Subgroups — C₂²×C₇×#498 #13237868

Order 1792.

2398 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	C₂²×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₇×#498	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₇×#106	C₂	1	4	yes	maximal
4	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇×Q₈	—	2	2	no	—
6	C₂³×C₇×C₈	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊Q₈	—	2	12	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	24	yes	—
9	C₂²×C₇×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₇×#498	C₂²	1	64	yes	direct product factor
13	C₂×C₇×#106	C₂²	1	48	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	12	yes	—
15	C₂²×#498	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇×Q₈	—	2	24	no	—
17	C₂²×C₇×Q₈	—	4	16	no	—
18	C₂³×C₄×C₇	—	2	4	no	—
19	C₂³×C₄×C₇	D₄	1	2	yes	—
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²×C₇×C₈	Q₈	1	16	yes	complemented
22	C₂²×C₇×C₈	C₂³	1	12	yes	—
23	C₇×C₄⋊Q₈	—	2	48	no	—
24	C₇×C₄⋊Q₈	C₂³	1	32	yes	—
25	C₂×C₇×C₄⋊C₄	—	2	24	no	—
26	C₂×C₇×C₄⋊C₄	—	2	12	no	—
27	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
28	C₂×C₄²×C₇	D₄	1	6	yes	—
29	C₂×C₄²×C₇	C₂³	1	6	yes	—
30	C₇×#106	C₂³	1	64	yes	—
31	C₄×C₇×C₈	C₂³	1	16	yes	—
32	C₂²×C₄⋊Q₈	C₂×C₇	1	2	yes	—
33	C₂×#498	C₂×C₇	1	24	yes	direct product factor
34	C₂²×#106	C₂×C₇	1	4	yes	—
35	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
36	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₇×Q₈	—	2	32	no	—
38	C₂×C₇×Q₈	—	4	96	no	—
39	C₂²×C₄×C₇	—	2	24	no	—
40	C₂²×C₄×C₇	—	4	16	no	—
41	C₂²×C₄×C₇	SD₁₆	1	16	yes	—
42	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
44	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
45	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
46	C₂×C₇×C₈	C₂×Q₈	1	48	yes	complemented
47	C₂×C₇×C₈	C₂⁴	1	8	yes	—
48	C₇×C₄⋊C₄	—	2	32	no	—
49	C₇×C₄⋊C₄	—	2	24	no	—
50	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
51	C₄²×C₇	C₂×D₄	1	12	yes	—
52	C₄²×C₇	C₂⁴	1	4	yes	—
53	C₂³×Q₈	—	2	2	no	—
54	C₂³×C₈	C₂²×C₇	1	2	yes	—
55	C₂×C₄⋊Q₈	—	2	12	no	—
56	C₂×C₄⋊Q₈	C₂²×C₇	1	24	yes	—
57	C₂²×C₄⋊C₄	—	2	2	no	—
58	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
59	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
60	#498	C₂²×C₇	1	64	yes	direct product factor
61	C₂×#106	C₂²×C₇	1	48	yes	—
62	C₂×C₄×C₈	C₂²×C₇	1	12	yes	—
63	C₂³×C₇	C₄⋊Q₈	1	4	yes	—
64	C₂³×C₇	C₂×SD₁₆	1	8	yes	—
65	C₂³×C₇	C₂²×D₄	1	3	yes	—
66	C₇×Q₈	—	4	64	no	—
67	C₂×C₄×C₇	—	2	16	no	—
68	C₂×C₄×C₇	—	4	48	no	—
69	C₂×C₄×C₇	C₄⋊Q₈	1	12	yes	—
70	C₂×C₄×C₇	C₂×SD₁₆	1	48	yes	—
71	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
72	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
73	C₂×C₄×C₇	C₂²×Q₈	1	12	yes	—
74	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
75	C₇×C₈	C₂²×Q₈	1	16	yes	complemented
76	C₂²×Q₈	—	2	24	no	—
77	C₂²×Q₈	—	4	16	no	—
78	C₂³×C₄	—	2	4	no	—
79	C₂³×C₄	C₇×D₄	1	2	yes	—
80	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
81	C₂²×C₈	C₇×Q₈	1	16	yes	complemented
82	C₂²×C₈	C₂³×C₇	1	12	yes	—
