×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
27	C₄×D₂₈	C₂×C₃	1	3	yes	complemented
28	#6435	C₂×C₃	1	3	yes	complemented
29	C₃×#520	—	7	1	no	maximal, Hall
30	C₂²×C₃×D₇	C₂³	1	3	yes	—
31	C₃×C₇×Q₈	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
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	1	yes	characteristic
38	C₃×D₂₈	C₂×C₄	1	12	yes	complemented
39	C₃×C₄×D₇	—	4	6	no	—
40	C₃×C₄×D₇	C₂×C₄	1	12	yes	complemented
41	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
42	#1161	C₃×C₄	1	8	yes	complemented
43	C₂×D₂₈	C₂²×C₃	1	3	yes	—
44	C₂×C₄×D₇	C₂²×C₃	1	6	yes	—
45	C₂×C₄×D₇	C₂²×C₃	1	3	yes	—
46	C₇×C₄⋊C₄	C₂²×C₃	1	3	yes	—
47	C₄²×C₇	C₂²×C₃	1	3	yes	—
48	(C₂×D₇)⋊C₄	C₂²×C₃	1	3	yes	—
49	(C₂×D₇)⋊C₄	C₂²×C₃	1	3	yes	—
50	C₄⋊Dic₇	C₂²×C₃	1	3	yes	—
51	Dic₇⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	Dic₇⋊C₄	C₂²×C₃	1	3	yes	—
53	C₄×Dic₇	C₂²×C₃	1	3	yes	—
54	C₂×C₃×#41	—	7	1	no	—
55	C₃×C₄×Q₈	—	7	1	no	—
56	C₃×C₄×Q₈	D₇	1	1	yes	characteristic, complemented
57	C₃×C₄×D₄	—	7	3	no	—
58	C₃×C₄×D₄	—	7	3	no	—
59	C₃×#117	—	7	3	no	—
60	C₂×C₃×C₄⋊C₄	—	7	3	no	—
61	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₃×D₇	—	2	3	no	—
63	C₂×C₃×D₇	—	2	3	no	—
64	C₂×C₃×D₇	C₂²×C₄	1	6	yes	—
65	C₃×C₄×C₇	—	2	3	no	—
66	C₃×C₄×C₇	—	2	1	no	—
67	C₃×C₄×C₇	C₂²×C₄	1	6	yes	—
68	C₃×Dic₇	—	2	3	no	—
69	C₃×Dic₇	—	2	1	no	—
70	C₃×Dic₇	C₂²×C₄	1	2	yes	—
71	#520	—	7	1	no	Sylow
72	C₂²×D₇	C₂³×C₃	1	3	yes	—
73	C₇×Q₈	C₂×C₃×C₄	1	4	yes	complemented
74	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
76	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
77	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
78	C₂×Dic₇	C₂³×C₃	1	3	yes	—
79	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
80	D₂₈	C₂×C₃×C₄	1	12	yes	complemented
81	C₄×D₇	—	4	6	no	—
82	C₄×D₇	C₂×C₃×C₄	1	12	yes	complemented
83	C₃×#41	—	7	8	no	—
84	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
85	C₂×C₃×D₄	—	7	3	no	—
86	C₂²×C₃×C₄	—	7	3	no	—
87	C₂²×C₃×C₄	—	7	6	no	—
88	C₃×C₄⋊C₄	—	7	3	no	—
89	C₃×C₄⋊C₄	—	7	1	no	—
90	C₃×C₄⋊C₄	—	7	3	no	—
91	C₃×C₄⋊C₄	C₂×D₇	1	3	yes	—
92	C₃×C₂²⋊C₄	—	7	3	no	—
93	C₃×C₂²⋊C₄	—	7	3	no	—
94	C₃×C₄²	—	7	3	no	—
95	C₃×C₄²	C₂×D₇	1	3	yes	—
96	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
99	C₃×D₇	—	2	6	no	—
100	C₂×#41	—	7	1	no	—
101	C₄×Q₈	—	7	1	no	—
102	C₄×Q₈	C₃×D₇	1	1	yes	characteristic, complemented
103	C₄×D₄	—	7	3	no	—
104	C₄×D₄	—	7	3	no	—
105	#117	—	7	3	no	—
106	C₂×C₄⋊C₄	—	7	3	no	—
107	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
108	C₂×D₇	—	2	3	no	—
109	C₂×D₇	—	2	3	no	—
110	C₂×D₇	C₂²×C₃×C₄	1	6	yes	—
111	C₄×C₇	—	2	3	no	—
112	C₄×C₇	—	2	1	no	—
113	C₄×C₇	C₂²×C₃×C₄	1	6	yes	—
114	Dic₇	—	2	3	no	—
115	Dic₇	—	2	1	no	—
116	Dic₇	C₂²×C₃×C₄	1	2	yes	—
117	C₂³×C₃	—	7	3	no	—
118	C₃×Q₈	C₄×D₇	1	4	yes	complemented
119	C₃×D₄	—	7	12	no	—
120	C₂×C₃×C₄	—	7	1	no	—
121	C₂×C₃×C₄	—	7	12	no	—
122	C₂×C₃×C₄	—	7	1	no	—
123	C₂×C₃×C₄	—	7	3	no	—
124	C₂×C₃×C₄	—	28	6	no	—
125	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
