Subgroups — C₂³×#14523977 #31719476

Order 17152.

3614 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
2	C₂²×#14523977	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₆₇	—	2	2	no	—
5	C₂²×C₄⋊Dic₆₇	—	2	14	no	—
6	C₂²×C₄⋊Dic₆₇	C₂²	1	28	yes	—
7	C₂×#14523977	C₂²	1	112	yes	direct product factor
8	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₄	1	16	yes	complemented
10	C₂³×C₈×C₆₇	C₂²	1	14	yes	—
11	C₂³×Dic₆₇	—	2	28	no	—
12	C₂³×Dic₆₇	—	4	16	no	—
13	C₂×C₄⋊Dic₆₇	—	2	84	no	—
14	C₂×C₄⋊Dic₆₇	C₂³	1	56	yes	—
15	#14523977	C₂³	1	64	yes	direct product factor
16	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
18	C₂³×C₄×C₆₇	D₄	1	7	yes	—
19	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	112	yes	complemented
22	C₂²×C₈×C₆₇	C₂³	1	28	yes	—
23	C₂²×Dic₆₇	—	2	56	no	—
24	C₂²×Dic₆₇	—	4	112	no	—
25	C₄⋊Dic₆₇	—	2	56	no	—
26	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
27	C₂⁴×C₆₇	C₄⋊C₄	1	8	yes	—
28	C₂⁴×C₆₇	D₈	1	8	yes	—
29	C₂⁴×C₆₇	Q₁₆	1	8	yes	—
30	C₂⁴×C₆₇	C₂×D₄	1	7	yes	—
31	C₂²×C₄×C₆₇	C₄⋊C₄	1	56	yes	—
32	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
33	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
34	C₂²×C₄×C₆₇	C₂×Q₈	1	28	yes	—
35	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
36	C₂×C₈×C₆₇	C₂²×C₄	1	112	yes	complemented
37	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
38	C₂³×C₆₇	#107	1	64	yes	complemented
39	C₂³×C₆₇	C₂×C₄⋊C₄	1	28	yes	—
40	C₂³×C₆₇	C₂×D₈	1	28	yes	—
41	C₂³×C₆₇	C₂×Q₁₆	1	28	yes	—
42	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
43	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	84	yes	—
44	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
45	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
46	C₂×C₄×C₆₇	C₂²×Q₈	1	14	yes	—
47	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
48	C₂×Dic₆₇	—	2	16	no	—
49	C₂×Dic₆₇	—	4	112	no	—
50	C₈×C₆₇	C₂³×C₄	1	16	yes	complemented
51	C₂²×C₆₇	C₂×#107	1	112	yes	complemented
52	C₂²×C₆₇	C₂²×C₄⋊C₄	1	14	yes	—
53	C₂²×C₆₇	C₂²×D₈	1	14	yes	—
54	C₂²×C₆₇	C₂²×Q₁₆	1	14	yes	—
55	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
56	C₄×C₆₇	C₂²×C₄⋊C₄	1	14	yes	—
57	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
58	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
59	Dic₆₇	—	4	16	no	—
60	C₂³×#107	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	C₂²×#107	1	28	yes	complemented
62	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
64	C₂×C₆₇	C₂³×Q₁₆	1	1	yes	characteristic
65	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
66	C₂³×C₄⋊C₄	—	67	2	no	—
67	C₂²×#107	—	67	28	no	—
68	C₆₇	C₂³×#107	1	1	yes	characteristic, complemented, Sylow
69	C₂⁴×C₄	—	134	2	no	—
70	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
71	C₂³×C₈	Dic₆₇	1	16	yes	complemented
72	C₂³×C₈	C₂×D₆₇	1	14	yes	—
73	C₂²×C₄⋊C₄	—	67	28	no	—
