Subgroups — C₂²×#90503 #14260381

Order 1792.

11230 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#90503	C₂	1	48	yes	maximal, direct product factor
3	C₂²×Q₈×Dic₇	C₂	1	8	yes	maximal, complemented
4	C₂²×#6489	C₂	1	6	yes	maximal
5	C₂³×C₇×Q₈	C₄	1	32	yes	complemented
6	C₂³×C₇×Q₈	C₂²	1	4	yes	—
7	C₂³×C₇×Q₈	C₂²	1	24	yes	—
8	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
9	C₂²×C₂²⋊Dic₇	C₂²	1	4	yes	—
10	#90503	C₂²	1	256	yes	direct product factor
11	C₂×Q₈×Dic₇	C₂²	1	192	yes	complemented
12	C₂×#6489	C₂²	1	144	yes	—
13	C₂²×C₄⋊Dic₇	C₂²	1	12	yes	—
14	C₂²×C₄×Dic₇	C₂²	1	12	yes	—
15	C₂²×#519	—	7	1	no	maximal, Sylow
16	C₂⁵×C₇	C₂³	1	1	yes	characteristic
17	C₂²×C₇×Q₈	C₂×C₄	1	64	yes	—
18	C₂²×C₇×Q₈	C₂×C₄	1	384	yes	complemented
19	C₂²×C₇×Q₈	C₂³	1	48	yes	—
20	C₂²×C₇×Q₈	C₂³	1	64	yes	—
21	C₂³×C₄×C₇	C₂×C₄	1	48	yes	—
22	C₂³×C₄×C₇	C₂³	1	6	yes	—
23	C₂³×C₄×C₇	C₂³	1	36	yes	—
24	C₂³×Dic₇	C₂³	1	8	yes	—
25	C₂×C₂²⋊Dic₇	C₂³	1	48	yes	—
26	Q₈×Dic₇	C₂³	1	512	yes	complemented
27	#6489	C₂³	1	384	yes	—
28	C₂×C₄⋊Dic₇	C₂³	1	144	yes	—
29	C₂×C₄×Dic₇	C₂³	1	144	yes	—
30	C₂⁴×Q₈	D₇	1	1	yes	characteristic
31	C₂×#519	—	7	48	no	—
32	C₂²×C₄×Q₈	—	7	8	no	—
33	C₂²×#117	—	7	6	no	—
34	C₂⁴×C₇	—	2	8	no	—
35	C₂⁴×C₇	C₂²×C₄	1	8	yes	—
36	C₂⁴×C₇	C₂⁴	1	6	yes	—
37	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
38	C₂×C₇×Q₈	C₂²×C₄	1	384	yes	—
39	C₂×C₇×Q₈	C₂²×C₄	1	512	yes	complemented
40	C₂×C₇×Q₈	C₂⁴	1	64	yes	—
41	C₂²×C₄×C₇	C₂²×C₄	1	48	yes	—
42	C₂²×C₄×C₇	C₂²×C₄	1	288	yes	—
43	C₂²×C₄×C₇	C₂⁴	1	36	yes	—
44	C₂²×C₄×C₇	C₂⁴	1	48	yes	—
45	C₂²×Dic₇	—	2	32	no	—
46	C₂²×Dic₇	C₂⁴	1	48	yes	—
47	C₂²⋊Dic₇	C₂⁴	1	64	yes	—
48	C₄⋊Dic₇	C₂⁴	1	192	yes	—
49	C₄×Dic₇	C₂⁴	1	192	yes	—
50	C₂³×Q₈	Dic₇	1	32	yes	complemented
51	C₂³×Q₈	C₂×D₇	1	4	yes	—
52	C₂³×Q₈	C₂×D₇	1	24	yes	—
53	C₂⁴×C₄	C₂×D₇	1	3	yes	—
54	#519	—	7	256	no	—
55	C₂×C₄×Q₈	—	7	192	no	—
56	C₂×#117	—	7	144	no	—
57	C₂²×C₄⋊C₄	—	7	12	no	—
58	C₂²×C₂²⋊C₄	—	7	4	no	—
59	C₂²×C₄²	—	7	12	no	—
60	C₂³×C₇	—	2	32	no	—
61	C₂³×C₇	—	2	24	no	—
62	C₂³×C₇	C₂³×C₄	1	24	yes	—
63	C₂³×C₇	C₂³×C₄	1	4	yes	—
64	C₂³×C₇	2₋¹⁺⁴	1	8	yes	—
65	C₂³×C₇	C₂⁵	1	4	yes	—
66	C₂³×C₇	C₂⁵	1	3	yes	—
67	C₇×Q₈	C₂³×C₄	1	256	yes	—
68	C₂×C₄×C₇	C₂³×C₄	1	144	yes	—
69	C₂×C₄×C₇	C₂³×C₄	1	192	yes	—
70	C₂×C₄×C₇	C₂⁵	1	24	yes	—
71	C₂×Dic₇	—	2	96	no	—
72	C₂×Dic₇	C₂⁵	1	32	yes	—
73	C₂⁵	C₂²×D₇	1	1	yes	characteristic
74	C₂²×Q₈	C₂×Dic₇	1	64	yes	—
75	C₂²×Q₈	C₂×Dic₇	1	384	yes	complemented
76	C₂²×Q₈	C₂²×D₇	1	48	yes	—
77	C₂²×Q₈	C₂²×D₇	1	64	yes	—
78	C₂³×C₄	—	7	8	no	—
79	C₂³×C₄	C₂×Dic₇	1	48	yes	—
80	C₂³×C₄	C₂²×D₇	1	36	yes	—
81	C₂³×C₄	C₂²×D₇	1	6	yes	—
82	C₄×Q₈	—	7	512	no	—
83	#117	—	7	384	no	—
84	C₂×C₄⋊C₄	—	7	144	no	—
