Subgroups — C₂×C₇×#1842 #13198005

Order 1792.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊C₈	C₂	1	6	yes	maximal
2	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#1842	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₇×C₈	C₂²	1	4	yes	—
5	C₂²×C₄²×C₇	C₄	1	3	yes	—
6	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₂²	1	3	yes	—
8	C₂×C₇×C₄⋊C₈	—	2	24	no	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	24	yes	—
10	C₄³×C₇	C₄	1	4	yes	—
11	C₄³×C₇	C₂²	1	4	yes	—
12	C₂×#1842	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
14	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
15	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
16	C₂²×C₇×C₈	—	2	24	no	—
17	C₂²×C₇×C₈	C₂³	1	8	yes	—
18	C₂×C₄²×C₇	—	2	12	no	—
19	C₂×C₄²×C₇	C₈	1	8	yes	complemented
20	C₂×C₄²×C₇	C₂×C₄	1	12	yes	—
21	C₂×C₄²×C₇	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇	D₄	1	12	yes	—
24	C₂×C₄²×C₇	Q₈	1	12	yes	—
25	C₂×C₄²×C₇	C₂³	1	6	yes	—
26	C₇×C₄⋊C₈	—	2	48	no	—
27	C₂²×C₄⋊C₈	C₂×C₇	1	6	yes	—
28	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
29	#1842	C₂×C₇	1	8	yes	direct product factor
30	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	—	2	12	no	—
32	C₂²×C₄×C₇	C₄⋊C₄	1	12	yes	—
33	C₂²×C₄×C₇	C₄⋊C₄	1	12	yes	—
34	C₂²×C₄×C₇	C₂×C₈	1	12	yes	—
35	C₂²×C₄×C₇	M₁₆	1	12	yes	—
36	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
40	C₂²×C₄×C₇	C₂×Q₈	1	6	yes	—
41	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
42	C₂×C₇×C₈	—	2	24	no	—
43	C₂×C₇×C₈	—	4	16	no	—
44	C₄²×C₇	—	2	24	no	—
45	C₄²×C₇	C₄⋊C₄	1	24	yes	—
46	C₄²×C₇	C₂×C₈	1	16	yes	complemented
47	C₄²×C₇	C₂×D₄	1	12	yes	—
48	C₄²×C₇	C₂×Q₈	1	12	yes	—
49	C₂³×C₈	C₂²×C₇	1	4	yes	—
50	C₂²×C₄²	C₄×C₇	1	3	yes	—
51	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
52	C₂²×C₄²	C₂²×C₇	1	3	yes	—
53	C₂×C₄⋊C₈	—	2	24	no	—
54	C₂×C₄⋊C₈	C₂²×C₇	1	24	yes	—
55	C₄³	C₄×C₇	1	4	yes	—
56	C₄³	C₂²×C₇	1	4	yes	—
57	C₂³×C₇	C₂×C₄⋊C₄	1	6	yes	—
58	C₂³×C₇	C₂²×C₈	1	2	yes	—
59	C₂³×C₇	C₂×M₁₆	1	6	yes	—
60	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
61	C₂×C₄×C₇	—	2	12	no	—
62	C₂×C₄×C₇	—	2	24	no	—
63	C₂×C₄×C₇	C₄⋊C₈	1	48	yes	complemented
64	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
65	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₇	C₂×C₄⋊C₄	1	12	yes	—
67	C₂×C₄×C₇	#127	1	4	yes	—
68	C₂×C₄×C₇	C₄⋊Q₈	1	12	yes	—
69	C₂×C₄×C₇	C₂²×C₈	1	12	yes	—
70	C₂×C₄×C₇	C₂×M₁₆	1	12	yes	—
71	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
72	C₂×C₄×C₇	C₂²×Q₈	1	3	yes	—
73	C₇×C₈	—	4	8	no	—
74	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
75	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
76	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
77	C₂²×C₈	—	2	24	no	—
78	C₂²×C₈	C₂³×C₇	1	8	yes	—
79	C₂×C₄²	—	2	12	no	—
80	C₂×C₄²	C₇×C₈	1	8	yes	complemented
81	C₂×C₄²	C₂×C₄×C₇	1	12	yes	—
82	C₂×C₄²	C₂×C₄×C₇	1	6	yes	—
83	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
84	C₂×C₄²	C₇×D₄	1	12	yes	—
85	C₂×C₄²	C₇×Q₈	1	12	yes	—
86	C₂×C₄²	C₂³×C₇	1	6	yes	—
87	C₄⋊C₈	—	2	48	no	—
88	C₂²×C₇	#384	1	4	yes	—