83	C₄⋊Q₈	—	2	48	no	—
84	C₄⋊Q₈	C₂³×C₇	1	32	yes	—
85	C₂×C₄⋊C₄	—	2	24	no	—
86	C₂×C₄⋊C₄	—	2	12	no	—
87	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
88	C₂×C₄²	C₇×D₄	1	6	yes	—
89	C₂×C₄²	C₂³×C₇	1	6	yes	—
90	#106	C₂³×C₇	1	64	yes	—
91	C₄×C₈	C₂³×C₇	1	16	yes	—
92	C₂²×C₇	#498	1	16	yes	direct product factor
93	C₂²×C₇	C₂×C₄⋊Q₈	1	6	yes	—
94	C₂²×C₇	C₂²×SD₁₆	1	12	yes	—
95	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
96	C₄×C₇	—	4	16	no	—
97	C₄×C₇	C₂×C₄⋊Q₈	1	6	yes	—
98	C₄×C₇	C₂²×SD₁₆	1	16	yes	—
99	C₄×C₇	C₂³×Q₈	1	2	yes	—
100	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
101	C₂×Q₈	—	2	32	no	—
102	C₂×Q₈	—	4	96	no	—
103	C₂²×C₄	—	2	24	no	—
104	C₂²×C₄	—	4	16	no	—
105	C₂²×C₄	C₇×SD₁₆	1	16	yes	—
106	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
107	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
108	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
109	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
110	C₂×C₈	C₂×C₇×Q₈	1	48	yes	complemented
111	C₂×C₈	C₂⁴×C₇	1	8	yes	—
112	C₄⋊C₄	—	2	32	no	—
113	C₄⋊C₄	—	2	24	no	—
114	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
115	C₄²	C₂×C₇×D₄	1	12	yes	—
116	C₄²	C₂⁴×C₇	1	4	yes	—
117	C₂×C₇	C₂×#498	1	12	yes	direct product factor
118	C₂×C₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₇	C₂³×SD₁₆	1	2	yes	—
120	C₂³	C₇×C₄⋊Q₈	1	4	yes	—
121	C₂³	C₂×C₇×SD₁₆	1	8	yes	—
122	C₂³	C₂²×C₇×D₄	1	3	yes	—
123	Q₈	—	4	64	no	—
124	C₂×C₄	—	2	16	no	—
125	C₂×C₄	—	4	48	no	—
126	C₂×C₄	C₇×C₄⋊Q₈	1	12	yes	—
127	C₂×C₄	C₂×C₇×SD₁₆	1	48	yes	—
128	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
129	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
130	C₂×C₄	C₂²×C₇×Q₈	1	12	yes	—
131	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
132	C₈	C₂²×C₇×Q₈	1	16	yes	complemented
133	C₇	C₂²×#498	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²	C₇×#498	1	16	yes	direct product factor
135	C₂²	C₂×C₇×C₄⋊Q₈	1	6	yes	—
136	C₂²	C₂²×C₇×SD₁₆	1	12	yes	—
137	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
138	C₄	—	4	16	no	—
139	C₄	C₂×C₇×C₄⋊Q₈	1	6	yes	—
140	C₄	C₂²×C₇×SD₁₆	1	16	yes	—
141	C₄	C₂³×C₇×Q₈	1	2	yes	—
142	C₂	C₂×C₇×#498	1	12	yes	direct product factor
143	C₂	C₂²×C₇×C₄⋊Q₈	1	1	yes	characteristic
144	C₂	C₂³×C₇×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#498	C₂	direct (N × H)
12. C₇×#498	C₂²	direct (N × H)
15. C₂²×#498	C₇	direct (N × H)
21. C₂²×C₇×C₈	Q₈	semidirect
33. C₂×#498	C₂×C₇	direct (N × H)
46. C₂×C₇×C₈	C₂×Q₈	semidirect
60. #498	C₂²×C₇	direct (N × H)
75. C₇×C₈	C₂²×Q₈	semidirect
81. C₂²×C₈	C₇×Q₈	semidirect
92. C₂²×C₇	#498	direct (N × H)
110. C₂×C₈	C₂×C₇×Q₈	semidirect
117. C₂×C₇	C₂×#498	direct (N × H)
132. C₈	C₂²×C₇×Q₈	semidirect
133. C₇	C₂²×#498	direct (N × H)
134. C₂²	C₇×#498	direct (N × H)
142. C₂	C₂×C₇×#498	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⋊Q₈	2
C₂×C₇×#498	2
C₂²×C₇×#106	2
C₂²×C₄×C₇×C₈	2
C₂²×#498	7

Subgroups — C22×C7×#498 #13237868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#498 #13237868