Subgroups — C₃×C₇×#408 #13056919

Order 1344.

316 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₇×#408	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₇×#41	—	2	2	no	—
7	C₃×C₇×#41	—	2	2	no	—
8	C₂×C₃×C₇×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₃×C₇×D₄	C₄	1	2	yes	—
10	C₂×C₃×C₇×D₄	C₄	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₃×C₇×M₁₆	—	2	2	no	—
14	C₂×C₃×C₇×C₈	—	2	2	no	—
15	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
17	C₂×C₇×#41	C₂×C₃	1	1	yes	characteristic
18	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
20	C₇×C₂²⋊C₈	C₂×C₃	1	4	yes	complemented
21	C₃×#408	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
23	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
24	C₃×C₇×Q₈	—	2	2	no	—
25	C₃×C₇×D₄	—	2	4	no	—
26	C₃×C₇×D₄	—	2	2	no	—
27	C₂×C₃×C₄×C₇	—	2	2	no	—
28	C₂×C₃×C₄×C₇	—	2	2	no	—
29	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₃×C₇×C₈	—	2	2	no	—
35	C₃×C₇×C₈	—	2	2	no	—
36	C₇×#41	—	2	2	no	—
37	C₇×#41	—	2	2	no	—
38	C₂×C₇×Q₈	C₃×C₄	1	1	yes	characteristic
39	C₂×C₇×D₄	C₃×C₄	1	2	yes	—
40	C₂×C₇×D₄	C₃×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
42	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
43	C₇×M₁₆	—	2	2	no	—
44	C₂×C₇×C₈	—	2	2	no	—
45	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
46	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
47	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic
48	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
49	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
50	C₃×C₂²⋊C₈	C₂×C₇	1	4	yes	complemented
51	C₂²×C₃×C₇	—	2	2	no	—
52	C₂²×C₃×C₇	—	2	2	no	—
53	C₂²×C₃×C₇	—	2	4	no	—
54	C₂²×C₃×C₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₇	—	2	2	no	—
57	C₃×C₄×C₇	C₂²⋊C₄	1	2	yes	—
58	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
59	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	#408	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
61	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
62	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
63	C₇×Q₈	—	2	2	no	—
64	C₇×D₄	—	2	4	no	—
65	C₇×D₄	—	2	2	no	—
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	2	2	no	—
68	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
69	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
71	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
72	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
73	C₇×C₈	—	2	2	no	—
74	C₇×C₈	—	2	2	no	—
75	C₃×#41	—	2	2	no	—
76	C₃×#41	—	2	2	no	—
77	C₂×C₃×Q₈	C₄×C₇	1	1	yes	characteristic
78	C₂×C₃×D₄	C₄×C₇	1	2	yes	—
79	C₂×C₃×D₄	C₄×C₇	1	1	yes	characteristic
80	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
81	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
82	C₃×M₁₆	—	2	2	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
85	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
86	C₂×C₃×C₇	—	2	2	no	—
87	C₂×C₃×C₇	—	4	2	no	—
88	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃×C₇	#131	1	2	yes	—
90	C₂×#41	C₂×C₃×C₇	1	1	yes	characteristic
91	C₂×M₁₆	C₂×C₃×C₇	1	1	yes	characteristic, complemented
92	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
93	C₂²⋊C₈	C₂×C₃×C₇	1	4	yes	complemented
94	C₂²×C₇	—	2	2	no	—
95	C₂²×C₇	—	2	2	no	—
96	C₂²×C₇	—	2	4	no	—
97	C₂²×C₇	C₃×C₂²⋊C₄	1	2	yes	—
98	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
99	C₄×C₇	—	2	2	no	—
100	C₄×C₇	C₃×C₂²⋊C₄	1	2	yes	—
101	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
102	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
103	C₂³×C₃	C₂×C₄×C₇	1	2	yes	—
104	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
105	C₃×Q₈	—	2	2	no	—
106	C₃×D₄	—	2	4	no	—
107	C₃×D₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
111	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
112	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
113	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
115	C₃×C₈	—	2	2	no	—
116	C₃×C₈	—	2	2	no	—
117	C₃×C₇	#408	1	1	yes	characteristic, direct product factor, Hall
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	C₂×Q₈	C₃×C₄×C₇	1	1	yes	characteristic
121	C₂×D₄	C₃×C₄×C₇	1	2	yes	—
122	C₂×D₄	C₃×C₄×C₇	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₃×C₇	1	2	yes	—
124	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
125	M₁₆	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
128	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
129	C₂×C₇	—	2	2	no	—
130	C₂×C₇	—	4	2	no	—
131	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₇	C₃×#131	1	2	yes	—
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	C₇×C₂²⋊C₄	1	2	yes	—
137	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
138	C₃×C₄	—	2	2	no	—
139	C₃×C₄	C₇×C₂²⋊C₄	1	2	yes	—
140	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
141	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
142	C₂³	C₂×C₃×C₄×C₇	1	2	yes	—
143	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
144	Q₈	—	2	2	no	—
145	D₄	—	2	4	no	—
146	D₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
150	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
151	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
152	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
153	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₇	C₃×#408	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	—	4	2	no	—
159	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₃	C₇×#131	1	2	yes	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	2	no	—
164	C₂²	C₃×C₇×C₂²⋊C₄	1	2	yes	—
165	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	C₃×C₇×C₂²⋊C₄	1	2	yes	—
168	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
169	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
170	C₃	C₇×#408	1	1	yes	characteristic, direct product factor, Sylow
171	C₂	—	2	2	no	—
172	C₂	—	4	2	no	—
173	C₂	C₂×C₃×C₇×C₂²⋊C₄	1	1	yes	characteristic
174	C₂	C₃×C₇×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₇	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₃×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×M₁₆	C₂	semidirect
3. C₂²×C₃×C₇×C₈	C₂	semidirect
4. C₃×C₇×C₂²⋊C₈	C₂	semidirect
5. C₇×#408	C₃	direct (N × H)
18. C₂×C₇×M₁₆	C₂×C₃	semidirect
19. C₂²×C₇×C₈	C₂×C₃	semidirect
20. C₇×C₂²⋊C₈	C₂×C₃	semidirect
21. C₃×#408	C₇	direct (N × H)
48. C₂×C₃×M₁₆	C₂×C₇	semidirect
49. C₂²×C₃×C₈	C₂×C₇	semidirect
50. C₃×C₂²⋊C₈	C₂×C₇	semidirect
60. #408	C₃×C₇	direct (N × H)
91. C₂×M₁₆	C₂×C₃×C₇	semidirect
92. C₂²×C₈	C₂×C₃×C₇	semidirect
93. C₂²⋊C₈	C₂×C₃×C₇	semidirect
117. C₃×C₇	#408	direct (N × H)
156. C₇	C₃×#408	direct (N × H)
170. C₃	C₇×#408	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×C₇×#41	2
C₂×C₃×C₇×M₁₆	2
C₂²×C₃×C₇×C₈	2
C₃×C₇×C₂²⋊C₈	2
C₇×#408	3
C₃×#408	7

Subgroups — C3×C7×#408 #13056919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#408 #13056919