Subgroups — C₂²×C₇×#500 #13237870

Order 1792.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 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×#500	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₇×#107	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×#500	C₂²	1	64	yes	direct product factor
13	C₂×C₇×#107	C₂²	1	48	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	12	yes	—
15	C₂²×#500	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	1	yes	characteristic
20	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²×C₇×C₈	Q₈	1	16	yes	complemented
23	C₂²×C₇×C₈	C₂³	1	12	yes	—
24	C₇×C₄⋊Q₈	—	2	48	no	—
25	C₇×C₄⋊Q₈	C₂³	1	32	yes	—
26	C₂×C₇×C₄⋊C₄	—	2	24	no	—
27	C₂×C₇×C₄⋊C₄	—	2	12	no	—
28	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
29	C₂×C₄²×C₇	D₄	1	6	yes	—
30	C₂×C₄²×C₇	C₂³	1	6	yes	—
31	C₇×#107	C₂³	1	64	yes	—
32	C₄×C₇×C₈	C₂³	1	16	yes	—
33	C₂²×C₄⋊Q₈	C₂×C₇	1	2	yes	—
34	C₂×#500	C₂×C₇	1	24	yes	direct product factor
35	C₂²×#107	C₂×C₇	1	4	yes	—
36	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
37	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₇×Q₈	—	2	32	no	—
39	C₂×C₇×Q₈	—	4	96	no	—
40	C₂²×C₄×C₇	—	2	24	no	—
41	C₂²×C₄×C₇	—	4	16	no	—
42	C₂²×C₄×C₇	D₈	1	8	yes	—
43	C₂²×C₄×C₇	Q₁₆	1	8	yes	—
44	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
45	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
46	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
47	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
48	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
49	C₂×C₇×C₈	C₂×Q₈	1	48	yes	complemented
50	C₂×C₇×C₈	C₂⁴	1	8	yes	—
51	C₇×C₄⋊C₄	—	2	32	no	—
52	C₇×C₄⋊C₄	—	2	24	no	—
53	C₇×C₄⋊C₄	C₂⁴	1	16	yes	—
54	C₄²×C₇	C₂×D₄	1	12	yes	—
55	C₄²×C₇	C₂⁴	1	4	yes	—
56	C₂³×Q₈	—	2	2	no	—
57	C₂³×C₈	C₂²×C₇	1	2	yes	—
58	C₂×C₄⋊Q₈	—	2	12	no	—
59	C₂×C₄⋊Q₈	C₂²×C₇	1	24	yes	—
60	C₂²×C₄⋊C₄	—	2	2	no	—
61	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
62	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
63	#500	C₂²×C₇	1	64	yes	direct product factor
64	C₂×#107	C₂²×C₇	1	48	yes	—
65	C₂×C₄×C₈	C₂²×C₇	1	12	yes	—
66	C₂³×C₇	C₄⋊Q₈	1	4	yes	—
67	C₂³×C₇	C₂×D₈	1	4	yes	—
68	C₂³×C₇	C₂×Q₁₆	1	4	yes	—
69	C₂³×C₇	C₂²×D₄	1	3	yes	—
70	C₇×Q₈	—	4	64	no	—
71	C₂×C₄×C₇	—	2	16	no	—
72	C₂×C₄×C₇	—	4	48	no	—
73	C₂×C₄×C₇	C₄⋊Q₈	1	12	yes	—
74	C₂×C₄×C₇	C₂×D₈	1	24	yes	—
75	C₂×C₄×C₇	C₂×Q₁₆	1	24	yes	—
76	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
77	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
78	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
79	C₂×C₄×C₇	C₂²×Q₈	1	12	yes	—
80	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
81	C₇×C₈	C₂²×Q₈	1	16	yes	complemented
82	C₂²×Q₈	—	2	24	no	—
83	C₂²×Q₈	—	4	16	no	—
84	C₂³×C₄	—	2	4	no	—
85	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
86	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
87	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
88	C₂²×C₈	C₇×Q₈	1	16	yes	complemented
89	C₂²×C₈	C₂³×C₇	1	12	yes	—
90	C₄⋊Q₈	—	2	48	no	—
91	C₄⋊Q₈	C₂³×C₇	1	32	yes	—
92	C₂×C₄⋊C₄	—	2	24	no	—
93	C₂×C₄⋊C₄	—	2	12	no	—
94	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
95	C₂×C₄²	C₇×D₄	1	6	yes	—
96	C₂×C₄²	C₂³×C₇	1	6	yes	—
