Subgroups — C₂×C₇×#1422 #13190845

Order 1792.

214 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1422	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₇×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
7	C₇×M₆₄	—	2	2	no	—
8	C₇×M₆₄	C₂²	1	4	yes	—
9	C₂×C₇×C₃₂	—	2	2	no	—
10	C₄×C₇×C₁₆	C₄	1	2	yes	—
11	C₄×C₇×C₁₆	C₂²	1	2	yes	—
12	C₂×#1422	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₇×C₈	C₈	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₇	C₈	1	1	yes	characteristic
16	C₂×C₇×C₁₆	—	2	1	no	—
17	C₂×C₇×C₁₆	—	2	1	no	—
18	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
23	C₂×C₇×C₁₆	Q₈	1	1	yes	characteristic
24	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
25	C₄×C₇×C₈	C₈	1	2	yes	—
26	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
27	C₇×C₃₂	—	2	4	no	—
28	C₂×M₆₄	C₂×C₇	1	2	yes	—
29	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic
30	#1422	C₂×C₇	1	4	yes	direct product factor
31	C₂²×C₄×C₇	—	2	1	no	—
32	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₇×C₈	C₁₆	1	2	yes	—
34	C₂×C₇×C₈	C₁₆	1	2	yes	—
35	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
38	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₇×C₈	M₁₆	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₇	—	2	1	no	—
43	C₄²×C₇	C₂×C₈	1	2	yes	—
44	C₇×C₁₆	—	2	2	no	—
45	C₇×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₇×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
48	C₇×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₁₆	C₄×C₇	1	1	yes	characteristic
50	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
51	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
52	M₆₄	—	2	2	no	—
53	M₆₄	C₂²×C₇	1	4	yes	—
54	C₂×C₃₂	—	2	2	no	—
55	C₄×C₁₆	C₄×C₇	1	2	yes	—
56	C₄×C₁₆	C₂²×C₇	1	2	yes	—
57	C₂³×C₇	M₃₂	1	1	yes	characteristic
58	C₂×C₄×C₇	—	2	4	no	—
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇	M₃₂	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
66	C₇×C₈	C₄⋊C₈	1	1	yes	characteristic
67	C₇×C₈	C₄⋊C₈	1	1	yes	characteristic
68	C₇×C₈	C₂×C₁₆	1	4	yes	—
69	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
71	C₂²×C₈	C₇×C₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
73	C₂×C₄²	C₇×C₈	1	1	yes	characteristic
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
77	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
78	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
79	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
80	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
81	C₂×C₁₆	C₇×Q₈	1	1	yes	characteristic
82	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
83	C₄×C₈	C₇×C₈	1	2	yes	—
84	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
85	C₃₂	—	2	4	no	—
86	C₂²×C₇	—	2	2	no	—
87	C₂²×C₇	C₄⋊C₁₆	1	1	yes	characteristic
88	C₂²×C₇	C₄⋊C₁₆	1	1	yes	characteristic
89	C₂²×C₇	C₂×M₃₂	1	1	yes	characteristic
90	C₄×C₇	—	2	4	no	—
91	C₄×C₇	C₄⋊C₁₆	1	1	yes	characteristic
92	C₄×C₇	C₄⋊C₁₆	1	1	yes	characteristic
93	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₇	C₂²×C₁₆	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
97	C₂×C₈	C₇×C₁₆	1	2	yes	—
98	C₂×C₈	C₇×C₁₆	1	2	yes	—
99	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
102	C₂×C₈	C₂×C₇×C₈	1	2	yes	—
103	C₂×C₈	C₇×M₁₆	1	1	yes	characteristic
104	C₂×C₈	C₇×M₁₆	1	1	yes	characteristic
105	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₂×C₇×C₈	1	2	yes	—
108	C₁₆	—	2	2	no	—
109	C₁₆	C₇×C₄⋊C₄	1	1	yes	characteristic
110	C₁₆	C₇×C₄⋊C₄	1	1	yes	characteristic
111	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
112	C₁₆	C₂×C₇×Q₈	1	1	yes	characteristic
113	C₂×C₇	—	2	1	no	—
114	C₂×C₇	—	2	1	no	—
115	C₂×C₇	#1422	1	2	yes	direct product factor
116	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
117	C₂³	C₇×M₃₂	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	C₇×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₄	C₇×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
123	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
124	C₂×C₄	C₇×M₃₂	1	1	yes	characteristic
125	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
126	C₈	C₇×C₄⋊C₈	1	1	yes	characteristic
127	C₈	C₇×C₄⋊C₈	1	1	yes	characteristic
128	C₈	C₂×C₇×C₁₆	1	4	yes	—
129	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
130	C₈	C₂×C₇×M₁₆	1	1	yes	characteristic
131	C₇	C₂×#1422	1	1	yes	characteristic, direct product factor, Sylow
132	C₂²	—	2	2	no	—
133	C₂²	C₇×C₄⋊C₁₆	1	1	yes	characteristic
134	C₂²	C₇×C₄⋊C₁₆	1	1	yes	characteristic
135	C₂²	C₂×C₇×M₃₂	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	C₇×C₄⋊C₁₆	1	1	yes	characteristic
138	C₄	C₇×C₄⋊C₁₆	1	1	yes	characteristic
139	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
140	C₄	C₂²×C₇×C₁₆	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	C₇×#1422	1	2	yes	direct product factor
144	C₂	C₂×C₇×C₄⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₁₆	D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₄⋊C₁₆
prime radical	C₂³×C₇	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₁₆ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1422	C₂	direct (N × H)
12. C₂×#1422	C₇	direct (N × H)
30. #1422	C₂×C₇	direct (N × H)
115. C₂×C₇	#1422	direct (N × H)
131. C₇	C₂×#1422	direct (N × H)
143. C₂	C₇×#1422	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×C₇×M₆₄	2
C₂×C₄×C₇×C₁₆	2
C₇×#1422	2
C₂×#1422	7

Subgroups — C2×C7×#1422 #13190845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1422 #13190845