74	C₂²×C₄⋊C₄	—	134	14	no	—
75	C₂×#107	—	67	112	no	—
76	C₂⁵	D₂₆₈	1	1	yes	characteristic
77	C₂³×C₄	—	268	16	no	—
78	C₂³×C₄	—	134	28	no	—
79	C₂³×C₄	Dic₁₃₄	1	8	yes	—
80	C₂³×C₄	D₂₆₈	1	7	yes	—
81	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
82	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
83	C₂²×C₈	C₂×Dic₆₇	1	112	yes	complemented
84	C₂²×C₈	C₂²×D₆₇	1	28	yes	—
85	C₂×C₄⋊C₄	—	67	56	no	—
86	C₂×C₄⋊C₄	—	134	84	no	—
87	#107	—	67	64	no	—
88	C₂⁴	C₄⋊Dic₆₇	1	8	yes	—
89	C₂⁴	D₅₃₆	1	8	yes	—
90	C₂⁴	Dic₂₆₈	1	8	yes	—
91	C₂⁴	C₂×D₂₆₈	1	7	yes	—
92	C₂²×C₄	—	268	112	no	—
93	C₂²×C₄	—	134	56	no	—
94	C₂²×C₄	C₄⋊Dic₆₇	1	56	yes	—
95	C₂²×C₄	C₂²×Dic₆₇	1	28	yes	—
96	C₂²×C₄	C₂×D₂₆₈	1	21	yes	—
97	C₂²×C₄	C₂×Dic₁₃₄	1	28	yes	—
98	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
99	C₂×C₈	C₂²×Dic₆₇	1	112	yes	complemented
100	C₂×C₈	C₂³×D₆₇	1	8	yes	—
101	C₄⋊C₄	—	67	16	no	—
102	C₄⋊C₄	—	134	56	no	—
103	C₂³	#14523977	1	64	yes	direct product factor
104	C₂³	C₂×C₄⋊Dic₆₇	1	28	yes	—
105	C₂³	C₂×D₅₃₆	1	28	yes	—
106	C₂³	C₂×Dic₂₆₈	1	28	yes	—
107	C₂³	C₂²×D₂₆₈	1	7	yes	—
108	C₂×C₄	—	268	112	no	—
109	C₂×C₄	—	134	16	no	—
110	C₂×C₄	C₂×C₄⋊Dic₆₇	1	84	yes	—
111	C₂×C₄	C₂³×Dic₆₇	1	14	yes	—
112	C₂×C₄	C₂²×D₂₆₈	1	7	yes	—
113	C₂×C₄	C₂²×Dic₁₃₄	1	14	yes	—
114	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
115	C₈	C₂³×Dic₆₇	1	16	yes	complemented
116	C₂²	C₂×#14523977	1	112	yes	direct product factor
117	C₂²	C₂²×C₄⋊Dic₆₇	1	14	yes	—
118	C₂²	C₂²×D₅₃₆	1	14	yes	—
119	C₂²	C₂²×Dic₂₆₈	1	14	yes	—
120	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
121	C₄	—	268	16	no	—
122	C₄	C₂²×C₄⋊Dic₆₇	1	14	yes	—
123	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
124	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
125	C₂	C₂²×#14523977	1	28	yes	direct product factor
126	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
127	C₂	C₂³×D₅₃₆	1	1	yes	characteristic
128	C₂	C₂³×Dic₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	D₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂³×#107
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#14523977	C₂	direct (N × H)
7. C₂×#14523977	C₂²	direct (N × H)
9. C₂³×C₈×C₆₇	C₄	semidirect
15. #14523977	C₂³	direct (N × H)
21. C₂²×C₈×C₆₇	C₂×C₄	semidirect
36. C₂×C₈×C₆₇	C₂²×C₄	semidirect
38. C₂³×C₆₇	#107	semidirect
50. C₈×C₆₇	C₂³×C₄	semidirect
51. C₂²×C₆₇	C₂×#107	semidirect
61. C₂×C₆₇	C₂²×#107	semidirect
68. C₆₇	C₂³×#107	semidirect
71. C₂³×C₈	Dic₆₇	semidirect
83. C₂²×C₈	C₂×Dic₆₇	semidirect
99. C₂×C₈	C₂²×Dic₆₇	semidirect
103. C₂³	#14523977	direct (N × H)
115. C₈	C₂³×Dic₆₇	semidirect
116. C₂²	C₂×#14523977	direct (N × H)
125. C₂	C₂²×#14523977	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₆₇	2
C₂²×#14523977	2
C₂⁴×C₈×C₆₇	2
C₂³×#107	67

Subgroups — C23×#14523977 #31719476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523977 #31719476