89	C₂²×C₇	C₂×C₄⋊C₈	1	12	yes	—
90	C₂²×C₇	#423	1	12	yes	—
91	C₂²×C₇	C₂²×C₄⋊C₄	1	3	yes	—
92	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
93	C₂²×C₇	C₂²×M₁₆	1	3	yes	—
94	C₄×C₇	—	2	12	no	—
95	C₄×C₇	#384	1	4	yes	—
96	C₄×C₇	C₂×C₄⋊C₈	1	24	yes	complemented
97	C₄×C₇	C₂×#127	1	1	yes	characteristic
98	C₄×C₇	C₂×C₄⋊Q₈	1	3	yes	—
99	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
100	C₂²×C₄	—	2	12	no	—
101	C₂²×C₄	C₇×C₄⋊C₄	1	12	yes	—
102	C₂²×C₄	C₇×C₄⋊C₄	1	12	yes	—
103	C₂²×C₄	C₂×C₇×C₈	1	12	yes	—
104	C₂²×C₄	C₇×M₁₆	1	12	yes	—
105	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
106	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
107	C₂²×C₄	C₂²×C₄×C₇	1	6	yes	—
108	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
109	C₂²×C₄	C₂×C₇×Q₈	1	6	yes	—
110	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
111	C₂×C₈	—	2	24	no	—
112	C₂×C₈	—	4	16	no	—
113	C₄²	—	2	24	no	—
114	C₄²	C₇×C₄⋊C₄	1	24	yes	—
115	C₄²	C₂×C₇×C₈	1	16	yes	complemented
116	C₄²	C₂×C₇×D₄	1	12	yes	—
117	C₄²	C₂×C₇×Q₈	1	12	yes	—
118	C₂×C₇	#1842	1	8	yes	direct product factor
119	C₂×C₇	C₂×#384	1	1	yes	characteristic
120	C₂×C₇	C₂²×C₄⋊C₈	1	3	yes	—
121	C₂×C₇	C₂×#423	1	3	yes	—
122	C₂³	C₂×C₇×C₄⋊C₄	1	6	yes	—
123	C₂³	C₂²×C₇×C₈	1	2	yes	—
124	C₂³	C₂×C₇×M₁₆	1	6	yes	—
125	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
126	C₂×C₄	—	2	24	no	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	C₇×C₄⋊C₈	1	48	yes	complemented
129	C₂×C₄	C₂×C₇×C₄⋊C₄	1	12	yes	—
130	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
131	C₂×C₄	C₂×C₇×C₄⋊C₄	1	12	yes	—
132	C₂×C₄	C₇×#127	1	4	yes	—
133	C₂×C₄	C₇×C₄⋊Q₈	1	12	yes	—
134	C₂×C₄	C₂²×C₇×C₈	1	12	yes	—
135	C₂×C₄	C₂×C₇×M₁₆	1	12	yes	—
136	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
137	C₂×C₄	C₂²×C₇×Q₈	1	3	yes	—
138	C₈	—	4	8	no	—
139	C₇	C₂×#1842	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²	C₇×#384	1	4	yes	—
141	C₂²	C₂×C₇×C₄⋊C₈	1	12	yes	—
142	C₂²	C₇×#423	1	12	yes	—
143	C₂²	C₂²×C₇×C₄⋊C₄	1	3	yes	—
144	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
145	C₂²	C₂²×C₇×M₁₆	1	3	yes	—
146	C₄	—	2	12	no	—
147	C₄	C₇×#384	1	4	yes	—
148	C₄	C₂×C₇×C₄⋊C₈	1	24	yes	complemented
149	C₄	C₂×C₇×#127	1	1	yes	characteristic
150	C₄	C₂×C₇×C₄⋊Q₈	1	3	yes	—
151	C₂	C₇×#1842	1	8	yes	direct product factor
152	C₂	C₂×C₇×#384	1	1	yes	characteristic
153	C₂	C₂²×C₇×C₄⋊C₈	1	3	yes	—
154	C₂	C₂×C₇×#423	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1842	C₂	direct (N × H)
12. C₂×#1842	C₇	direct (N × H)
19. C₂×C₄²×C₇	C₈	semidirect
29. #1842	C₂×C₇	direct (N × H)
46. C₄²×C₇	C₂×C₈	semidirect
63. C₂×C₄×C₇	C₄⋊C₈	semidirect
80. C₂×C₄²	C₇×C₈	semidirect
96. C₄×C₇	C₂×C₄⋊C₈	semidirect
115. C₄²	C₂×C₇×C₈	semidirect
118. C₂×C₇	#1842	direct (N × H)
128. C₂×C₄	C₇×C₄⋊C₈	semidirect
139. C₇	C₂×#1842	direct (N × H)
148. C₄	C₂×C₇×C₄⋊C₈	semidirect
151. C₂	C₇×#1842	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×C₄⋊C₈	2
C₂×C₄³×C₇	2
C₇×#1842	2
C₂×#1842	7

Subgroups — C2×C7×#1842 #13198005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1842 #13198005