85	C₂×C₂²⋊C₄	—	7	48	no	—
86	C₂×C₄²	—	7	144	no	—
87	C₂²×C₇	—	2	8	no	—
88	C₂²×C₇	—	2	48	no	—
89	C₂²×C₇	#519	1	16	yes	complemented
90	C₂²×C₇	C₂⁴×C₄	1	8	yes	—
91	C₂²×C₇	C₂⁴×C₄	1	6	yes	—
92	C₂²×C₇	C₂×2₋¹⁺⁴	1	12	yes	—
93	C₂²×C₇	C₂⁶	1	1	yes	characteristic
94	C₄×C₇	C₂⁴×C₄	1	48	yes	—
95	Dic₇	—	2	32	no	—
96	C₂⁴	—	2	8	no	—
97	C₂⁴	C₂²×Dic₇	1	8	yes	—
98	C₂⁴	C₂³×D₇	1	6	yes	—
99	C₂⁴	C₂³×D₇	1	1	yes	characteristic
100	C₂×Q₈	C₂²×Dic₇	1	384	yes	—
101	C₂×Q₈	C₂²×Dic₇	1	512	yes	complemented
102	C₂×Q₈	C₂³×D₇	1	64	yes	—
103	C₂²×C₄	—	7	48	no	—
104	C₂²×C₄	—	14	32	no	—
105	C₂²×C₄	C₂²×Dic₇	1	48	yes	—
106	C₂²×C₄	C₂²×Dic₇	1	288	yes	—
107	C₂²×C₄	C₂³×D₇	1	36	yes	—
108	C₂²×C₄	C₂³×D₇	1	48	yes	—
109	C₄⋊C₄	—	7	192	no	—
110	C₂²⋊C₄	—	7	64	no	—
111	C₄²	—	7	192	no	—
112	C₂×C₇	—	2	8	no	—
113	C₂×C₇	C₂×#519	1	12	yes	complemented
114	C₂×C₇	C₂⁵×C₄	1	1	yes	characteristic
115	C₂×C₇	C₂²×2₋¹⁺⁴	1	2	yes	—
116	C₂³	—	2	32	no	—
117	C₂³	—	2	24	no	—
118	C₂³	#6551	1	8	yes	—
119	C₂³	C₂³×Dic₇	1	24	yes	—
120	C₂³	C₂³×Dic₇	1	4	yes	—
121	C₂³	C₂⁴×D₇	1	4	yes	—
122	C₂³	C₂⁴×D₇	1	3	yes	—
123	Q₈	C₂³×Dic₇	1	256	yes	—
124	C₂×C₄	—	7	32	no	—
125	C₂×C₄	—	14	96	no	—
126	C₂×C₄	C₂³×Dic₇	1	144	yes	—
127	C₂×C₄	C₂³×Dic₇	1	192	yes	—
128	C₂×C₄	C₂⁴×D₇	1	24	yes	—
129	C₇	C₂²×#519	1	1	yes	characteristic, complemented, Sylow
130	C₂²	—	2	8	no	—
131	C₂²	—	2	48	no	—
132	C₂²	#90503	1	16	yes	direct product factor
133	C₂²	C₂×#6551	1	12	yes	—
134	C₂²	C₂⁴×Dic₇	1	8	yes	—
135	C₂²	C₂⁴×Dic₇	1	6	yes	—
136	C₂²	C₂⁵×D₇	1	1	yes	characteristic
137	C₄	—	14	32	no	—
138	C₄	C₂⁴×Dic₇	1	48	yes	—
139	C₂	—	2	8	no	—
140	C₂	C₂×#90503	1	12	yes	direct product factor
141	C₂	C₂²×#6551	1	2	yes	—
142	C₂	C₂⁵×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇
commutator	C₂×C₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₇
Fitting	C₂⁴×C₇×Q₈	C₂
socle	C₂⁴×C₇	C₂⁴
prime radical	C₂⁵×C₇	C₂³
cosocle	C₂²×C₇	C₂⁶
hypercenter	C₂⁴×Q₈	D₇
hypocenter	C₇	C₂²×#519
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#90503	C₂	direct (N × H)
3. C₂²×Q₈×Dic₇	C₂	semidirect
5. C₂³×C₇×Q₈	C₄	semidirect
10. #90503	C₂²	direct (N × H)
11. C₂×Q₈×Dic₇	C₂²	semidirect
18. C₂²×C₇×Q₈	C₂×C₄	semidirect
26. Q₈×Dic₇	C₂³	semidirect
39. C₂×C₇×Q₈	C₂²×C₄	semidirect
50. C₂³×Q₈	Dic₇	semidirect
75. C₂²×Q₈	C₂×Dic₇	semidirect
89. C₂²×C₇	#519	semidirect
101. C₂×Q₈	C₂²×Dic₇	semidirect
113. C₂×C₇	C₂×#519	semidirect
129. C₇	C₂²×#519	semidirect
132. C₂²	#90503	direct (N × H)
140. C₂	C₂×#90503	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×Q₈	2
C₂×#90503	2
C₂²×Q₈×Dic₇	2
C₂²×#6489	2
C₂²×#519	7

Subgroups — C22×#90503 #14260381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#90503 #14260381