Subgroups — C₂³×#14523953 #31719213

Order 17152.

4126 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14523953	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₆₇	C₄	1	2	yes	—
6	C₂³×C₆₇⋊C₈	C₄	1	16	yes	complemented
7	C₂³×C₆₇⋊C₈	C₂²	1	14	yes	—
8	C₂²×C₄×Dic₆₇	C₄	1	14	yes	—
9	C₂²×C₄×Dic₆₇	C₂²	1	14	yes	—
10	C₂×#14523953	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₄	1	16	yes	complemented
13	C₂³×C₈×C₆₇	C₂²	1	14	yes	—
14	C₂³×Dic₆₇	—	2	16	no	—
15	C₂³×Dic₆₇	C₂×C₄	1	28	yes	—
16	C₂²×C₆₇⋊C₈	C₂×C₄	1	112	yes	complemented
17	C₂²×C₆₇⋊C₈	C₂³	1	28	yes	—
18	C₂×C₄×Dic₆₇	C₂×C₄	1	84	yes	—
19	C₂×C₄×Dic₆₇	C₂³	1	28	yes	—
20	#14523953	C₂³	1	64	yes	direct product factor
21	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₆₇	C₂×C₄	1	7	yes	—
23	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	112	yes	complemented
27	C₂²×C₈×C₆₇	C₂³	1	28	yes	—
28	C₂²×Dic₆₇	—	2	112	no	—
29	C₂²×Dic₆₇	C₂²×C₄	1	56	yes	—
30	C₂×C₆₇⋊C₈	C₂²×C₄	1	112	yes	complemented
31	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
32	C₄×Dic₆₇	C₂²×C₄	1	56	yes	—
33	C₄×Dic₆₇	C₂⁴	1	8	yes	—
34	C₂⁴×C₆₇	C₄²	1	8	yes	—
35	C₂⁴×C₆₇	M₁₆	1	16	yes	—
36	C₂⁴×C₆₇	C₂²×C₄	1	7	yes	—
37	C₂²×C₄×C₆₇	C₄²	1	56	yes	—
38	C₂²×C₄×C₆₇	C₂²×C₄	1	21	yes	—
39	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
40	C₂²×C₄×C₆₇	C₂²×C₄	1	28	yes	—
41	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	112	yes	complemented
43	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
44	C₂³×C₆₇	#97	1	64	yes	complemented
45	C₂³×C₆₇	C₂×C₄²	1	28	yes	—
46	C₂³×C₆₇	C₂×M₁₆	1	56	yes	—
47	C₂³×C₆₇	C₂³×C₄	1	7	yes	—
48	C₂×C₄×C₆₇	C₂×C₄²	1	84	yes	—
49	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
50	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
51	C₂×C₄×C₆₇	C₂³×C₄	1	14	yes	—
52	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
53	C₂×Dic₆₇	—	2	112	no	—
54	C₂×Dic₆₇	C₂³×C₄	1	16	yes	—
55	C₈×C₆₇	C₂³×C₄	1	16	yes	complemented
56	C₆₇⋊C₈	C₂³×C₄	1	16	yes	complemented
57	C₂²×C₆₇	C₂×#97	1	112	yes	complemented
58	C₂²×C₆₇	C₂²×C₄²	1	14	yes	—
59	C₂²×C₆₇	C₂²×M₁₆	1	28	yes	—
60	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
61	C₄×C₆₇	C₂²×C₄²	1	14	yes	—
62	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
63	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
64	Dic₆₇	—	2	16	no	—
65	C₂³×#97	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	C₂²×#97	1	28	yes	complemented
67	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
68	C₂×C₆₇	C₂³×M₁₆	1	2	yes	—
69	C₂⁴×C₈	—	67	1	no	—
70	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
71	C₂³×C₄²	—	67	1	no	—
72	C₂²×#97	—	67	28	no	—
73	C₆₇	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
74	C₂⁴×C₄	—	67	2	no	—
75	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
76	C₂³×C₈	—	67	16	no	—
77	C₂³×C₈	—	67	14	no	—
78	C₂³×C₈	Dic₆₇	1	16	yes	complemented
79	C₂³×C₈	C₂×D₆₇	1	14	yes	—
80	C₂²×C₄²	—	67	14	no	—
81	C₂²×C₄²	—	67	14	no	—
82	C₂×#97	—	67	112	no	—