127	C₂×C₃×C₄	C₂²×D₇	1	3	yes	—
128	C₃×C₇	#520	1	1	yes	characteristic, complemented, Hall
129	#41	—	7	8	no	—
130	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
131	C₂×D₄	—	7	3	no	—
132	C₂²×C₄	—	7	3	no	—
133	C₂²×C₄	—	7	6	no	—
134	C₄⋊C₄	—	7	3	no	—
135	C₄⋊C₄	—	7	1	no	—
136	C₄⋊C₄	—	7	3	no	—
137	C₄⋊C₄	C₂×C₃×D₇	1	3	yes	—
138	C₂²⋊C₄	—	7	3	no	—
139	C₂²⋊C₄	—	7	3	no	—
140	C₄²	—	7	3	no	—
141	C₄²	C₂×C₃×D₇	1	3	yes	—
142	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
143	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
145	D₇	—	2	6	no	—
146	C₂²×C₃	—	7	6	no	—
147	C₂²×C₃	—	14	3	no	—
148	C₂²×C₃	—	14	3	no	—
149	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
150	C₃×C₄	—	2	1	no	—
151	C₃×C₄	—	2	3	no	—
152	C₃×C₄	—	7	2	no	—
153	C₃×C₄	—	14	1	no	—
154	C₃×C₄	—	14	3	no	—
155	C₃×C₄	C₂×C₄×D₇	1	6	yes	—
156	C₂³	—	7	3	no	—
157	Q₈	C₃×C₄×D₇	1	4	yes	complemented
158	D₄	—	7	12	no	—
159	C₂×C₄	—	7	1	no	—
160	C₂×C₄	—	7	12	no	—
161	C₂×C₄	—	7	1	no	—
162	C₂×C₄	—	7	3	no	—
163	C₂×C₄	—	28	6	no	—
164	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
165	C₂×C₄	C₂²×C₃×D₇	1	3	yes	—
166	C₂×C₄	C₂²×C₃×D₇	1	3	yes	—
167	C₇	C₃×#520	1	1	yes	characteristic, complemented, Sylow
168	C₂×C₃	—	14	6	no	—
169	C₂×C₃	C₂²×C₄×D₇	1	1	yes	characteristic
170	C₂×C₃	#6551	1	1	yes	characteristic
171	C₂×C₃	#6553	1	1	yes	characteristic
172	C₂²	—	7	6	no	—
173	C₂²	—	14	3	no	—
174	C₂²	—	14	3	no	—
175	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	3	no	—
178	C₄	—	7	2	no	—
179	C₄	—	14	1	no	—
180	C₄	—	14	3	no	—
181	C₄	C₂×C₃×C₄×D₇	1	6	yes	—
182	C₃	#90265	1	1	yes	characteristic, direct product factor, Sylow
183	C₂	—	14	6	no	—
184	C₂	C₂²×C₃×C₄×D₇	1	1	yes	characteristic
185	C₂	C₃×#6551	1	1	yes	characteristic
186	C₂	C₃×#6553	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	C₂×C₃×#1161	C₂
cosocle	C₂²×C₇	C₂⁴×C₃
hypercenter	C₃×C₄×Q₈	D₇
hypocenter	C₇	C₃×#520
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
4. C₃×#6456	C₂	semidirect
5. C₃×D₇×C₄⋊C₄	C₂	semidirect
6. C₃×C₄×D₂₈	C₂	semidirect
7. C₃×#6435	C₂	semidirect
8. #90265	C₃	direct (N × H)
10. C₃×#1161	C₄	semidirect
23. C₄×C₇×Q₈	C₂×C₃	semidirect
24. Q₈×Dic₇	C₂×C₃	semidirect
25. #6456	C₂×C₃	semidirect
26. D₇×C₄⋊C₄	C₂×C₃	semidirect
27. C₄×D₂₈	C₂×C₃	semidirect
28. #6435	C₂×C₃	semidirect
31. C₃×C₇×Q₈	C₂×C₄	semidirect
38. C₃×D₂₈	C₂×C₄	semidirect
40. C₃×C₄×D₇	C₂×C₄	semidirect
42. #1161	C₃×C₄	semidirect
56. C₃×C₄×Q₈	D₇	semidirect
73. C₇×Q₈	C₂×C₃×C₄	semidirect
80. D₂₈	C₂×C₃×C₄	semidirect
82. C₄×D₇	C₂×C₃×C₄	semidirect
102. C₄×Q₈	C₃×D₇	semidirect
118. C₃×Q₈	C₄×D₇	semidirect
128. C₃×C₇	#520	semidirect
157. Q₈	C₃×C₄×D₇	semidirect
167. C₇	C₃×#520	semidirect
182. C₃	#90265	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
C₃×C₄×C₇×Q₈	2
C₃×Q₈×Dic₇	2
C₃×#6456	2
C₃×D₇×C₄⋊C₄	2
C₃×C₄×D₂₈	2
C₃×#6435	2
#90265	3
C₃×#520	7

Subgroups — C3×#90265 #13060916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90265 #13060916