97	#107	C₂³×C₇	1	64	yes	—
98	C₄×C₈	C₂³×C₇	1	16	yes	—
99	C₂²×C₇	#500	1	16	yes	direct product factor
100	C₂²×C₇	C₂×C₄⋊Q₈	1	6	yes	—
101	C₂²×C₇	C₂²×D₈	1	6	yes	—
102	C₂²×C₇	C₂²×Q₁₆	1	6	yes	—
103	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
104	C₄×C₇	—	4	16	no	—
105	C₄×C₇	C₂×C₄⋊Q₈	1	6	yes	—
106	C₄×C₇	C₂²×D₈	1	8	yes	—
107	C₄×C₇	C₂²×Q₁₆	1	8	yes	—
108	C₄×C₇	C₂³×Q₈	1	2	yes	—
109	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
110	C₂×Q₈	—	2	32	no	—
111	C₂×Q₈	—	4	96	no	—
112	C₂²×C₄	—	2	24	no	—
113	C₂²×C₄	—	4	16	no	—
114	C₂²×C₄	C₇×D₈	1	8	yes	—
115	C₂²×C₄	C₇×Q₁₆	1	8	yes	—
116	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
117	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
118	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
119	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
120	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
121	C₂×C₈	C₂×C₇×Q₈	1	48	yes	complemented
122	C₂×C₈	C₂⁴×C₇	1	8	yes	—
123	C₄⋊C₄	—	2	32	no	—
124	C₄⋊C₄	—	2	24	no	—
125	C₄⋊C₄	C₂⁴×C₇	1	16	yes	—
126	C₄²	C₂×C₇×D₄	1	12	yes	—
127	C₄²	C₂⁴×C₇	1	4	yes	—
128	C₂×C₇	C₂×#500	1	12	yes	direct product factor
129	C₂×C₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₇	C₂³×D₈	1	1	yes	characteristic
131	C₂×C₇	C₂³×Q₁₆	1	1	yes	characteristic
132	C₂³	C₇×C₄⋊Q₈	1	4	yes	—
133	C₂³	C₂×C₇×D₈	1	4	yes	—
134	C₂³	C₂×C₇×Q₁₆	1	4	yes	—
135	C₂³	C₂²×C₇×D₄	1	3	yes	—
136	Q₈	—	4	64	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	4	48	no	—
139	C₂×C₄	C₇×C₄⋊Q₈	1	12	yes	—
140	C₂×C₄	C₂×C₇×D₈	1	24	yes	—
141	C₂×C₄	C₂×C₇×Q₁₆	1	24	yes	—
142	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
143	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
144	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
145	C₂×C₄	C₂²×C₇×Q₈	1	12	yes	—
146	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
147	C₈	C₂²×C₇×Q₈	1	16	yes	complemented
148	C₇	C₂²×#500	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²	C₇×#500	1	16	yes	direct product factor
150	C₂²	C₂×C₇×C₄⋊Q₈	1	6	yes	—
151	C₂²	C₂²×C₇×D₈	1	6	yes	—
152	C₂²	C₂²×C₇×Q₁₆	1	6	yes	—
153	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
154	C₄	—	4	16	no	—
155	C₄	C₂×C₇×C₄⋊Q₈	1	6	yes	—
156	C₄	C₂²×C₇×D₈	1	8	yes	—
157	C₄	C₂²×C₇×Q₁₆	1	8	yes	—
158	C₄	C₂³×C₇×Q₈	1	2	yes	—
159	C₂	C₂×C₇×#500	1	12	yes	direct product factor
160	C₂	C₂²×C₇×C₄⋊Q₈	1	1	yes	characteristic
161	C₂	C₂³×C₇×D₈	1	1	yes	characteristic
162	C₂	C₂³×C₇×Q₁₆	1	1	yes	characteristic

	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×#500	C₂	direct (N × H)
12. C₇×#500	C₂²	direct (N × H)
15. C₂²×#500	C₇	direct (N × H)
22. C₂²×C₇×C₈	Q₈	semidirect
34. C₂×#500	C₂×C₇	direct (N × H)
49. C₂×C₇×C₈	C₂×Q₈	semidirect
63. #500	C₂²×C₇	direct (N × H)
81. C₇×C₈	C₂²×Q₈	semidirect
88. C₂²×C₈	C₇×Q₈	semidirect
99. C₂²×C₇	#500	direct (N × H)
121. C₂×C₈	C₂×C₇×Q₈	semidirect
128. C₂×C₇	C₂×#500	direct (N × H)
147. C₈	C₂²×C₇×Q₈	semidirect
148. C₇	C₂²×#500	direct (N × H)
149. C₂²	C₇×#500	direct (N × H)
159. C₂	C₂×C₇×#500	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×#500	2
C₂²×C₇×#107	2
C₂²×C₄×C₇×C₈	2
C₂²×#500	7

Subgroups — C22×C7×#500 #13237870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#500 #13237870