83	C₂⁵	C₄×D₆₇	1	1	yes	characteristic
84	C₂³×C₄	—	67	28	no	—
85	C₂³×C₄	—	134	16	no	—
86	C₂³×C₄	C₄×D₆₇	1	7	yes	—
87	C₂³×C₄	C₄×D₆₇	1	8	yes	—
88	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
89	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
90	C₂²×C₈	—	67	112	no	—
91	C₂²×C₈	—	67	28	no	—
92	C₂²×C₈	C₂×Dic₆₇	1	112	yes	complemented
93	C₂²×C₈	C₂²×D₆₇	1	28	yes	—
94	C₂×C₄²	—	67	84	no	—
95	C₂×C₄²	—	67	28	no	—
96	#97	—	67	64	no	—
97	C₂⁴	C₄×Dic₆₇	1	8	yes	—
98	C₂⁴	#11761611	1	16	yes	—
99	C₂⁴	C₂×C₄×D₆₇	1	7	yes	—
100	C₂²×C₄	—	67	56	no	—
101	C₂²×C₄	—	134	112	no	—
102	C₂²×C₄	C₄×Dic₆₇	1	56	yes	—
103	C₂²×C₄	C₂×C₄×D₆₇	1	21	yes	—
104	C₂²×C₄	C₂×C₄×D₆₇	1	28	yes	—
105	C₂²×C₄	C₂²×Dic₆₇	1	28	yes	—
106	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
107	C₂×C₈	—	67	112	no	—
108	C₂×C₈	—	67	8	no	—
109	C₂×C₈	C₂²×Dic₆₇	1	112	yes	complemented
110	C₂×C₈	C₂³×D₆₇	1	8	yes	—
111	C₄²	—	67	56	no	—
112	C₄²	—	67	8	no	—
113	C₂³	#14523953	1	64	yes	direct product factor
114	C₂³	C₂×C₄×Dic₆₇	1	28	yes	—
115	C₂³	C₂×#11761611	1	56	yes	—
116	C₂³	C₂²×C₄×D₆₇	1	7	yes	—
117	C₂×C₄	—	67	16	no	—
118	C₂×C₄	—	134	112	no	—
119	C₂×C₄	C₂×C₄×Dic₆₇	1	84	yes	—
120	C₂×C₄	C₂²×C₄×D₆₇	1	7	yes	—
121	C₂×C₄	C₂²×C₄×D₆₇	1	14	yes	—
122	C₂×C₄	C₂³×Dic₆₇	1	14	yes	—
123	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
124	C₈	—	67	16	no	—
125	C₈	C₂³×Dic₆₇	1	16	yes	complemented
126	C₂²	C₂×#14523953	1	112	yes	direct product factor
127	C₂²	C₂²×C₄×Dic₆₇	1	14	yes	—
128	C₂²	C₂²×#11761611	1	28	yes	—
129	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
130	C₄	—	134	16	no	—
131	C₄	C₂²×C₄×Dic₆₇	1	14	yes	—
132	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
133	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
134	C₂	C₂²×#14523953	1	28	yes	direct product factor
135	C₂	C₂³×C₄×Dic₆₇	1	1	yes	characteristic
136	C₂	C₂³×#11761611	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₇
commutator	C₂×C₆₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂³×#97
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#14523953	C₂	direct (N × H)
6. C₂³×C₆₇⋊C₈	C₄	semidirect
10. C₂×#14523953	C₂²	direct (N × H)
12. C₂³×C₈×C₆₇	C₄	semidirect
16. C₂²×C₆₇⋊C₈	C₂×C₄	semidirect
20. #14523953	C₂³	direct (N × H)
26. C₂²×C₈×C₆₇	C₂×C₄	semidirect
30. C₂×C₆₇⋊C₈	C₂²×C₄	semidirect
42. C₂×C₈×C₆₇	C₂²×C₄	semidirect
44. C₂³×C₆₇	#97	semidirect
55. C₈×C₆₇	C₂³×C₄	semidirect
56. C₆₇⋊C₈	C₂³×C₄	semidirect
57. C₂²×C₆₇	C₂×#97	semidirect
66. C₂×C₆₇	C₂²×#97	semidirect
73. C₆₇	C₂³×#97	semidirect
78. C₂³×C₈	Dic₆₇	semidirect
92. C₂²×C₈	C₂×Dic₆₇	semidirect
109. C₂×C₈	C₂²×Dic₆₇	semidirect
113. C₂³	#14523953	direct (N × H)
125. C₈	C₂³×Dic₆₇	semidirect
126. C₂²	C₂×#14523953	direct (N × H)
134. C₂	C₂²×#14523953	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×C₆₇⋊C₈	2
C₂³×C₄×Dic₆₇	2
C₂²×#14523953	2
C₂⁴×C₈×C₆₇	2
C₂³×#97	67

Subgroups — C23×#14523953 #31719213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523953 #31719213